SOCIAL NETWORK ANALYSIS: METHODS AND APPLICATIONS
社交網路分析：方法與應用

STANLEY WASSERMAN
斯坦利·沃瑟曼

2011

Abstract
抽象

The advantages of a Galois lattice for representing an affiliation network are the focus on subsets, and the complementary relationships between the actors and the events that are displayed in the diagram. The focus on subsets is especially appropriate for representing affiliation networks. In addition patterns in the relationships between
伽羅瓦格子表示隸屬網路的優點是關注子集，以及參與者與圖中顯示的事件之間的互補關係。對子集的關注特別適合表示隸屬關係網路。此外，模式在關係中
8.6 (j9Analysis of Actors and Events
8.6 （j9參與者和事件分析

Synopsis
概要

A Galois lattice is beneficial for representing affiliation networks due to its emphasis on subsets and the complementary relationships between actors and events displayed in the diagram.
伽羅瓦格子有利於表示隸屬網路，因為它強調子集以及圖中顯示的參與者和事件之間的互補關係。

The text discusses social groups, role structures, network analysis, statistical approaches, and methods for studying social networks.
本文討論了社會群體、角色結構、網路分析、統計方法和研究社交網路的方法。
It covers topics such as structural equivalence, centrality, and n-cliques in networks.
它涵蓋了網路中的結構等價性、中心性和 n 集團等主題。
The text also mentions the importance of replication in network analysis and the development of general propositions about social network structures.
本文還提到了複製在網路分析中的重要性以及關於社會網路結構的一般命題的發展。
Additionally, it touches on the use of graph theory in identifying important actors in social networks and provides examples of affiliation networks and their properties.
此外，它還涉及圖論在識別社交網路中重要參與者中的應用，並提供了隸屬網路及其屬性的示例。
The text discusses various applications of affiliation networks, methods for gathering social network data, and techniques for analyzing social networks.
本文討論隸屬關係網路的各種應用、收集社交網路數據的方法以及分析社交網路的技術。
It covers topics such as hypergraphs, data collection techniques, nodal degree, graph density, cohesive subgroups, two-mode networks, and role structures.
它涵蓋了超圖、數據收集技術、節點度、圖密度、內聚子組、雙模網路和角色結構等主題。
The text also mentions the use of graph theory in social network analysis and the evaluation of methods for positional analysis.
本文還提到了圖論在社會網路分析中的應用以及位置分析方法的評估。
Additionally, it touches on topics like transitivity in social networks, n-cliques, and n-clans.
此外，它還涉及社交網路、n 集團和 n 氏族中的傳遞性等主題。
The text discusses various concepts related to social network analysis, including cohesive subgroups, graph theory, and multirelational blockmodels.
本文討論了與社交網路分析相關的各種概念，包括內聚子群、圖論和多關係塊模型。
It also covers methods for analyzing social networks, such as structural equivalence, hubbell's approach, and the k-plex definition.
它還涵蓋了分析社交網路的方法，例如結構等價性、Hubbell 方法和 k-plex 定義。
The text emphasizes the importance of studying patterns of ties and relationships within social networks to understand social structures and interactions.
該文本強調了研究社交網路中的聯繫和關係模式以了解社會結構和互動的重要性。
The text discusses various concepts and methods related to social network analysis, including formal network role analysis, sampling methods, structural balance, matrix operations, centrality indices, cohesive subgroups, and balance in signed digraphs.
本文討論了與社會網路分析相關的各種概念和方法，包括形式網路角色分析、抽樣方法、結構平衡、矩陣運算、中心性指數、內聚子群和符號二元圖中的平衡。
It also covers the importance of actor attributes, Bayesian estimation, and the use of sociometric and graph theoretic notations.
它還涵蓋了參與者屬性、貝葉斯估計以及社會計量和圖論符號的使用的重要性。
The text emphasizes the mathematical definitions provided by social network analysis and the significance of network data in understanding social relationships.
本文強調了社會網路分析提供的數學定義以及網路數據在理解社會關係中的重要性。
The text discusses various concepts related to social network analysis, including equivalence, relations, graphs, and structural properties of actors and groups.
本文討論了與社會網路分析相關的各種概念，包括參與者和群體的等價性、關係、圖形和結構屬性。
It also covers the use of Bayesian ideas, maximum likelihood estimates, and matrix permutation analysis in network analysis.
它還涵蓋了貝葉斯思想、最大似然估計和矩陣排列分析在網路分析中的應用。
The text emphasizes the importance of structural variables in network data sets and explores methods for comparing ego algebras and measuring equivalence between actors in a network.
本文強調了結構變數在網路數據集中的重要性，並探討了比較自我代數和測量網路中參與者之間等價性的方法。
Additionally, it mentions the application of these methods to real-world problems and the concept of blockmodels in network analysis.
此外，它還提到了這些方法在實際問題中的應用以及網路分析中塊模型的概念。
It covers methods for analyzing social networks, such as composition of relations, regular equivalence, and measures of structural equivalence.
它涵蓋了分析社交網路的方法，例如關係的組成、規則等價和結構等價的度量。
The text also touches on the importance of network theory and measurement in understanding social structures.
本文還談到了網路理論和測量在理解社會結構中的重要性。
Additionally, it mentions the use of graph theory and formal definitions to translate theoretical concepts into formal measures for network analysis.
此外，它還提到了使用圖論和形式定義將理論概念轉化為網路分析的形式度量。
The text discusses various methods and concepts related to social network analysis, including the use of correlation matrices, multidimensional scaling, hierarchical clustering, network software, stochastic blockmodels, goodness-of-fit statistics, and clusterability in signed graphs.
本文討論了與社交網路分析相關的各種方法和概念，包括相關矩陣的使用、多維縮放、分層聚類、網路軟體、隨機塊模型、擬合優度統計和有符號圖中的聚類性。
It also covers topics such as affiliation networks, transitivity, graph theory, matrix operations, and random directed graph distributions.
它還涵蓋了隸屬網路、傳遞性、圖論、矩陣運算和隨機有向圖分佈等主題。
The text emphasizes the importance of understanding network properties and measures for social and behavioral science research.
本文強調了理解網路屬性和測量對社會和行為科學研究的重要性。

Key points
要點

Social Group and Subgroup 7.2 Subgroups Based on Complete Mutuality 7.2.1 Definition of a Clique 7.2.2 An Example 7.3.2 An Example 7.3.4 n-clans and n-clubs
社會群體和子群體 7.2 基於完全互惠的子群體 7.2.1 集團的定義 7.2.2 示例 7.3.2 示例 7.3.4 n氏族和n氏族

When multidimensional scaling is used to study network positions using measures of structural equivalence as input, the results show which subsets of actors are more, and which are less, structurally equivalent
當使用多維縮放來研究使用結構等價度量作為輸入的網路位置時，結果顯示哪些參與者子集在結構上等價更多，哪些參與者更少

We have described how to compare role structures
我們已經描述了如何比較角色結構

This usage is apparent in a statement such as, "a person takes on the role of leader in a group." A theoretical statement is provided by Homans (1967), who defines role as
這種用法在諸如「一個人在團隊中扮演領導者的角色」之類的陳述中很明顯。Homans（1967）提供了一個理論陳述，他將角色定義為

Goodenough (1969) makes the important distinctions between status, pOSition, and role. He argues that the fact that many authors have not carefully distinguished between status and position has led to unfortunate confusions
Goodenough（1969）對地位、pOSition和角色進行了重要的區分。他認為，許多作者沒有仔細區分地位和地位，這一事實導致了不幸的混淆

The difference between the G2 statistics for two hierarchically nested models is approximately asymptotically distributed as a X2 random variable. This difference can be compared to tabled values of X2 to establish whether the model with more parameters fits significantly better than the simpler model
兩個分層嵌套模型的 G2 統計量之間的差值近似地呈漸近分佈，作為 X2 隨機變數。可以將此差異與 X2 的表值進行比較，以確定具有更多參數的模型是否明顯優於更簡單的模型

By restrictive we mean that if one equivalence definition is more restrictive than another, any actors who are equivalent by the first definition are equivalent by the second definition, though actors who are equivalent by the second may not be equivalent by the first
我們所說的限制性是指，如果一個等價定義比另一個定義更具限制性，則任何與第一個定義等價的行為者在第二個定義下都是等價的，儘管在第二個定義上等價的行為者可能與第一個定義不等價

Summary
總結

15.8 Goodness-of-fit statistics for the fabricated network
15.8 預製網路的擬合優度統計
15.10 Parameter estimates for Krackhardt's high-tech managers 632. 15.11 The W-array for the second-graders using friendship and age (the first subset consists of the 7-year-old children, Eliot, Keith, and Sarah, and the second subset consists of the 8-year-old children, Allison, Drew, and Ross.).
15.10 克拉克哈特高科技管理人員的參數估計 632.15.11 使用友誼和年齡的二年級學生的 W 陣列（第一個子集由 7 歲的孩子 Eliot、Keith 和 Sarah 組成，第二個子集由 8 歲的孩子 Allison、Drew 和 Ross 組成）。

15.10 Parameter estimates for Krackhardt's high-tech managers 632.
15.10 克拉克哈特高科技管理人員的參數估計 632.

15.11 The W-array for the second-graders using friendship and age.
15.11 使用友誼和年齡的二年級學生的 W 陣列。

15.12 The W-arrays for Krackhardt's high-tech managers, using tenure, and age and tenure.
15.12 Krackhardt高科技經理的W陣列，使用任期、年齡和任期。

15.13 Parameters, models, and associated margins for models for attribute variables.
15.13 屬性變數模型的參數、模型和相關邊距。

15.14 Goodness-of-fit statistics for the fabricated network, using attribute variables.
15.14 使用屬性變數對預製網路進行擬合優度統計。

15.16 Goodness-of-fit statistics for Krackhardt's managers and the advice relation, with attribute variables.
15.16 Krackhardt經理的擬合優度統計和建議關係，以及屬性變數。

15.17 Goodness-of-fit statistics for Krackhardt's managers and the friendship relation, with attribute variables
15.17 Krackhardt經理和友誼關係的擬合優度統計，以及屬性變數

Dendrogram for complete link hierarchical clustering of correlation coefficients on the advice relation for
用於對建議關係的相關係數進行完整連結分層聚類的樹狀圖
9.9 Multidimensional scaling of correlation coefficients on the advice relation for Krackhardt's high-tech managers agers permuted according to positions from hierarchical clustering of correlations.
9.9 Krackhardt高科技經理人諮詢關係的相關係數的多維縮放，根據相關性分層聚類中的位置進行置換。

9.11 Density table for the advice relation from Krackhardt's high-tech managers, positions identified by hierarchical clustering of correlations.
9.11 Krackhardt高科技經理的建議關係密度表，通過相關性的分層聚類確定職位。

9.12 Image matrix for the advice relation from Krackhardt's high-tech managers, positions identified by hierarchical clustering of correlations.
9.12 來自Krackhardt高科技經理的建議關係的圖像矩陣，通過相關性的分層聚類確定的職位。

9.13 Reduced graph for the advice relation from Krackhardt's high-tech managers, positions identified by hierarchical clustering of correlations
9.13 Krackhardt高科技經理的建議關係簡化圖，通過相關性的分層聚類確定職位

12.11 Ego algebras for the example of two relations
12.11 以兩個關係為例的自我代數
12.12 Distances between ego algebras for a hypothetical example of two relations. 12.13 Distances between ego algebras computed on advice and friendship relations for Krackhardt's high-tech managers 500.
12.12 自我代數之間的距離，用於兩個關係的假設示例。12.13 根據建議和友誼關係計算的自我代數之間的距離 克拉克哈特的高科技經理 500.

12.12 Distances between ego algebras for a hypothetical example of two relations.
12.12 自我代數之間的距離，用於兩個關係的假設示例。

12.13 Distances between ego algebras computed on advice and friendship relations for Krackhardt's high-tech managers 500.
12.13 根據建議和友誼關係計算的自我代數之間的距離 克拉克哈特的高科技經理 500.

12.14 Hierarchical clustering of distances between ego algebras on the two relations for Krackhardt's high-tech managers
12.14 自我代數之間距離的層次聚類對克拉克哈特高科技經理的兩種關係的影響

Part I
第一部分
The notion of a social network and the methods of social network analysis have attracted considerable interest and curiosity from the social and behavioral science community in recent decades.
近幾十年來，社交網路的概念和社交網路分析方法引起了社會和行為科學界的極大興趣和好奇心。

Network analysis enters into the process or model development, specification, and testing in a number of ways: 10 express relationally defined theoretical concepts by providing formal definitions, measures and descriptions, to evaluate models and theories in which key concepts and propositions are expressed as relational processes or structural outcomes, or to provide statistical analyses of multirelational systems.
網路分析以多種方式進入過程或模型開發、規範和測試：10 通過提供正式的定義、度量和描述來表達關係定義的理論概念，以評估關鍵概念和命題表示為關係過程或結構結果的模型和理論，或提供多重關係系統的統計分析。

In this first, descriptive context, network analysis provides a vocabulary and set of formal definitions for expressing theoretical concepts and properties.
在第一個描述性上下文中，網路分析提供了用於表達理論概念和屬性的詞彙表和一組形式定義。

Usefulness of one or more attributes for predicting the level of another attribute, the social network perspective views characteristics of the social units as arising out of structural or Social Ne/wo/," Analysis in the Social and Behavioral Sciences relational processes 01' focuses on properties of the relational systems themselves.
社會網路視角認為社會單位的特徵源於結構或社會 Ne/wo/，“社會和行為科學關係過程分析 01' 側重於關係系統本身的屬性。

Bronfenbenner (1943) and Moreno and Jennings (1945) were the first to study such tendencies quantitatively
Bronfenbenner（1943）和Moreno和Jennings（1945）是第一個定量研究這種趨勢的人

Theoretical Motivations
理論動機
Theoretical notions have provided impetus for development of network methods. Here, the authors explore some of the theoretical concepts that have motivated the development of specific network analysis methods.
理論概念為網路方法的發展提供了動力。在這裡，作者探討了一些理論概念，這些概念推動了特定網路分析方法的發展。

Balanced relations were quite common in empirical work; theorists were quick to pose theories about why such things occurred so frequently
平衡關係在實證工作中很常見;理論家們很快就提出了關於為什麼這樣的事情如此頻繁發生的理論

This concept led to a very active thirty-year period of empirical, theoretical, and quantitative research on triples of individuals.
這個概念導致了對三倍個體的三十年非常活躍的實證、理論和定量研究。

Mathematicians had long been interested in graphs and distributions for graphs, and the more mathematical social network analysts were quick to pick up models and methods from the mathematicians
長期以來，數學家一直對圖和圖的分佈感興趣，而更懂數學的社交網路分析師很快就從數學家那裡學習了模型和方法

Graph theory provides both an appropriate representa~ tion of a social network and a set of concepts that can be used to study formal properties of social networks.
圖論既提供了社交網路的適當表示~，也提供了一組可用於研究社交網路的形式屬性的概念。

Empirical and theoretical work on balance theory and transitivity motivated a variety of mathematicians and statisticians to formulate mathematical models for behavior of triples of actors.
關於平衡理論和傳遞性的實證和理論工作促使各種數學家和統計學家為三重參與者的行為制定數學模型。

Such theories argue that units are not acting independently from one another, but rather influence each other
這些理論認為，單位不是彼此獨立行動，而是相互影響

Focusing on such structural variables opens up a different range of possibilities for, and constraints on, data analysis and model building.
關注這些結構變數為數據分析和模型構建開闢了不同的可能性和限制。

The historical examination of empirical, theoretical, and mathematical developments in network research should convince the reader that social
對網路研究的經驗、理論和數學發展的歷史考察應該讓讀者相信，社會

Fundamental Concepts in Network
網路的基本概念
Analysis network analysis is far more than an intuitively appealing vocabulary, metaphor, or set of images for discussing social, behavioral, political, or economic relationships.
分析 網路分析遠不止是用於討論社會、行為、政治或經濟關係的直觀吸引人的詞彙、隱喻或一組圖像。

Social network analysis provides a precise way to define important social concepts, a theoretical alternative to the assumption of independent social actors, and a framework for testing theories about structured social relationships.
社會網路分析提供了一種定義重要社會概念的精確方法，是獨立社會行為者假設的理論替代方案，也是測試結構化社會關係理論的框架。

The methods of network analysis provide explicit formal statements and measures of social structural properties that might otherwise be defined only in metaphorical terms.
網路分析方法提供了社會結構屬性的明確形式陳述和測量，否則這些屬性可能只能用隱喻術語來定義。

Such phrases as webs of relationships, closely knit networks of relations, social role, social position, group, clique, popularity, isolation, prestige, prominence, and so on are given mathematical definitions by social network analysis.
諸如關係網、緊密聯繫的關係網路、社會角色、社會地位、群體、集團、受歡迎程度、孤立、聲望、突出等短語通過社會網路分析給出了數學定義。

There are several key concepts at the heart of network analysis that are fundamental to the discussion of social networks
網路分析的核心有幾個關鍵概念，這些概念是討論社交網路的基礎

These concepts are: actor, relational tie, dyad, triad, subgroup, group, relation, and network.
這些概念是：參與者、關係關係、二元組、三元組、子組、組、關係和網路。

Many important social network methods and models focus on the triad; a subset of three actors and the tie(s) among them.
許多重要的社交網路方法和模型都集中在三元組上;三個參與者的子集以及他們之間的聯繫。

It is important to note that a relation refers to the collection of ties of a given kind measured on pairs of actors from a specified actor set.
需要注意的是，關係是指在特定參與者集中的成對的參與者上測量的給定類型關係的集合。

Having defined actor, group, and relation the authors can give a more explicit definition of social network.
在定義了行動者、群體和關係之後，作者可以對社交網路給出更明確的定義。

Distinctive
獨特
Features social network analysis requires a specialized vocabulary, and deals with conceptual entities and research problems that are quite difficult to pursue using a more traditional statistical and data analytic framework.
特徵 社交網路分析需要專門的詞彙，並處理概念實體和研究問題，而這些問題很難使用更傳統的統計和數據分析框架進行研究。

The authors turn to some of the distinctive features of network analysis
作者轉向網路分析的一些顯著特徵

Distinctive Features of Network Theory and Measurement
網路理論與測量的顯著特點
It is quite important to note the key features that distinguish network theory, and network measurement, from the more usual data analytic framework common in the social and behavioral sciences.
重要的是要注意將網路理論和網路測量與社會和行為科學中常見的更常見的數據分析框架區分開來的關鍵特徵。

Such features provide the necessary motivation for the topics discussed in this book.
這些特點為本書所討論的主題提供了必要的動力。

Many network analysis methods provide formal definitions and descriptions of structural properties of actors, subgroups of actors, or groups
許多網路分析方法提供了對參與者、參與者、參與者的子組或組的結構屬性的正式定義和描述

These methods translate core concepts in social and behavioral theories into formal definitions expressed in relational terms.
這些方法將社會和行為理論中的核心概念轉化為以關係術語表達的正式定義。

Social network data require measurements on ties among social units; attributes of the actors may be collected
社交網路數據需要測量社會單位之間的聯繫;可以收集參與者的屬性

Such data sets need social network methods for analysis.
此類數據集需要社交網路方法進行分析。

Part 11 presents graph theory, develops the vocabulary and concepts that are widely used in network analysis, and relies heavily on examples
第 11 部分介紹了圖論，發展了網路分析中廣泛使用的詞彙和概念，並大量依賴示例

It discusses simple actor and group properties.
它討論了簡單的參與者和組屬性。

The methods presented in these three parts of the book assume specific descriptive models for the structure of a network, and primarily present descriptive techniques for network analysis which translate theoretical concepts.into formal measures.
本書的這三個部分介紹的方法假設了網路結構的特定描述性模型，並主要提出了將理論概念轉化為形式度量的網路分析的描述性技術。

Introduction
介紹
Social network data consist of at least one structural variable measured on a set of actors.
社交網路數據由至少一個在一組參與者上測量的結構變數組成。

The substantive concerns and theories motivating a specific network study usually determine which variables to measure, and often which techniques are most appropriate for their measurement.
激發特定網路研究的實質性關注點和理論通常決定了要測量哪些變數，以及通常哪種技術最適合其測量。

If one is studying economic transactions between countries, one cannot rely on observational techniques; one would probably use archival records to obtain information on such transactions.
如果一個人正在研究國家之間的經濟交易，就不能依賴觀察技術;人們可能會使用檔案記錄來獲取有關此類交易的資訊。

Friendships among people are most likely studied using questionnaires or interviews, rather than using archival or historical records.
人與人之間的友誼最有可能通過問卷調查或訪談來研究，而不是使用檔案或歷史記錄。

The nature of the study determines whether the entire set of actors can be surveyed or whether a sample of the actors must be taken.
研究的性質決定了是否可以對整套行為者進行調查，或者是否必須對行為者進行抽樣調查。

The nature of the structural variables determines which analytic methods are appropriate for their study.
結構變數的性質決定了哪些分析方法適合於他們的研究。

It is crucial to understand the nature of these variables.
了解這些變數的性質至關重要。

The data collection techniques described here determine, to some degree, the characteristics of the relations
這裡描述的數據收集技術在某種程度上決定了關係的特徵

Structural and Composition Variables
結構和組成變數
There are two types of variables that can be included in a network data set: structural and composition.
網路數據集中可以包含兩種類型的變數：結構變數和組合變數。

Structural variables are measured on pairs of actors and are the cornerstone of social network data sets.
結構變數是在成對的參與者上測量的，是社交網路數據集的基石。

Structural variables can measure business transactions between corporations, friendships between people, or trade between nations.
結構變數可以衡量公司之間的商業交易、人與人之間的友誼或國家之間的貿易。

Actors comprising these pairs usually belong to a single set of actors.
組成這些對的 actor 通常屬於一組 actor。

Structural variables measured on a single set of actors give rise to one-mode networks.
在一組參與者上測量的結構變數產生了單模網路。

There are types of structural variables that are measured on two sets of entities.
有一些類型的結構變數是在兩組實體上測量的。

A two-mode network data set contains measurements on which actors from one of the sets have ties to actors in the other set.
雙模式網路數據集包含來自一個集合的參與者與另一個集合中的參與者有聯繫的度量。

The second mode in an affiliation network is a set of events to which the actors belong.
隸屬關係網路中的第二種模式是參與者所屬的一組事件。

In affiliation network data the two modes are the actors and the events.
在隸屬網路數據中，兩種模式是參與者和事件。

In such data, the events are defined not on pairs of actors, but on subsets of actors.
在此類數據中，事件不是在成對的參與者上定義的，而是在參與者的子集上定義的。

Boundary Specification and
邊界規範和
Sampling identify the population to be studied, and if sampling is necessary, worry about how to sample actors and relations.
抽樣確定要研究的總體，如果需要抽樣，請擔心如何對參與者和關係進行抽樣。

The second example comes from the study of community leaders by Laumann and Pappi (1973)
第二個例子來自Laumann和Pappi（1973）對社區領袖的研究

They asked community leaders to define the boundary by identifying the elite actors in the community of Altneustadt.
他們要求社區領導人通過確定Altneustadt社區的精英行為者來定義邊界。

Name all persons [who] are in general very influential in Altneustadt
說出所有在阿爾特諾伊施塔特非常有影響力的人

From these lists, each of which can be considered a sample of the relevant actors in the elite network, the actor set was enumerated.
從這些名單中，每個名單都可以被視為精英網路中相關參與者的樣本，列舉了參與者集。

Examples of social network studies using snowball sampling include: Johnson (1990) and Johnson, Boster, and Holbert (1989) on commercial fishermen; Moore (1979) and Alba and Moore (1978) on elite networks
使用滾雪球抽樣的社交網路研究的例子包括：Johnson（1990）和Johnson，Boster和Holbert（1989）關於商業漁民;Moore （1979） 和 Alba and Moore （1978） 在精英網路上

Most network studies focus on well-defined, completely enumerated sets, rather than on samples of actors from larger popuiations.
大多數網路研究都集中在定義明確、完全列舉的集合上，而不是來自較大人口的參與者樣本。

Methodology for the latter situation is considerably different from methods for the former.
后一種情況的方法與前一種情況的方法有很大不同。

One example is data arising from an ego-centered network design
一個例子是以自我為中心的網路設計產生的數據

Data on such networks are gathered using special sampling strategies that allow the researcher to focus on a specific set of respondents, and the ties that these respondents have to particular others.
這些網路上的數據是使用特殊的抽樣策略收集的，這些策略使研究人員能夠專注於一組特定的受訪者，以及這些受訪者與特定其他人的聯繫。

The authors turn to a discussion of one-mode, two-mode, and affiliational, and egocentric and special networks
作者轉而討論單模式、雙模式、從屬關係、自我中心和特殊網路

One-Mode Networks
單模網路
Suppose the network under study is one-mode, and involves measurements on just a single set of actors.
假設所研究的網路是單模的，並且只涉及對一組參與者的測量。

The relations measured on the single set of actors in a one-mode network are usually viewed as representing specific substantive connections, or "relational contents" (Knoke and Kuklinski 1982).
在單模式網路中，在一組參與者上測量的關係通常被視為代表特定的實質性聯繫或“關係內容”（Knoke and Kuklinski 1982）。

These connections, measured at the level of pairs of actors, can be of many types.
這些連接，在成對的參與者的水平上測量，可以有多種類型。

In addition to relational information, social network data sets can contain measurements on the characteristics of the actors.
除了關係資訊外，社交網路數據集還可以包含對參與者特徵的測量。

Such measurements of actor attribute variables constitute the composition of the social network.
這種對行為者屬性變數的測量構成了社交網路的組成。

Relations in a two-mode network measure ties between the actors in one set and actors in a second set.
雙模網路中的關係衡量一個集合中的參與者和第二個集合中的參與者之間的聯繫。

The type of two-mode social network, which the authors refer to as an affiliation network, arises when one set of actors is measured with respect to attendance at, or affiliation with, a set of events or activities.
雙模式社交網路的類型，作者稱之為從屬網路，當一組參與者被衡量為一組事件或活動的出席或從屬關係時，就會出現。

Social network data consist of one relations measured among a set of actors.
社交網路數據由一組參與者之間測量的一種關係組成。

The authors could record the dollar value of manufactured goods that are exported from One country to a second country, giving rise to a valued relation
作者可以記錄從一個國家出口到第二個國家的製成品的美元價值，從而產生價值關係

Collection
收集
There are a variety of ways in which social network data can be gathered. These techniques are: Questionnaires; Interviews; Observations; Archival records; Experiments; Other techniques, including ego-centered, small world, and diaries.
有多種方法可以收集社交網路數據。這些技術是：問卷調查;採訪;觀察;檔案記錄;實驗;其他技術，包括以自我為中心、小世界和日記。

An example of a complete rank order design is the study by Bernard, Killworth, and Sailer (1980)
完整排名順序設計的一個例子是 Bernard、Killworth 和 Sailer （1980） 的研究

They asked each of forty members of a social science research office to report the amount of communication with each other member of the office using the following procedure: ...
他們要求社會科學研究辦公室的四十名成員中的每一個人使用以下程式報告與辦公室其他成員的交流量：......

Observing interactions among actors is another way to collect network data
觀察參與者之間的交互是收集網路數據的另一種方法

This method has been widely used in field research to study relatively small groups of people who have faceto-face interactions (Roethlisberger and Dickson 1961; Kapferer 1969; Hammer, Polgar, and Salzinger 1969; Thurman 1980; Bernard and Killworth 1977; Killworth and Bernard 1976; Bernard, Killworth, and Sailer 1980, 1982; Freeman and Romney 1987; Freeman, Romney, and Freeman 1987; Freeman, Freeman, and Michaelson 1988, 1989).
這種方法已被廣泛用於實地研究，以研究具有面對面互動的相對較小的人群（Roethlisberger and Dickson 1961;Kapferer 1969 年;Hammer、Polgar 和 Salzinger 1969 年;瑟曼 1980;伯納德和基爾沃思 1977 年;Killworth 和 Bernard 1976;Bernard、Killworth 和 Sailer 1980、1982;弗里曼和羅姆尼 1987;弗里曼、羅姆尼和弗里曼 1987 年;Freeman、Freeman 和 Michaelson 1988、1989）。

Many network studies employ a variety of data collection methods for recording ties, in addition to gathering actor attribute information.
許多網路研究除了收集參與者屬性資訊外，還採用多種數據收集方法來記錄關係。

Another way to gather social network data is to ask each respondent to keep a continuous record of the other people with whom they interact.
收集社交網路數據的另一種方法是要求每個受訪者連續記錄與他們互動的其他人。

The data were gathered by Taba (1955), who focused on the differences and similarities between boy-boy and girl-girl choices, and "mixed gender" ties
這些數據是由Taba（1955）收集的，他專注於男孩和女孩女孩選擇之間的差異和相似之處，以及“混合性別”關係

Measurement Validity, Reliability, Accuracy, Error
測量有效性、可靠性、準確度、誤差
As Freeman and Romney (1987) note, "social structure refers to a relatively prolonged and stable pattern of interpersonal relations" (1987, pages 330-331)
正如Freeman和Romney（1987）所指出的，“社會結構是指一種相對長期和穩定的人際關係模式”（1987年，第330-331頁）

In their discussion of measurement error in sociometry, Holland and Leinhardt (1973) refer to this pattern as the true structure, in contrast to the observed structure contained in the measured network data, which might contain error.
Holland和Leinhardt（1973）在討論社會測量學中的測量誤差時，將這種模式稱為真實結構，與測量網路數據中包含的觀察到的結構形成鮮明對比，後者可能包含誤差。

Important concerns in social network measurement are the validity, reliability, and measurement error in these data.
社交網路測量中的重要問題是這些數據的有效性、可靠性和測量誤差。

Little work has been done on the issues of validity, reliability, and measurement error in social network data.
關於社交網路數據的有效性、可靠性和測量誤差問題，幾乎沒有做過任何工作。

They found that what people report about their interactions is related to the long-range social structure, rather than to particular instances
他們發現，人們報告的互動與長期社會結構有關，而不是與特定實例有關

Another issue related to the accuracy of network data occurs when the actors in the network are organizations but information on ties is collected from individuals as representatives of the organization.
當網路中的行為者是組織，但有關關係的資訊是從作為組織代表的個人那裡收集的時，就會發生與網路數據準確性有關的另一個問題。

The construct validity of social network measures can be studied by examining how these measures behave in a range of theoretical propositions (Mouton, Blake, and Fruchter 1955b; Burt, Marsden, and Rossi 1985).
社會網路測量的結構有效性可以通過檢查這些測量在一系列理論命題中的行為來研究（Mouton， Blake， and Fruchter 1955b;Burt、Marsden 和 Rossi 1985）。

Of particular importance in the discussion presented by Holland and Leinhardt is the error that arises in fixed choice data collection designs.
在Holland和Leinhardt提出的討論中，特別重要的是固定選擇數據收集設計中出現的錯誤。

Kraekhardt's High-teeh Managers
Kraekhardt的高端經理
This is a one-mode network, with three relations measured on a set of people
這是一個單模式網路，在一組人身上測量了三種關係

These data were gathered by Krackhardt (1987a) in a small manufacturing organization on the west coast of the U.S This organization had been in existence for ten years and produced high-tech machinery for other companies.
這些數據是由Krackhardt（1987a）在美國西海岸的一個小型製造組織中收集的，該組織已經存在了十年，併為其他公司生產高科技機械。

This is a one-mode network with two relations measured among a set of families
這是一個單模式網路，在一組族之間測量了兩個關係

These multirelational network data, compiled by Padgett, consist of the marriage and business ties among 16 families in 15th century Florence, Italy.
這些由帕吉特編製的多關係網路數據包括 15 世紀義大利佛羅倫薩 16 個家庭之間的婚姻和商業關係。

There are three actor attributes: net wealth in 1427; number of priors from 1282-1344; and number of business or marriage ties in the total network
有三個參與者屬性：1427年的凈財富;1282-1344 年的先驗數量;以及整個網路中的商業或婚姻關係數量

This is a one-mode network with two relations measured on a set of people.
這是一個單模式網路，在一組人身上測量了兩種關係。

These data come from a computer conference among researchers working in the emerging scientific specialty of social network research, organized by Freeman, and sponsored by the National Science Foundation.
這些數據來自由弗里曼組織並由美國國家科學基金會贊助的社交網路研究新興科學專業研究人員之間的計算機會議。

Social network data consist of measurements on a variety of relations for one or more sets of actors.
社交網路數據包括對一組或多組參與者的各種關係的測量。

OMultiple Relations
OMultiple 關係
Graph theoretic notation can be generalized to multirelatiorial networks, which could include both directional and nondirectional relations.
圖論符號可以推廣到多關係網路，其中可以包括方向關係和非方向關係。

Each of these relations can be represented as a graph or directed graph; each has associated with it a set of lines or arcs, specifying which lines are present in the graph for the relation (or, which pairs are "relating").
這些關係中的每一個都可以表示為圖或有向圖;每個都與它關聯了一組線或弧，指定關係圖中存在哪些線（或者，哪些線對是“相關的”）。

Each relation has a corresponding set of arcs, fi'" which contains Lr ordered pairs of actors as elements.
每個關係都有一組相應的弧線 fi'“，其中包含作為元素的 Lr 有序的參與者對。

The subscript r ranges from 1 to R, the total number of relations
下標 r 的範圍從 1 到 R，關係總數

Each of these R sets defines a directed graph on the nodes in .AI.
這些 R 集中的每一個都定義了 中節點上的有向圖。人工智慧。

For a non,directional relation, such as "lives near," measurements are made on unordered rather than ordered pairs.
對於非方向性關係，例如「生命附近」，測量是在無序而不是有序對上進行的。

The authors use (., .) to denote pairs of actors for whom a tie is present on a nondirectional relation, and use < .,.
作者使用 （.， .） 來表示在非方向關係中存在平局的參與者對，並使用 < .，。

Since "lives near" is nondirected, there are no arrowheads on these lines
由於「附近生活」是無向的，因此這些行上沒有箭頭

Sociometric notation is general enough to handle valued relations
社會計量符號足夠通用，可以處理價值關係

Sociometric Notation
社會計量符號
Sociometry is the study of positive and negative affective relations, such as liking/disliking and friends/enemies, among a set of people.
社會測量學是研究一組人之間的積極和消極情感關係，例如喜歡/不喜歡和朋友/敵人。

A social network data set consisting of people and measured affective relations between people is often referred to as sociometric.
由人與人之間測量的情感關係組成的社交網路數據集通常被稱為社會計量學。

A sociomatrix for a dichotomous relation is exactly the adjacency matrix for the graph quantifying the ties between the actors for the relation in question
二分關係的社會矩陣正是圖的鄰接矩陣，用於量化相關關係的參與者之間的聯繫

This notation can be viewed as complementary to graph theoretic notation described .
這種符號可以看作是對所描述的圖論符號的補充。

Festinger (1949) applied matrix multiplication to sociomatrices and described how products of a sociomatrix can be used to find cliques or subgroups of similar actors
Festinger（1949）將矩陣乘法應用於社會矩陣，並描述了如何使用社會矩陣的乘積來尋找相似參與者的集團或子群體

Since such powers have simple graph theoretic interpretation-fsee-Chap~ ter 4's discussion of 2- and 3-step walks), this research helped begin the era of graph theoretic approaches to social network analysis.
由於這種冪具有簡單的圖論解釋-fsee-Chap~ ter 4 對 2 步和 3 步行走的討論），這項研究有助於開啟圖論方法用於社交網路分析的時代。

Define Xij as the value of the tie from the ith actor to the jth actor on the single relation.
將 Xij 定義為單個關係上從第 i 個 actor 到第 j 個 actor 的領帶值。

The authors place these measurements into a sociomatrix.
作者將這些測量結果放入社會矩陣中。

Since there are g actors, the matrix is of size g x g
由於有 g 參與者，因此矩陣的大小為 g x g

Sociometric notation uses such matrices to denote measurements on ties.
社會計量符號使用此類矩陣來表示領帶的測量值。

Friendship at Beginning of Year
年初的友誼
Sarah represented by the arc 11 there is an arc from Allison to Drew in the sociogram for the first relation, indicating that Allison chooses Drew as a friend at the beginning of the school year.
由弧線 11 表示的莎拉在第一個關係的社會圖中有一個從艾莉森到德魯的弧線，表明艾莉森在學年開始時選擇德魯作為朋友。

This arc is how this tie is denoted by graph theoretic notation
這個弧線就是用圖論符號表示這種關係的方式

Consider how this single tie is coded with sociometric notation.
考慮一下這個單一的領帶是如何用社會計量符號編碼的。

Consider the entry which quantifies Allison as a sender and Drew (n2) as a receiver on relation ,q(1
考慮將 Allison 量化為發送者，將 Drew （n2） 量化為關係的接收者，q（1

This entry is in the (1,2) cell of this sociomatrix, and contains a 1 indicating that xl21 the value of the tie from nl to n2 on relation f!ll.
此條目位於此社會矩陣的 （1,2） 單元格中，包含一個 1，表示 xl21 關係 f！ll 上從 nl 到 n2 的領帶值。

As the authors have mentioned, there are network data sets for which sociometric notation is more difficult to use - those which contain information on the attributes of the actors.
正如作者所提到的，有些網路數據集更難使用社會計量符號 - 那些包含有關參與者屬性的資訊的數據集。

The authors would record the tie implied by "child i chooses child j as a friend at the beginning of the school year" as iFj. In sociometric notation, iFj means that XijF = 1, and implies that there is a "I" in the cell at row i and column j of the sociomatrix for this relation.
作者將「孩子我在學年開始時選擇孩子j作為朋友」所暗示的領帶記錄為iFj。在社會計量符號中，iFj 表示 XijF = 1，並暗示此關係的社會矩陣第 i 行和第 j 列的單元格中有一個“I”。

OTwo Sets of
OTwo 集
Actors presents no problem for us, since the models that use algebraic notation are specific to dichotomous relations.
Actors 對我們來說沒有問題，因為使用代數符號的模型是特定於二分關係的。

If the relation is defined on a single set of actors, both actors in the pair can be senders and both can be receivers.
如果關係是在一組參與者上定義的，則對中的兩個參與者都可以是發送者，也可以是接收者。

The authors return to the collection of six secondgrade children, and consider a second set of actors, vii, consisting of h = 4 adults.
作者回到了六個二年級兒童的集合，並考慮了第二組演員，vii，由h = 4個成年人組成。

In homogeneous pairs the senders and receivers are from the same set, while in heterogeneous pairs actors are from different sets.
在同質對中，發送者和接收者來自同一集合，而在異質對中，參與者來自不同的集合。

Assuming the relation for the heterogeneous pairs is directional, the originating actor must belong to a different set than the receiving actor.
假設異構對的關係是定向的，則原始參與者必須屬於與接收參與者不同的集合。

Since there are two sets of actors, the authors get two kinds of heterogeneous pairs: Sender belongs to % and Receiver belongs to vii;.
由於有兩組 actor，作者得到了兩種異構對：Sender 屬於 %，Receiver 屬於 vii;。

If the relation is defined for actors from different sets, in general, its sociomatrix will not be square.
如果為來自不同集合的參與者定義關係，則通常其社會矩陣不會是正方形的。

This relation is defined for the heterogeneous pairs consisting of a child as the seI,lder and an adult as a receiver.
這種關係是為異質對定義的，這些異構對由一個子作為 seI，lder 和一個作為接收者的成人組成。

Putting It All Together children that are taught by each teacher
把每個老師教的孩子放在一起
Note how this array codes the information in the directional relation between two sets of actors.
請注意此陣列如何對兩組參與者之間的方向關係中的資訊進行編碼。

The authors will use the symbols "ni ~ n/, as shorthand notation for nj "chooses" nj on the single relation in question; that is, the arc from nj to nj is contained in the set 2, so that there is a tie present for the ordered pair < ni.
作者將使用符號「ni ~ n/」作為 nj 在所討論的單個關係上「選擇」nj 的簡寫符號;也就是說，從 NJ 到 NJ 的弧包含在集合 2 中，因此 Ni <有序對存在領帶。

Nodes and arcs are the basic building blocks for graph theoretic notation
節點和圓弧是圖論符號的基本構建塊

To relate these concepts to the elements of sociometric notation, the authors consider again the collection of all ordered pairs of actors in ..¥.
為了將這些概念與社會計量符號的元素聯繫起來，作者再次考慮了 .. 中所有有序的參與者對的集合。¥.

Sometimes this collection is denoted ..¥ x ..¥, a Cartesian product of sets.
有時這個集合表示為..¥ x ..¥，集合的笛卡爾乘積。

Freeman (1989) views the triple consisting of the algebraic structure S, the directed graph or sociogram '#d, and the adjacency matrix or sociomatrix X as a social network: g = < S, '#d, X>
Freeman（1989）認為由代數結構S、有向圖或社會圖'#d 以及鄰接矩陣或社會矩陣X組成的三元組是一個社交網络：g = < S， '#d， X>

This triple provides a nice abstract definition of the central concept of this book.
這三重奏為本書的中心概念提供了一個很好的抽象定義。

In the final section of this chapter the authors define and illustrate basic matrix operations that are used in social network analysis, and show how many of these matrix operations can be used to study the graph theoretic concepts discussed of this chapter
在本章的最後一節中，作者定義並說明了用於社會網路分析的基本矩陣運算，並展示了這些矩陣運算中有多少可用於研究本章討論的圖論概念

Why Grapbs?
為什麼選擇Grapbs？
Graph theory has been useful in social network analysis for many reasons. Among these reasons are the following (see Harary, Norman, and Cartwright 1965, page 3).
圖論在社交網路分析中很有用，原因有很多。這些原因如下（參見 Harary、Norman 和 Cartwright 1965，第 3 頁）。

The authors will illustrate the graph theoretic concepts discussed on small, simple social networks
作者將說明在小型、簡單的社交網路上討論的圖論概念

Most of these examples will consist of hypothetical data created to demonstrate specific properties of graphs.
這些範例中的大多數將包含為演示圖形的特定屬性而創建的假設數據。

A graph is a model for a social network with an undirected dichotomous relation; that is, a tie is either present or absent between each pair of actors
圖是具有無向二分關係的社交網路的模型;也就是說，每對演員之間存在或不存在領帶

Nondirectional relations include such things as co-membership in formal organizations or informal groups, some kinship relations such as "is married to," "is a blood relative of," proximity relations such as "lives near," and interactions such as "works with." In a graph, nodes represent actors and lines represent ties between actors.
非定向關係包括正式組織或非正式團體的共同成員身份，一些親屬關係，如“已婚”、“是血親”，親近關係，如“住在附近”，以及互動，如“一起工作”。在圖形中，節點表示參與者，線表示參與者之間的聯繫。

In a graph of a social network with a single nondirectional dichotomous relation, the nodes represent actors, and the lines represent the ties that exist between pairs of actors on the relation.
在具有單個非方向二分關係的社交網路圖中，節點表示參與者，線表示關係中成對參與者之間存在的聯繫。

A line lk = is included in the set of lines, 2, if there is a tie present between the two actors in the network who are represented by nodes nj and nj in the graph.
如果網路中由圖中的節點 nj 和 nj 表示的兩個參與者之間存在平局，則線 lk = 包含在線集 2 中。

The sets of nodes and lines are listed
列出了節點和線的集

The authors turn to an example to demonstrate nodal degree and graph density
作者舉了一個例子來證明節點度和圖密度

Example
例
Padgett's Florentine families network includes a set of sixteen Italian families in the early 15th century.
帕吉特的佛羅倫薩家庭網路包括 15 世紀初的 16 個義大利家庭。

The authors define and illustrate properties that are used to study the connectivity of graphs, to define the distance between pairs of nodes, and to identify nodes and lines that are critical for the connectivity of the graph
作者定義並說明了用於研究圖的連通性、定義節點對之間的距離以及識別對圖的連通性至關重要的節點和線的屬性

These properties are important in themselves, but are building blocks for later properties.
這些屬性本身很重要，但也是後續屬性的構建塊。

If the authors consider a network of communications among people in which lines in a graph represent channels for transmission of messages between people, if two actors are reachable, it is possible for a message to travel from one actor to the other by passing the message through intermediaries.
如果作者考慮人與人之間的通信網路，其中圖中的線表示人與人之間傳遞消息的管道，如果可以到達兩個參與者，則消息可以通過仲介傳遞消息從一個參與者傳播到另一個參與者。

The two components in this graph are the subgraphs generated by the subsets: let them consider the paths between a pair of nodes.
此圖中的兩個分量是由子集生成的子圖：讓它們考慮一對節點之間的路徑。

The authors use the ideas of reachability between pairs of nodes, the concept of a connected graph, and components in a disconnected graph to define nodes and lines that are critical for the connectivity of a graph.
作者使用節點對之間的可達性、連接圖的概念以及斷開連接圖中的元件來定義對圖的連通性至關重要的節點和線。

One can consider the extent of connectivity in a graph in terms of the number of nodes or the number of lines that must be removed in order to leave the graph disconnected.
人們可以根據節點數或必須刪除的線數來考慮圖中的連通性程度，以使圖斷開連接。

The authors will focus on the most important directed graph concepts including the nodal degrees, walks, paths, reachability, and connectivity
作者將重點介紹最重要的有向圖概念，包括節點度、步行、路徑、可達性和連通性

Nodal lndegree and Outdegree
淋巴結 lndegree 和 outdegree
The degree of a node is the number of nodes adjacent to it. In a digraph, a node can be either adjacent to, or adjacent from another node, depending on the "direction" of the arc.
節點的度數是與其相鄰的節點數。在二合圖中，一個節點可以與另一個節點相鄰，也可以與另一個節點相鄰，具體取決於圓弧的“方向”。

These two numbers are equal, since they are considering the same set of arcs, but from different "directions." The authors will denote the mean indegree as ch, and the mean outdegree as do.
這兩個數位是相等的，因為它們考慮的是同一組弧線，但來自不同的“方向”。作者將平均度數表示為ch，將平均度數表示為ch。

The indegrees and outdegrees of the nodes in a directed graph can be used to distinguish four different kinds of nodes based on the possible ways that arcs can be incident with the node.
有向圖中節點的內度和外度可用於根據弧入射到節點的可能方式來區分四種不同類型的節點。

The density of a directed graph is equal to the proportion of arcs present in the digraph.
有向圖的密度等於二元圖中存在的弧的比例。

It is calculated as the number of arcs, L, divided by the possible number of arcs.
它的計算方法是弧數 L 除以可能的弧數。

Let them illustrate nodal indegree and outdegree, and the density of a directed graph on the example of friendships among Krackhardt's hightech managers.
讓他們以 Krackhardt 的高科技經理之間的友誼為例來說明節點內度和外度，以及有向圖的密度。

Walks and related concepts in graphs can be defined for digraphs, but one must consider the direction of the arcs.
圖中的行走和相關概念可以為二圖定義，但必須考慮弧線的方向。

The length of a path is the number of arcs in it
路徑的長度是其中的弧數

Dir,ected Graphs
Dir，ected 圖表
Directed walk Directed path Semipath Cycle Semicyc)e ns "I n2 "3 "4 "2 "3 ns 114 "2 "3.
定向行走 定向路徑 半路徑 迴圈 Semicyc）e ns “I n2 ”3 “4 ”2 “3 ns 114 ”2 “3.

The authors will consider walks and paths in which the arc between previous and subsequent nodes in the sequence may go in either direction.
作者將考慮序列中前一個節點和後續節點之間的弧線可能沿任一方向移動的步道和路徑。

A semiwalk joining nodes nj and nj is a sequence of nodes and arcs in which successive pairs of nodes are incident with an arc from the first to the second, or by an arc from the second to the first.
連接節點 nj 和 nj 的半漫遊是節點和弧的序列，其中連續的節點對從第一個節點到第二個節點以弧入射，或從第二個節點到第一個節點以弧入射。

A semipath joining nodes nj and nj is a sequence of distinct nodes, where all successive pairs of nodes are connected by an arc from the first to the second, or by an arc from the second to the first for all successive .
連接節點 nj 和 nj 的半徑是不同節點的序列，其中所有連續的節點對都通過從第一個節點到第二個節點的弧連接，或者通過從第二個節點到第一個節點的弧連接所有連續節點。

A cycle in a directed graph is a closed directed walk of at least three nodes in which all nodes except the first and last are distinct.
有向圖中的迴圈是至少三個節點的閉合有向遊走，其中除第一個節點和最後一個節點外的所有節點都是不同的。

A semicycle in a directed graph is a closed directed semiwalk of at least three nodes in which all nodes except the first and last are distinct.
有向圖中的半圓是至少三個節點的閉合有向半遊走，其中除第一個節點和最後一個節點外的所有節點都是不同的。

Reachability and Connectivity in Digraphs
二合體中的可達性和連通性
Using the ideas of paths and semipaths, the authors can define reachability and connectivity of pairs of nodes, and the connectedness of a directed graph.
使用路徑和半路徑的思想，作者可以定義節點對的可達性和連通性，以及有向圖的連通性。

A pair of nodes, nj, nj, is: (i) Weakly connected if they are joined by a semipath (ii) Unilaterally connected if they are joined by a path from nj to nj, or a path from nj to nj (iii) Strongly connected if there is a path from nj to nj, and a path from nj to nj; the path from nj to nj may contain different nodes and arcs than the path from nj to nj (iv) Recursively connected if they are strongly connected, and the path from nj to nj uses the same nodes and arcs as the path from nj to nj, in reverse order
一對節點 nj， nj 是： （i） 如果它們由半路徑連接，則為弱連接 （ii） 如果它們由從 nj 到 nj 的路徑連接，或從 nj 到 nj 的路徑連接，則為單邊連接 （iii） 如果存在從 nj 到 nj 的路徑，以及從 nj 到 nj 的路徑，則為強連接;從 NJ 到 NJ 的路徑可能包含與從 NJ 到 NJ 的路徑不同的節點和弧 （iv） 如果它們強連接，則遞歸連接，並且從 NJ 到 NJ 的路徑使用與從 NJ 到 NJ 的路徑相同的節點和弧，順序相反

Notice that these forms of connectivity are increasingly strict, and that any strict form implies connectivity of any less strict form.
請注意，這些形式的連接越來越嚴格，任何嚴格的形式都意味著任何不太嚴格的形式的連接。

The diameter of a weakly or unilaterally connected directed graph is undefined
弱連接或單邊連接的有向圖的直徑是未定義的

OSpecia/ Kinds of Directed Graphs
OSpecia/ 有向圖的種類
The authors describe several kinds of digraphs with important properties. The authors begin by defining digraph complement and digraph converse.
作者描述了幾種具有重要性質的二合字。作者首先定義了二合補語和雙合字對立。

The arcs in the converse connect the same pairs of nodes as the arcs in the digraph, but all arcs are reversed in direction.
相反的弧線連接與二合圖中的弧線相同的節點對，但所有弧線的方向都是相反的。

In the digraph representing the relation of friendship the arc < nj, nj > means i "chooses" j as a friend.
在代表友誼關係的二元圖中，弧線< nj，nj > 表示我“選擇”j 作為朋友。

In the digraph representing the complement of the relation of friendship, the arc < ni, nj > means i "does not choose" j as a friend.
在代表友誼關係補語的二元圖中，弧< ni，nj >表示我“不選擇”j 作為朋友。

One other special type of a digraph is a tournament, which mathematically represents a set of actors competing in some event(s) and a relation indicating superior performances or "beats" in competition.
二元圖的另一種特殊類型是錦標賽，它在數學上表示一組參加某些賽事的演員，以及表示在比賽中表現出色或“節拍”的關係。

Such tournaments can be modeled as round robin designs (Kenny 1981; Kenny and LaVoie 1984; Wong 1982)
這樣的比賽可以建模為迴圈賽設計（Kenny 1981;Kenny 和 LaVoie 1984;Wong 1982）

These competitive records form a special type of digraph, because each pair of nodes is connected by exactly one arc.
這些競爭記錄形成了一種特殊類型的二元圖，因為每對節點都恰好由一條弧線連接。

Many relations are valued; that is, the ties indicate the strength or intensity of the tie between each pair of actors.
許多關係都受到重視;也就是說，領帶表示每對演員之間領帶的強度或強度。

The graph for a valued relation must convey more information by representing the strength of an arc or a line.
值關係的圖形必須通過表示弧或線的強度來傳達更多資訊。

Signed Graphs and Signed Directed Graphs
有符號圖和有符號有向圖
Relations are measured in which the ties can be interpreted as being either positive or negative in affect, evaluation, or meaning.
關係是衡量關係的，在這種關係中，關係可以被解釋為在影響、評估或意義上是積極的或消極的。

A signed graph is a graph whose lines carry the additional information of a valence: a positive or negative sign.
有符號圖是一種圖，其線條帶有價的附加資訊：正號或負號。

A complete signed graph is a signed graph in which all unordered pairs of nodes are included in the set of lines.
完整的有符號圖是一個有符號圖，其中所有無序的節點對都包含在線集中。

Since all lines are present in a complete signed graph, and all lines have a valence either "+" or "-", each unordered pair of nodes is assigned either "+" or "-".
由於所有線都存在於一個完整的有符號圖中，並且所有線都具有“+”或“-”的化合價，因此每個無序節點對都被分配為“+”或“-”。

In a complete signed graph, a triad may be in one of four possible states, depending on whether zero, one, two, or three positive lines are present among the three nodes.
在完整的有符號圖中，三元組可能處於四種可能的狀態之一，具體取決於三個節點之間是否存在零、一條、兩條或三條正線。

In a signed directed graph the most general cycles are usually referred to as semicycles.
在有符號有向圖中，最一般的迴圈通常稱為半週期。

Signed graphs and signed directed graphs generalize graphs and directed graphs by allowing the lines or arcs to have valences.
有符號圖和有向圖通過允許直線或圓弧具有價來概括圖和有向圖。

Examples of valued relations include the frequency of interaction among pairs of people, the dollar amount of trade between nations, or the rating of friendship between people in a group
有價值的關係的例子包括成對的人之間互動的頻率、國家之間的貿易金額或群體中人與人之間的友誼等級

Such relations cannot be fully represented using a graph or a directed graph, since lines or arcs in a graph or directed graph are only present or absent.
這種關係不能用圖或有向圖完全表示，因為圖或有向圖中的線或弧只存在或不存在。

In a valued directed graph, the arc from node nj to node nj is not the
在值有向圖中，從節點 nj 到節點 nj 的弧不是

Valued Graphs and Valued Directed
值圖和值定向圖
Graphs same as the arc from node nj to node nj
圖形與從節點 nj 到節點 nj 的弧相同

In a graph or digraph, nodal degree is equal to the number of lines incident with the node or the number of arcs incident to it or from it.
在圖形或二合圖中，節點度等於與節點入射的線數或入射到節點或從節點入射的弧數。

One way to generalize the notion of degree to valued graphs and digraphs is to average the values over all lines incident with a node, or all arcs incident to or from a node.
將度的概念推廣到值圖和二元圖的一種方法是對入射到節點的所有線或入射到節點或從節點入射的所有弧的值取平均值。

In order to define these concepts for valued graphs, the authors must consider the values attached to each of the lines in a path.
為了定義值圖的這些概念，作者必須考慮附加到路徑中每條線的值。

The authors define a path at level c as a path between a pair of nodes such that each and every line in the path has a value greater than or equal to c; that is, VI ~ c for all VI in the path (Doreian 1969, 1974).
作者將 c 級的路徑定義為一對節點之間的路徑，使得路徑中的每一行都具有大於或等於 c 的值;也就是說，路徑中所有VI的VI~c（Doreian 1969,1974）。

Directed graphs are used for representing directional relations and generalize graphs by considering the direction of the arcs between pairs of nodes.
有向圖用於表示方向關係，並通過考慮節點對之間的弧的方向來概括圖。

If more than one relation is measured on the same set of actors, the graph representing this network must allow each pair of nodes to be connected in more than one way.
如果在同一組參與者上測量了多個關係，則表示此網路的圖形必須允許每對節點以多種方式連接。

The entry in cell (i, j) of X records the strength of the tie from actor i to actor j
X 的儲存格 （i， j） 中的項目記錄了從演員 i 到演員 j 的領帶強度

OMatrices 101' Hypergraphs
OMatrices 101' 超圖
The matrix for a hypergraph, denoted by A, is a g by h matrix that records which points are contained within which edges.
超圖的矩陣（用 A 表示）是一個 g x h 矩陣，它記錄了哪些點包含在哪些邊內。

A sociomatrix for a network with a single set of actors and one relation has g rows and g columns, and is of size g x g.
具有一組參與者和一個關係的網路的社會矩陣具有 g 行和 g 列，大小為 g x g。

Each entry in a matrix is called a cell, and is denoted by its row index and column index.
矩陣中的每個條目稱為儲存格，由其行索引和列索引表示。

The sociomatrix for a digraph is not necessarily symmetric, since the arc < ni,nj > is not the same as the arc < ni,nj >, and the entry in cell Xij is not necessarily the same as the entry in ceIl Xji. The authors turn to some important matrix operations, including matrix permutation, the transpose of a matrix, matrix addition and subtraction, matrix multiplication, and Boolean matrix multiplication.
二合字的社會矩陣不一定是對稱的，因為ni，nj > <弧與ni，nj > <弧不同，並且單元格 Xij中的條目不一定與ceIl Xji中的條目相同。作者轉向一些重要的矩陣運算，包括矩陣排列、矩陣的轉置、矩陣加減法、矩陣乘法和布爾矩陣乘法。

In a sociomatrix, the order of the rows and columns indexing the actors in the network or the nodes in the graph is arbitrary, so long as the rows and columns are indexed in the same order.
在社會矩陣中，索引網路中的參與者或圖形中的節點的行和列的順序是任意的，只要行和列以相同的順序進行索引即可。

Sometimes the patterns of ties between actors is not clear until the authors permute both the rows and the columns of the matrix.
有時，在作者排列矩陣的行和列之前，參與者之間的聯繫模式並不清晰。

In the transpose of the sociomatrix, a 1 in cell (i,j) indicates that row actor i received a tie from column actor j.
在社會矩陣的轉置中，單元格 （i，j） 中的 1 表示行 actor i 從列 actor j 那裡獲得了平局。

In general, we define XP (X to the pth power) as the matrix product of X times itself, p times
通常，我們將 XP（X 的 p 次方）定義為 X 乘以本身的矩陣乘積，即 p 乘以
The authors will first describe how to use matrix multiplication to study walks and reachability in a graph and show how properties of matrices can be used to quantify nodal degree and graph density.
作者將首先描述如何使用矩陣乘法來研究圖中的遊走和可達性，並展示如何使用矩陣的屬性來量化節點度和圖密度。

Since every path is a walk, the authors can study reachability of pairs of nodes by considering the powers of the matrix X that count walks of a given length.
由於每條路徑都是步行，因此作者可以通過考慮計算給定長度的步行的矩陣 X 的冪來研究節點對的可達性。

Since any two nodes that are reachable are connected by a path of length g - 1 or less, non-zero entries in the matrix X[rl indicate pairs of nodes that are reachable.
由於可訪問的任意兩個節點都通過長度為 g - 1 或更小的路徑連接，因此矩陣 X[rl 中的非零條目表示可訪問的節點對。

The entry in cell (i,j) of X[Rl is equal to 1 if nodes ni and Hj are reachable, and equal to 0 otherwise
如果節點 ni 和 Hj 可訪問，則 X[Rl 的儲存格 （i，j） 中的條目等於 1，否則等於 0

The authors can calculate these values by looking at the elements of X[I:l, and noting which ones are non~zero.
作者可以通過查看 X[I：l 的元素並注意哪些元素是非~零來計算這些值。

The authors can consider directed walks of any length by studying powers of the matrix X.
作者可以通過研究矩陣 X 的冪來考慮任何長度的有向遊走。

The entries of the matrix XP give the total number of directed walks of length p beginning at row node ni and ending at column node nj.
矩陣 XP 的條目給出了長度為 p 的有向遊走總數，從行節點 ni 開始，到列節點 nj 結束。

(g - 1), the authors obtain a matrix, denoted by X£l:l, whose entries give the total number of directed walks from row node nj to column node nj> of any length less than or equal to g-1.
（g - 1），作者得到一個矩陣，用 X£l：l 表示，其條目給出了從行節點 nj 到列節點 nj 的有向遊走總數>長度小於或等於 g-1。

These representations are quite useful, as Katz (1947) first realized
正如Katz（1947）首先意識到的那樣，這些表示非常有用

4.10 Properties of Graphs, Relations, and Matrices
4.10 圖形、關係和矩陣的性質
The authors have noted three important properties of social networks: reflexivity, symmetry, and transitivity.
作者指出了社交網路的三個重要特性：自反性、對稱性和傳遞性。

The authors show how they can be studied by examining matrices, relations, and graphs.
作者展示了如何通過檢查矩陣、關係和圖形來研究它們。

In the discussion of graphs the authors have focused on simple graphs, which, by definition, exclude loops.
在對圖的討論中，作者將重點放在簡單的圖上，根據定義，這些圖不包括迴圈。

A simple graph is irreflexive, since no.
一個簡單的圖是不可反身的，因為沒有。

If all loops are present, the graph represents a reflexive relation.
如果所有迴圈都存在，則圖形表示自反關係。

In a sociomatrix loops are coded by the entries along the main diagonal of the matrix, Xii for all i.
在社會矩陣中，迴圈由沿矩陣主對角線的條目編碼，Xii 表示所有 i。

A relation is reflexive if, in the sociomatrix, Xij = 1 for all i.
如果在社會矩陣中，所有 i 的 Xij = 1，則關係是自反關係。

An irreflexive relation has entries on the main diagonal of the sociomatrix that are undefined.
非反身關係在社會矩陣的主對角線上具有未定義的條目。

4.11 Summary that is not reflexive (also not irreflexive) has some, but not all, values of
4.11 非反身（也不是非反身）的摘要具有一些（但不是全部）值
The sociomatrix for a symmetric relation is symmetric; Xij = Xji for all distinct i and j.
對稱關係的社會矩陣是對稱的;Xij = Xji 表示所有不同的 i 和 j。

Transitivity is a property that considers patterns of triples of actors in a network or triples of nodes in a graph.
傳遞性是一種屬性，它考慮網路中參與者的三元組或圖中節點的三元組的模式。

In order for the relation to be transitive, whenever x};l ;::: 1, xij must equal 1.
為了使關係是可傳遞的，只要 x};l ;：：： 1，xij 必須等於 1。

Graph theory is a useful way to represent network data.
圖論是表示網路數據的有用方法。

Actors in a network are represented as nodes of a graph.
網路中的參與者表示為圖形的節點。

Nondirectional ties between actors are represented as lines between the nodes of a graph.
參與者之間的非方向性聯繫表示為圖形節點之間的線。

Directed ties between actors are represented as arcs between the nodes in a digraph.
參與者之間的有向聯繫表示為二圖中節點之間的弧。

Harary (1969) and Bondy and Murty (1976) are excellent mathematical introductions to graph theory, with coverage ranging from proofs of many of the statements the authors have made, to solutions to a variety of applied problems.
Harary（1969）和Bondy和Murty（1976）是圖論的優秀數學入門，涵蓋範圍從作者所做的許多陳述的證明到各種應用問題的解決方案。

The excellent text by Frank (1971) is more mathematically advanced and focuses on social networks.
弗蘭克（Frank，1971）的優秀著作在數學上更先進，專注於社交網路。

Roberts (1976, 1978) and Hage and Harary (1983) provide very readable, elementary introductions to graph theory, with many concepts illustrated on anthropological network data.
Roberts（1976,1978）和Hage和Harary（1983）對圖論進行了非常可讀的基本介紹，其中許多概念都說明瞭人類學網路數據。

A more mathematical discussion of tournaments can be found in Moon (1968). Berge (1989) discusses hypergraphs in detail
關於錦標賽的更多數學討論可以在Moon（1968）中找到。Berge（1989）詳細討論了超圖

Part III
第三部分
One of the primary uses of graph theory in social network analysis is the identification of the "most important" actors in a social network.
圖論在社交網路分析中的主要用途之一是識別社交網路中“最重要”的參與者。

Among the definitions that the authors will discuss are those based on degree, closeness, betweenness, iriformation, and the differential status or rank of the actors
作者將討論的定義包括那些基於程度、親密程度、中間性、虹膜形成以及演員的不同地位或等級的定義

These definitions yield actor indices which attempt to quantify the prominence of an individual actor embedded in a network.
這些定義產生了行為者指數，這些指數試圖量化嵌入網路中的單個行為者的突出程度。

Measures such as outdegree and indegree are quite likely to be different, and prestigious actors are usually those with large indegrees, or "choices" received
學位和學位等衡量標準很可能是不同的，而有聲望的演員通常是那些擁有大量學位或獲得“選擇”的人

Both centrality and prestige indices are examples of measures of the prominence or importance of the actors in a social network.
中心性和聲望指數都是衡量社會網路中參與者的突出性或重要性的例子。

Bavelas (1950), Flament (1963), Beauchamp (1965), and Sabidussi (1966) state that very centralized graphs are compact, in the sense that the distances between pairs of nodes are small
Bavelas （1950）、Flament （1963）、Beauchamp （1965） 和 Sabidussi （1966） 指出，非常集中的圖是緊湊的，從某種意義上說，節點對之間的距離很小

These authors proposed an index of actor centrality based on closeness, as the authors will discuss later .
這些作者提出了一個基於親密關係的演員中心性指數，作者將在後面討論。

The variance is .recommended as a group-level index of centrality by Snijders (1981a, 1981b), reflecting the view of H0ivik and Gleditsch (1975) that centralization is synonymous with the dispersion or heterogeneity of an actor index
Snijders（1981a，1981b）推薦將方差作為群體層面的中心性指數，反映了H0ivik和Gleditsch（1975）的觀點，即中心化是參與者指數的離散性或異質性的同義詞

This index attains its minimum value of 0 when all degrees are equal or when the graph is regular.
當所有度數相等或圖形正則時，該指數達到其最小值 0。

Brazil, Czechoslovakia, and Argentina are linked directly to other prestigious countries
巴西、捷克斯洛伐克和阿根廷與其他著名國家直接相連

Comparisons and Extensions
比較和擴展
Several authors have compared the performance of the many centrality and prestige indices discussed either on real or simulated data, or both
幾位作者比較了在真實數據或類比數據上討論的許多中心性和聲望指數的性能，或兩者兼而有之

Earlier researchers, such as Stogdill (1951), concentrated on different measures of actor degrees, focusing attention on only one centrality index.
早期的研究者，如Stogdill（1951），專注於不同的演員度測量，只關注一個中心性指數。

Freeman lists all thirty-four possible graphs with g = 5 nodes, and compares actor- and group-level degree, closeness, and betweenness centrality measures across the graphs.
Freeman 列出了 g = 5 節點的所有 34 個可能的圖，並比較了圖中的參與者和組級程度、接近度和仲介中心性度量。

The study of structural balance in a social network, consisting of a relation measured for a set of actors, requires that the ties have a sign or a valence.
對社會網路中結構平衡的研究，包括為一組參與者測量的關係，要求這些關係具有符號或效價。

A relation must be representable as a signed graph or digraph in order to be studied using ideas of balance: positive ties as well as negative ties must be possible.
關係必須可以表示為有符號圖或二元圖，以便使用平衡思想進行研究：正向關係和負向關係必須是可能的。

Sociologists and social psychologists have used the term "structural balance" to refer to groups of people and affective relations that substantively are "pleasing" or lack intrapersonal psychological "tension." The authors will formally define a triple of nodes, and the lines between them, as balanced if the cycle has a positive sign.
社會學家和社會心理學家使用「結構平衡」一詞來指代實質上「令人愉悅」或缺乏人內心理」張力“的人群和情感關係。作者將正式定義一個三元組節點，以及它們之間的線，如果週期有一個正號，則為平衡。

A balanced signed graph important generalization of this idea first mentioned by Heider: that balanced triples have actor partitions for which positive ties occur within and negative between.
平衡有符號圖是Heider首先提到的這個想法的重要概括：平衡三元組具有參與者分區，其中正連接發生，負連接之間發生。

The authors will return to clusterability as a generalization of structural balance later
作者稍後將回到聚類性，作為結構平衡的推廣

Signed Directional Relations
簽署方向關係
Suppose that the relation under investigation is directional, so that the relevant representation is a signed digraph.
假設所研究的關係是方向性的，因此相關表示是帶符號的二合體。

To generalize balance to such structures requires some care, since there are a number of ways to examine cycles in directed graphs.
將平衡推廣到這種結構需要一些小心，因為有許多方法可以檢查有向圖中的週期。

Consider the triple shown, which has one negative arc, and two positive arcs
考慮所示的三元組，它有一個負弧和兩個正弧

This digraph does not contain a cycle, since the arc from nl to n2 is oriented in the wrong direction.
這個二合圖不包含迴圈，因為從 nl 到 n2 的弧的方向是錯誤的。

Reversing the direction of this arc would give them a digraph with a cycle of length 3, nln3n2nJ, with a negative sign, and the digraph appears to be an unbalanced structure.
反轉這個弧線的方向會給它們一個週期長度為 3 的二合圖，nln3n2nJ，帶有負號，並且二合圖似乎是一個不平衡的結構。

To formally define balance in signed digraphs, the authors consider not paths and cycles, but semipaths and semicycles.
為了正式定義有符號二合字中的平衡，作者考慮的不是路徑和迴圈，而是半路徑和半迴圈。

With these definitions, the authors can state: Definition 6.2 A signed digraph is balanced if and only if all semicycles have positive signs.
根據這些定義，作者可以說明： 定義 6.2 當且僅當所有半週期都有正符號時，有符號二合符是平衡的。

In a balanced signed digraph, all semicycles must have an even number of negative signs attached to the arcs.
在平衡有符號二合符中，所有半環都必須有偶數個負符號附著在弧上。

The semicycle, n2nln3n2, has a sign "-", so this digraph is not balanced.
半週期 n2nln3n2 有一個符號“-”，所以這個二元圖是不平衡的。

The authors should note that there is a very comprehensive set theoretic approach to structural balance given by Flament (1963), similar to Freeman's (1989) representation for social network data discussed at the end of Chapter 3
作者應該注意到，Flament （1963） 給出了一個非常全面的結構平衡理論方法，類似於Freeman （1989） 在第3章末尾討論的社交網路數據表示

OChecking for Balance
平衡
A single unbalanced cycle or semicycle insures that the graph or digraph is not balanced.
單個不平衡迴圈或半迴圈確保圖形或二元圖不平衡。

It is natural to consider how many cycles or semicycles in a graph or digraph do not have positive signs.
很自然地要考慮圖或二合圖中有多少個週期或半周期沒有正號。

From this consideration, one can develop graph-level indices measuring the amount of unbalance in a structure.
從這個考慮出發，人們可以開發圖形級指數來衡量結構中的不平衡量。

Cycles have a maximum length of g, so the authors need not raise the sociomatrix to any power greater than g
迴圈的最大長度為 g，因此作者不需要將社會矩陣提高到任何大於 g 的冪

The authors note that the numbers on the diagonals of the power sociomatrices for balanced graphs are the sums of the signs of closed walks, with lengths equal to the powers of the respective matrices.
作者指出，平衡圖的冪社會矩陣對角線上的數位是閉合行走符號的總和，長度等於各個矩陣的冪。

To quantify how "unbalanced" an unbalanced graph or digraph is, one first must count the number of cycles or the number of semicycles that have negative Signs.
要量化不平衡圖或二元圖的“不平衡”程度，首先必須計算具有負號的週期數或半週期數。

An index such as this is usually referred to as a cycle index for balance.
這樣的指數通常被稱為平衡周期指數。

References to its use in practice and theory abound - Taylor (1970), who presents both a text for readers on balance and social interaction, and critically reviews the literature, cites nearly 200 papers and books. Hage and Harary (1983), in their chapter on signed graphs, and Hage and Harary (1991) cite many anthropological studies of balance in networks. Davis (1963, 1967, 1968b) takes a variety of very important studies and formulates a large number of propositions
關於它在實踐和理論中的使用比比皆是——泰勒（Taylor，1970）為讀者提供了關於平衡和社會互動的文本，並批判性地回顧了文獻，引用了近200篇論文和書籍。Hage和Harary（1983）在他們關於符號圖的章節中，以及Hage和Harary（1991）引用了許多關於網路平衡的人類學研究。Davis（1963,1967,1968b）進行了各種非常重要的研究，並提出了大量的命題

Clusterability about social structure from the writings of these theorists
從這些理論家的著作中關於社會結構的聚類性
The studies are Durkheim (1947), Stouffer Suchman, DeVinney, Star, and Williams (1949), Merton and Kitt (1950), Homans (1950, 1961), Festinger (1954, 1957), Berelson, Lazarsfeld, and McPhee (1954), Lazarsfeld and Merton (1954), Katz and Lazarsfeld (1955), Lipset, Trow, and Coleman (1956), Bott (1957), Coleman (1957), Fiedler (1958), and Davis (1959).
這些研究是塗爾干（1947），Stouffer Suchman，DeVinney，Star和Williams（1949），Merton和Kitt（1950），Homans（1950,1961），Festinger（1954,1957），Berelson，Lazarsfeld和McPhee （1954），Lazarsfeld 和 Merton（1954），Katz 和 Lazarsfeld（1955），Lipset，Trow 和 Coleman（1956），Bott（1957），Coleman（1957），Fiedler（1958）和Davis（1959）。

Structural balance, as noted by Granovetter (1979) need not apply to the behavior of actors outside of small group settings
正如Granovetter（1979）所指出的，結構平衡不一定適用於小團體環境之外的參與者的行為

Some ties, especially those that makecycles have negative signs, may be reinforced by a wide variety of institutional, economic, and political constraints.
一些關係，特別是那些具有負面信號的聯繫，可能會因各種制度、經濟和政治限制而得到加強。

The most important aspect of structural balance is that the nodes in a balanced graph can be partitioned into two subsets or clusters
結構平衡最重要的方面是平衡圖中的節點可以劃分為兩個子集或集群

This fact follows directly from the original theorem for balance involving the signs of cycles, and allows one to consider clusters of actors among whom all ties are possible.
這一事實直接來自涉及週期符號的原始平衡定理，並允許人們考慮所有聯繫都可能的參與者集群。

It allowed researchers, in the 1950's and 1960's, to consider ways to generalize structural balance, so that actors could possibly be partitioned into more than two subsets.
它允許研究人員在1950年代和1960年代考慮推廣結構平衡的方法，以便參與者可以被劃分為兩個以上的子集。

Harary (1954) proved that balanced signed graphs have partitions of nodes into two clusters or subsets such that only positive lines join nodes in the same cluster and only negative lines join nodes in different clusters.
Harary（1954）證明，平衡有符號圖將節點劃分為兩個簇或子集，使得只有正線連接同一簇中的節點，只有負線連接不同簇中的節點。

These two theorems give the conditions under which a signed graph has a clustering; that is, under what conditions on the
這兩個定理給出了有符號圖具有聚類的條件;也就是說，在什麼條件下

Clllsterability cycles of a graph will the ,graph be clusterable?
圖的可聚性循環 ，圖是否可聚類？
The second theorem is more specific than the first, since it is appropriate only for complete signed graphs, where all nodes are adjacent.
第二個定理比第一個定理更具體，因為它只適用於所有節點相鄰的完整有符號圖。

It is important since it shows that for complete signed graphs, one need only look at cycles of length 3 to determine clusterability.
這很重要，因為它表明，對於完整的有符號圖，只需要查看長度為 3 的週期即可確定聚類性。

None of these cycles contains exactly one line with a sign of "-", so, by the theorem, the graph is clusterable.
這些迴圈中沒有一個只包含一條帶有符號“-”的線，因此，根據定理，該圖是可聚類的。

If there is a triple of actors in a cycle containing three negative lines, these three actors can be partitioned into three different clusters.
如果一個包含三條負線的迴圈中存在三重參與者，則這三個參與者可以劃分為三個不同的集群。

The following theorem extends clusterability to complete signed graphs; its last condition is very important
以下定理將聚類性擴展到完整的有符號圖;它的最後一個條件非常重要

It comes directly from Davis (1967).
它直接來自Davis（1967）。

Davis' clustering theorems, coupled with Flament's (1963) finding that the properties of triples were sufficient to assess the balance of a complete signed graph, led to nearly two decades of research on statistical and deterministic models for triples.
Davis的聚類定理，加上Flament（1963）發現三元組的性質足以評估完整有符號圖的平衡性，導致了近二十年對三元組的統計和確定性模型的研究。

Through these theorems, the properties of the triples of nodes in a graph tell them whether theoretically important structural properties are present.
通過這些定理，圖中節點三元組的性質告訴它們是否存在理論上重要的結構性質。

The clusters of actors appeared to be ranked, or hierarchical in nature, with the actors "on the bottom" choosing those "at the top"
這些行為者群體似乎是按等級劃分的，或者說是等級分明的，“在底層”的在“在底層”在底層的

ORanked Clusterability
ORanked 可聚類性
Davis and Leinhardt (1968) presented a concept of ranked clusters, for complete signed directed graphs.
Davis 和 Leinhardt （1968） 提出了一個排序聚類的概念，用於完整的有符號有向圖。

Ranked c1l1sterability, in which the positive arcs emanating to or from the nodes in [+-j dyads are postulated to "point" in the same direction, states that the triples numbered 2, 10, 11, 12, 13, 14, 15, and 16 of Figure 6.6 should not occur in practice.
排序 c1l1sterability，其中 [+-j 二元組中向或從節點發出的正弧被假定為“指向”同一方向，指出圖 6.6 中編號為 2、10、11、12、13、14、15 和 16 的三元組在實踐中不應出現。

These "miserable" eight (Davis 1979) depart from both clusterability and ranked cIusterability.
這“悲慘的”八個（Davis 1979）偏離了可聚類性和排名cIusterability。

As Davis (1979) notes, there was strong empirical evidence for 6/8th's of a theorem
正如Davis（1979）所指出的，有很強的經驗證據證明一個定理的6/8

These two triples are quite common in positive affect relations which are in an "early" development stage; that is, assuming that the relation under study will change over time, these triples contain dyads which might evolve into triples which are not prohibited.
這兩個三元組在處於「早期」發展階段的積極影響關係中很常見;也就是說，假設所研究的關係會隨著時間的推移而改變，這些三元組包含二元組，這些二元組可能會演變成不被禁止的三元組。

This ranked clusterability model was quite elegant, but little used.
這種排序的可聚類性模型非常優雅，但很少使用。

To turn ranked cIusterability for complete signed digraphs into an equivalent idea for digraphs
將完整有符號二合字的排序 cIusterability 轉化為二合字的等價思想

Transitivity without signs is quite simple
沒有符號的傳遞性非常簡單
The authors take the idea of ranked clusters for complete signed digraphs, and do not consider arcs with negative signs.
作者將排序簇的思想用於完整的有符號二合符，並且不考慮帶有負號的弧。

The assumption is that the relation under study is the "positive" part of the signed relation - for example, the authors study only "like," "not like," and "dislike." Figure 6.7 shows the triples of Figure 6.6, without the negative arcs.
假設所研究的關係是有符號關係的“積極”部分——例如，作者只研究“喜歡”、“不喜歡”和“不喜歡”。圖 6.7 顯示了圖 6.6 的三元組，沒有負弧。

Holland and Leinhardt showed that ranked c1usterability is a special case of a more general set of theorems which naturally blend balance, clusterability, and ranked c1usterability.
Holland 和 Leinhardt 表明，排序 c1usterability 是一組更一般的定理的特例，這些定理自然地融合了平衡性、聚類性和排序 c1usterability。

Their partially ordered clusterability leads naturally to a consideration of the concept of transitivity.
它們的部分有序聚類性自然導致了對傳遞性概念的考慮。

One can obtain balanced, clusterable, and ranked c1usterable graphs by making various assumptions about reciprocity and asymmetry of choices.
通過對選擇的互惠性和不對稱性做出各種假設，可以獲得平衡的、可聚類的和可排序的 c1usterable 圖。

During the past two decades, evidence has accumulated that transitivity is a compelling force in the organization of social groups.
在過去的二十年中，越來越多的證據表明，傳遞性是社會群體組織中的一股令人信服的力量。

Transitivity
傳遞
The authors turn the attention to a triple of actors, i, j, and k, and the ties between them. The authors state: Definition 6.4 The triad involving actors i. j. and k is transitive if whenever i --+ j and j --+ k i --+ k.
作者將注意力轉向了三重演員，i、j 和 k，以及它們之間的聯繫。作者指出： 定義 6.4 涉及行為者 i. j. 和 k 的三元組是及物的，如果每當 i --+ j 和 j --+ k i --+ k。

The number of transitive and/or intransitive triples within a particular type of triad is very important when quantitatively and statistically assessing the amount of transitivity in a digraph.
在定量和統計地評估二元圖中的傳遞性量時，特定類型的三元組中傳遞和/或不傳遞三元組的數量非常重要。

The authors note in conclusion that while this small set of graph theorists, sociologists, social psychologists, and statisticians were working on mathematical models of balance, clusterability, and transitivity, other methodologists were busy studying about cliques and cohesive subgroups.
作者在結論中指出，當這一小群圖理論家、社會學家、社會心理學家和統計學家正在研究平衡、聚類性和傳遞性的數學模型時，其他方法學家正忙於研究集團和有凝聚力的子群體。

This area of research is described .
描述了這一研究領域。

The the authors discuss methods for finding cohesive sub~oups of actors within a social network.
作者討論了在社交網路中尋找有凝聚力的子~角色的方法。

Social Group and Subgroup
社會群體和子群體
Many authors have discussed the role of social cohesion in social explanations and theories (Burt 1984; Collins 1988; Erickson 1988; Friedkin 1984).
許多作者討論了社會凝聚力在社會解釋和理論中的作用（Burt 1984;柯林斯 1988;埃裡克森 1988;Friedkin 1984）。

Many network researchers who have developed or reviewed methods for cohesive subgroups in social networks have noted that these methods attempt to formalize the notion of social group (Seidman and Foster 1978a, 1978b; Alba and Moore 1978; Mokken 1979; Burt 1980; Freeman 1984, 1992a; Sailer and Gaulin 1984)
許多開發或審查了社交網路中凝聚性子群體方法的網路研究人員注意到，這些方法試圖將社會群體的概念形式化（Seidman and Foster 1978a， 1978b;Alba 和 Moore 1978;莫肯 1979;伯特 1980;弗里曼 1984， 1992a;Sailer 和 Gaulin 1984）

According to these authors, the concept of social group can be studied by looking at properties of subsets of actors within a network.
根據這些作者的說法，社會群體的概念可以通過觀察網路中參與者子集的屬性來研究。

The authors' discussion is divided into sections, each of which takes up methods that are primarily motivated by one of these ideas
作者的討論分為幾個部分，每個部分都採用主要受這些想法之一啟發的方法

In contrast to these ideas that focus on ties between pairs of actors in one-mode networks, some cohesive subgroup ideas are concerned with the linkages that are established among individuals by virtue of their common membership in collectivities.
與這些側重於單一模式網路中成對行為者之間聯繫的思想相反，一些有凝聚力的子群體思想關注的是個人之間由於集體中的共同成員身份而建立的聯繫。

Subgroups Based on Complete
基於完成的子組
Mutuality length of a path is the number of lines in it. A shortest path between two nodes is called a geodesic, and the (geodesic) distance between two nodes, denoted by dU, j), is the length of a shortest path between them.
路徑的互數長度是其中的行數。兩個節點之間的最短路徑稱為測地線，兩個節點之間的（測地線）距離（用 dU、j 表示）是它們之間最短路徑的長度。

Luce and Perry and Festinger proposed that a clique for a relation of positive affect is a subset of people among whom all choices are mutual, and no other people can be added to the subset who have mutual choices with all members of the subset
Luce、Perry 和 Festinger 提出，積極影響關係的集團是所有選擇都是相互的人群的子集，並且不能將與該子集的所有成員有共同選擇的其他人添加到子集中

This definition of a clique is appropriate for a directional dichotomous relation.
集團的這種定義適用於方向性二分關係。

The authors will use the example of the relations of marriage and business among Padgett's Florentine families to illustrate cohesive subgroups throughout this chapter.
作者將以帕吉特的佛羅倫薩家庭之間的婚姻和商業關係為例，來說明本章中具有凝聚力的亞群體。

Recall that both of these relations are dichotomous and nondirectionaL The authors used the network analysis programs GRADAP 2.0.
回想一下，這兩種關係都是二分和非方向性的aL作者使用了網路分析程式GRADAP 2.0。

Two different structural properties have been used to relax the clique notion: first, Luce (1950), and later Alba (1973) and Mokken (1979), have used properties of reachability, path distance, and diameter to extend the clique definition; second, Seidman and Foster (1978a) and Seidman (1981b, 1983b) used nodal degree to propose alternative cohesive subgroup ideas.
使用了兩種不同的結構屬性來放鬆集團概念：首先，Luce（1950），後來Alba（1973）和Mokken（1979）使用可達性，路徑距離和直徑的屬性來擴展集團定義;其次，Seidman和Foster（1978a）和Seidman（1981b，1983b）使用節點度提出了替代的內聚子群思想。

Subgroups Based on Reachability and Diameter
基於可達性和直徑的子組
Reachability is the motivation for the first cohesive subgroup ideas that extend the notion of a clique.
可達性是第一個有凝聚力的子群體思想的動機，這些思想擴展了集團的概念。

These alternative subgroup ideas are useful if the researcher hypothesizes that important social processes occur through intermediaries.
如果研究人員假設重要的社會過程是通過中介發生的，那麼這些替代子群體的想法是有用的。

Recall that the geodesic distance between two nodes, denoted by dei, j), is the length of a shortest path between them.
回想一下，兩個節點之間的測地線距離（用 dei， j） 表示）是它們之間最短路徑的長度。

Cohesive subgroups based on reachability require that the geodesic distances among members of a subgroup be small.
基於可達性的內聚子組要求子組成員之間的測地線距離很小。

The authors can specify some cutoff value, n, as the maximum length of geodesics connecting pairs of actors within the cohesive subgroup.
作者可以指定一些截止值 n，作為連接內聚子群內參與者對的測地線的最大長度。

Restricting geodesic distance among subgroup members is the basis for the definition of an n-clique (Alba 1973; Luce 1950).
限制子組成員之間的測地線距離是定義 n 集團的基礎（Alba 1973;盧斯 1950 年）。

An n-clique is a maximal subgraph in which the largest geodesic distance between any two nodes is no greater than n.
n 群是一個最大子圖，其中任意兩個節點之間的最大測地線距離不大於 n。

2-cliques are subgraphs in which all members need not be adjacent, but all members are reachable through at most one intermediary.
2-cliques 是子圖，其中所有成員不需要相鄰，但最多可以通過一個仲介訪問所有成員。

In this graph, there are two 2-cliques: {1,2,3,4,5} and {2,3,4,5,6}.
在此圖中，有兩個 2 集團：{1,2,3,4,5} 和 {2,3,4,5,6}。

2-c1iques: {1, 2, 3, 4, 5} and {2, 3,4, 5,6} 2-cIan: {2,3,4,5,6} 2-c1ubs: {1,2,3,4}, {1,2,3,5}, and {2,3,4,5,6}
2-c1iques： {1， 2， 3， 4， 5} 和 {2， 3,4， 5,6} 2-cIan： {2,3,4,5,6} 2-c1ubs： {1,2,3,4}， {1,2,3,5} 和 {2,3,4,5,6}

An Example
一個例子
Let them return to the example of marriage and business relations among Padgett's Florentine families to illustrate n-cliques.
讓他們回到帕吉特的佛羅倫薩家庭之間的婚姻和商業關係的例子來說明n-cliques。

There are thirteen 2-cliques in the marriage relation: Acciaiuoli Albizzi BarbadoriMedici Ridolfi Salviati Tornabuoni;.
婚姻關係中有 13 個 2 派系：Acciaiuoli Albizzi BarbadoriMedici Ridolfi Salviati Tornabuoni;。

. Albizzi Ginori Guadagni Medici; Albizzi Guadagni Medici Ridolfi Tornabuoni;.
.阿爾比齊·吉諾里·瓜達尼·美第奇;阿爾比齊·瓜達尼·美第奇·里多爾菲·托爾納波尼;

Guadagni Medici Ridolfi Strozzi Tornabuoni; Medici Pazzi Salviati.
收入醫生：里多爾菲·斯特羅齊·托爾納波尼;美第奇·帕齊·薩爾維亞蒂。

There are four 2-cliques on the business relation: Barbadori Bischeri Castellani Lamberteschi Peruzzi;.
業務關係有四個 2 集團：Barbadori Bischeri Castellani Lamberteschi Peruzzi;。

. Barbadori Ginori Medici Pazzi Salviati Tornabuoni;.
.巴巴多里、吉諾里、美第奇、帕齊、薩爾維亞蒂、龍捲風;

. Bischeri Castellani Guadagni Lamberteschi Peruzzi Notice that the 2-cliques are both larger and more numerous than the cliques found for both the marriage and business relations.
.比舍里·卡斯特拉尼 瓜達尼·蘭貝特斯基·佩魯齊 請注意，2 個集團比婚姻和商業關係的集團更大、更多。

Since the definition of an n-clique is less restrictive than the definition of a clique, when n is greater than 1 it is likely that a network will contain more n-cliques than cliques.
由於 n 集團的定義比集團的定義限制性更小，因此當 n 大於 1 時，網路可能包含的 n 集團多於集團。

It is likely that the n-cliques will be larger than the cliques
n-集團可能會比集團大

Considerations
考慮
There are several important properties of n-cliques, some of which limit the usefulness of this cohesive subgroup definition.
n 小集團有幾個重要屬性，其中一些限制了這個有凝聚力的子群定義的有用性。

Since n-cliques are defined for geodesic paths that can include any nodes in the graph, two problems might arise: first, an n-clique, as a subgraph, may have a diameter greater than n, and second, an n-clique might be disconnected.
由於 n 社區是為可以包含圖中任何節點的測地線路徑定義的，因此可能會出現兩個問題：首先，作為子圖的 n 社區可能具有大於 n 的直徑，其次，n 個社區可能會斷開連接。

Mokken (1979) has described two logical ways to do this
Mokken（1979）描述了兩種合乎邏輯的方法

The first, which he calls an n-clan, starts with the n-cliques that are identified in a network and excludes those n-cliques that have a diameter greater than n.
第一種，他稱之為 n 氏族，從網路中識別的 n 氏族開始，排除直徑大於 n 的 n 氏族。

This example is taken from Alba (1973) and Mokken (1979), and illustrates the difference between n-cliques, nclans, and n-clubs
這個例子取自 Alba （1973） 和 Mokken （1979），說明瞭 n-cliques、nclans 和 n-clubs 之間的區別

For this graph, taking n = 2 results in the following sets: 2-cliques: {1,2,3,4,5} and {2,3,4,5,6}; 2-clan: {2, 3, 4, 5, 6}; 2-clubs: {1, 2, 3,4}, {1, 2, 3, 5}, and {2, 3,4, 5, 6}.
對於此圖，取 n = 2 會得到以下集合：2 個集團：{1,2,3,4,5} 和 {2,3,4,5,6};2 族：{2、3、4、5、6};2 桿：{1、2、3,4}、{1、2、3、5} 和 {2、3、4、5、6}。

Five of the 2-cliques have diameter greater than 2, so they are excluded from the list of 2-clans
2 個集團中有 5 個的直徑大於 2，因此它們被排除在 2 集團名單之外

This leaves eight 2-c1ans: Acciaiuoli Albizzi Barbadori Medici Ridolfi Salviati Tornabuoni;.
這剩下八個 2-c1ans：Acciaiuoli Albizzi Barbadori Medici Ridolfi Salviati Tornabuoni;。

The diameter of the 2-clique {Barbadori, Medici, Ridolfi, Strozzi, Tornabuoni} is greater than 2, since the geodesic between Strozzi and Barbadori includes Castellani.
2 集團 {巴巴多里、美第奇、里多爾菲、斯特羅齊、龍捲風} 的直徑大於 2，因為斯特羅齊和巴巴多里之間的測地線包括卡斯特拉尼。

An n-c1ique may be seen as too loose a definition of cohesive subgroup, and restrictions requiring geodesic paths to remain
n-c1ique 可能被視為對內聚子群的定義過於寬鬆，並且需要保留測地線路徑的限制

Subgroups Based on Nodal
基於節點的子組
Degree within the subgroup can be applied by requiring the subgraph to have a given maximum diameter. n-c1ubs and n-c1ans are two possible definitions that have the desired restrictions.
可以通過要求子圖具有給定的最大直徑來應用子組內的度數。N-C1UBS 和 N-C1ANS 是具有所需限制的兩個可能定義。

In studying network processes such as information diffusion that "flow" through intermediaries, cohesive subgroups based on indirect connections of relatively short paths provide a reasonable approach.
在研究網路過程（例如通過仲介“流動”的信息擴散）時，基於相對較短路徑的間接連接的內聚子群提供了一種合理的方法。

Hubbell's approach relies on measures of influence based on a weighting of adjacencies and paths of influence, and a partitioning of actors based on the degree to which subgroup members mutually influence each other.
Hubbell 的方法依賴於基於鄰接和影響路徑權重的影響力衡量，以及基於子群體成員相互影響程度的參與者劃分。

Since the number of actors adjacent to a given actor is quantified by the degree of the node in a graph, these subgroup methods focus on nodal degree.
由於與給定參與者相鄰的參與者的參與者的數量由圖中節點的度數量化，因此這些子組方法側重於節點度。

One measures robustness by considering "the degree to which the structure is vulnerable to the removal of any given individual" (Seidman and Foster 1978, page 142).
人們通過考慮“結構在多大程度上容易受到任何給定個體的移除”來衡量魯棒性（Seidman and Foster 1978，第 142 頁）。

The possible lack of robustness of n-cliques was one consideration that led to the proposal of an alternative subgroup definition
n-集團可能缺乏穩健性是導致提出替代亞組定義的一個考慮因素

This alternative definition, the k-plex, builds on the notion that cohesive subgroups should contain sets of actors among whom there are relatively numerous adjacencies (Seidman 1978; Seidman and Foster 1978).
這種替代定義，即 k-plex，建立在這樣一種概念之上，即有凝聚力的子群應該包含一組參與者，其中有相對多的相鄰關係（Seidman 1978;Seidman 和 Foster 1978）。

Borgatti, Everett, and Shirey (1990) have extended the notion of an LS set
Borgatti， Everett， and Shirey （1990） 擴展了 LS 集合的概念

Their approach, which they call a lambda set, is motivated by the idea that a cohesive subset should be relatively robust in terms of its connectivity.
他們的方法，他們稱之為lambda集，其動機是，一個有凝聚力的子集在其連通性方面應該是相對穩健的。

Several researchers have proposed measures for the extent to which ties are concentrated within a subgroup, rather than between subgroups
一些研究人員提出了衡量關係集中在一個亞組內而不是亞組之間的程度的措施

Measures of Subgroup
亞組的措施
The authors discuss extensions of cohesive subgroups to relations that are valued or directional
作者討論了將有凝聚力的子群擴展到有價值的或方向性的關係

These extensiOIlN allow the cohesive subgroup ideas discussed to be applied to a much wider range of social network data.
這些擴展OIlN允許所討論的有凝聚力的子組思想應用於更廣泛的社交網路數據。

Recall that the definition of a clique originally proposed by Festinger (1949) and Luce and Perry (1949) focused on directional affective relations and required that all ties between all pairs of clique members be reciprocated.
回想一下，最初由Festinger（1949）和Luce和Perry（1949）提出的集團定義側重於定向情感關係，並要求所有集團成員之間的所有聯繫都是相互的。

A more flexible way to extend cohesive subgroup ideas to directional relations uses definitions of semipaths and connectivity for directed graphs.
將內聚子群思想擴展到方向關係的一種更靈活的方法使用有向圖的半路徑和連通性的定義。

In a valued relation the authors can study cohesive subgroups that vary in the strength of ties among members.
在價值關係中，作者可以研究成員之間聯繫強度不同的有凝聚力的子群體。

An n-clique at level c requires that geodesics between subgroup members contain lines that have values that are all c or greater.
級別 c 的 n 組要求子組成員之間的測地線包含值均為 c 或更大的線。

One way to study cohesive subgroups in valued relations is to define one or more derived dichotomous relations based on the strength of the ties in the original valued relation (Doreian 1969).
研究價值關係中內聚子群的一種方法是根據原始價值關係中關係的強度定義一個或多個派生的二分關係（Doreian 1969）。

Every derived dichotomous relation defines a graph that can be analyzed using methods for finding cohesive subgroups, described above.
每個派生的二分關係都定義了一個圖，可以使用上面描述的查找內聚子群的方法進行分析。

The numbers of actors in the subgroups and the degree to which these subgroups overlap can be used to describe the structure of the network as a whole
子組中參與者的數量以及這些子組的重疊程度可用於描述整個網路的結構

7.10 Other Approaches
7.10 其他方法
All of the cohesive subgroup ideas discussed define specific graph theoretic properties that should be satisfied in order to identify a subset of actors as a cohesive subgroup.
所討論的所有內聚子群思想都定義了特定的圖論屬性，這些屬性應該滿足這些屬性，以便將參與者的子集識別為一個內聚的子群。

For all of these approaches, the analytic problem is to examine a set of social network data to see whether any subsets of actors meet the specified subgroup definition.
對於所有這些方法，分析問題是檢查一組社交網路數據，以查看是否有任何參與者子集滿足指定的子組定義。

The result is the possible assignment of actors to one or more cohesive subgroups.
其結果是可能將參與者分配到一個或多個有凝聚力的子組。

More exploratory, approach to cohesion in social networks seeks to represent the group structure in a network as a whole.
更具探索性的是，社交網路中的凝聚力方法試圖將整個網路中的群體結構表示為一個整體。

Collections of actors among whom there are relatively strong ties can become more visible by displaying functions or rearrangements of the graphs or sociomatrices.
通過顯示函數或圖形或社會矩陣的重新排列，可以更明顯地顯示具有相對緊密聯繫的參與者的集合。

7.10.1 Matrix Permutation Approaches
7.10.1 矩陣排列方法
The earliest contributions to cohesive subgroup analysis of social networks were concerned with systematic ways for ordering rows and columns of a sociomatrix to reveal the subgroup structure of a network (Forsyth and Katz 1946; Katz 1947).
對社交網路的內聚子群分析的最早貢獻是關於對社會矩陣的行和列進行排序以揭示網路的子群結構的系統方法（Forsyth and Katz 1946;Katz 1947年）。

If the value of equation (7.5) is small, the ordering of rows and columns in the sociomatrix places actors among whom there are relatively strong ties close to each other, as is desired.
如果等式 （7.5） 的值很小，則社會矩陣中行和列的順序會根據需要放置彼此之間有相對強聯繫的參與者。

Beum and Brundage (1950), Coleman and MacRae (1960), and Arabie, Hubert, and Schleutermann (1990) suggest strategies for reordering rows and columns of the sociomatrix so that i and j corresponding to large values of xij are moved closer together
Beum 和 Brundage （1950）、Coleman 和 MacRae （1960） 以及 Arabie、Hubert 和 Schleutermann （1990） 提出了對社會矩陣的行和列進行重新排序的策略，以便對應於 xij 的大值的 i 和 j 靠得更近

This problem of sociomatrix permutation to optimize a given quantity is an instance of the more general analysis problem of combinatorial optimization.
這種用於優化給定量的社會矩陣排列問題是組合優化的更一般分析問題的一個實例。

The result of a matrix permutation analysis is a reordering of the rows and columns of the sociomatrix so that actors that are close in the sociomatrix tend to have relatively strong ties.
矩陣排列分析的結果是對社會矩陣的行和列進行重新排序，以便社會矩陣中接近的參與者往往具有相對較強的聯繫。

To study cohesive subsets of actors in a network the input to multidimensional scaling should be some measure of pairwise proximity among actors, such as the geodesic distance between each pair of actors.
為了研究網路中參與者的內聚子集，多維縮放的輸入應該是參與者之間成對接近的某種度量，例如每對參與者之間的測地線距離。

Factor analysis can be used to study cohesive subgroups in an exploratory way, the most influential and important cohesive subgroup ideas are those that express specific formal properties of cohesive subgroups and locate such subgroups that might exist within a network data set
因數分析可用於以探索性的方式研究內聚子群，最有影響力和最重要的內聚子群思想是那些表達內聚子群的特定形式屬性並定位網路數據集中可能存在的此類子群的思想

7.11 Summary
7.11 總結
The authors have presented methods for studying cohesive subgroups in social networks, for dichotomous nondirectional relations, directional relations, and valued relations.
作者提出了研究社交網路中內聚子群的方法，包括二分非方向關係、方向關係和價值關係。

These methods are motivated by theoretically important properties of cohesive subgroups, and present alternative ways of quantifying the idea of social group using social networks.
這些方法受到內聚子群體理論上重要特性的激勵，並提出了使用社交網路量化社會群體概念的替代方法。

The authors presented methods for assessing the cohesiveness of subgroups.
作者介紹了評估亞組凝聚力的方法。

The authors discuss methods for analyzing a special kind of twomode social network that represents the affiliation of a set of actors with a set of social occasions.
作者討論了分析一種特殊的雙模式社交網路的方法，該社交網路代表了一組參與者與一組社交場合的隸屬關係。

The authors will refer to these data as affiliation network data, or measurements on an affiliation variable.
作者將這些數據稱為隸屬關係網路數據，或對隸屬關係變數的測量。

This kind of two-mode network has been called a membership network (Breiger 1974, 1990a) or hypernetwork (McPherson 1982), and the affiliation relation has been referred to as an involvement relation (Freeman and White 1993)
這種雙模網路被稱為隸屬網路（Breiger 1974,1990a）或超網路（McPherson 1982），隸屬關係被稱為參與關係（Freeman and White 1993）

Affiliation Networks
聯盟網路
Affiliation networks differ in several important ways from the types of social networks the authors have discussed so far.
隸屬關係網路在幾個重要方面與作者迄今為止討論的社交網路類型不同。

Affiliation networks are two-mode networks, consisting of a set of actors and a set of events.
隸屬網路是雙模式網路，由一組參與者和一組事件組成。

Affiliation networks describe collections of actors rather than ties between pairs of actors.
隸屬關係網路描述的是參與者的集合，而不是成對參與者之間的聯繫。

Both of these features of affiliation networks make their analysis and interpretation somewhat distinct from the analysis and interpretation of one-mode networks, and lead them to the special set of methods discussed .
隸屬網路的這兩個特徵使它們的分析和解釋與單模網路的分析和解釋有些不同，並將它們引向所討論的一組特殊方法。

. Affiliation networks consist of subsets of actors, rather than pairs of actors;.
.隸屬關係網路由參與者的子集組成，而不是成對的參與者;

Affiliation networks allow one to study the dual perspectives of the actors and the events.
隸屬關係網路允許人們研究行為者和事件的雙重視角。

Many of the methods the authors discuss are concerned with representing affiliation networks using graph theoretic and related ideas, rather than with analyzing these networks.
作者討論的許多方法都涉及使用圖論和相關思想來表示隸屬網路，而不是分析這些網路。

The authors discuss how affiliation networks establish linkages among the entities in each of the modes.
作者討論隸屬關係網路如何在每種模式的實體之間建立聯繫。

The authors examine what the affiliation network implies about the association between the actors and the events, and present two approaches for analyzing the two modes simultaneously
作者研究隸屬關係網路對行為者與事件之間關聯的含義，並提出了兩種同時分析這兩種模式的方法

Background
背景
The authors review some of the more influential theoretical and substantive contributions to the study of affiliation networks.
作者回顧了對隸屬關係網路研究的一些較有影響力的理論和實質性貢獻。

The following list is a small sample: membership on a corporate board of directors (Allen 1982; Bearden and Mintz 1987; Burt 1978/79b; Fennema and Schijf 1978/79; Levine 1972; Mariolis 1975; Mintz and Schwartz 1981a, 1981b; Mizruchi 1984; Mokken and Stokman 1978/79; Sonquist and Koenig 1975), records of the club memberships of a set of community decision makers or elites (Domhoff 1975; Galaskiewicz 1985), memberships in voluntary organizations (McPherson 1982), records of the academic institutions with which researchers have been affiliated (Freeman 1980b), ceremonial events attended by members of a village (Foster and Seidman 1984), committees on which university faculty sit (Atkin 1974, 1976), social events people attend (Breiger 1974; Davis, Gardner, and Gardner 1941; Homans 1950), high school clubs (Bonacich 1978), observations of collections of individuals' interactions (Bernard, KiUworth, and Sailer 1980, 1982; Freeman and Romney 1987; Freeman, Romney, and Freeman 1987; Freeman, Freeman, and Michaelson 1988), trade partners of major oil exporting nations (Breiger 1990b), the overlap of subspecialties within an academic discipline (Cappell and Guterbock 1992; Ennis 1992), and the fate of Chinese political figures (Schweizer 1990)
以下清單是一個小樣本：公司董事會成員（Allen 1982;Bearden 和 Mintz 1987;伯特 1978/79b;Fennema 和 Schijf 1978/79;萊文 1972;馬里奧利斯 1975;Mintz 和 Schwartz 1981a， 1981b;Mizruchi 1984 年;Mokken 和 Stokman 1978/79;Sonquist 和 Koenig 1975），一組社區決策者或精英的俱樂部成員記錄（Domhoff 1975;Galaskiewicz 1985）、志願組織的成員資格（McPherson 1982）、研究人員所屬的學術機構的記錄（Freeman 1980b）、村民參加的儀式活動（Foster and Seidman 1984）、大學教師參加的委員會（Atkin 1974、1976）、人們參加的社會活動（Breiger 1974;大衛斯、加德納和加德納 1941 年;Homans 1950）、高中俱樂部（Bonacich 1978）、對個人互動集合的觀察（Bernard、KiUworth 和 Sailer 1980、1982;弗里曼和羅姆尼 1987;弗里曼、羅姆尼和弗里曼 1987 年;Freeman， Freeman， and Michaelson 1988）、主要石油出口國的貿易夥伴（Breiger 1990b）、學科內亞專業的重疊（Cappell and Guterbock 1992;Ennis 1992）和中國政治人物的命運（Schweizer 1990）

Given this wide range of applications, it is useful to note three primary rationales for studying affiliation networks.
鑒於如此廣泛的應用，有必要注意研究隸屬關係網路的三個主要理由。

McPherson (1982) has used hypergraphs to examine participation in voluntary organizations, and has discussed issues of sampling and estimation. Berge (1973, 1989) presents a mathematical discussion of graphs and hypergraphs
McPherson（1982）使用超圖來檢查志願組織的參與情況，並討論了抽樣和估計問題。Berge（1973,1989）提出了對圖和超圖的數學討論

OSimplices and Simplicial Complexes
OSimplices 和 Simplicial 複合體
Simplices and simplicial complexes provide yet another way to represent affiliation networks using ideas from algebraic topology.
單純複合和單純複數提供了另一種使用代數拓撲思想來表示隸屬網路的方法。

This approach draws heavily on the work by Atkin (1972, 1974), and exploits a more geometric, or topological, interpretation of the relationship between the actors and the events.
這種方法在很大程度上借鑒了阿特金（1972,1974）的工作，並利用了對參與者和事件之間關係的更幾何或拓撲的解釋。

A simplicial complex is useful for studying the overlaps among the subsets and the connectivity of the network, and can be used to define the dimensionality of the network in a precise mathematical way.
簡單複數可用於研究子集之間的重疊和網路的連通性，並可用於以精確的數學方式定義網路的維數。

Simplicial complexes can be used to study the internal structure of the onemode networks implied by the affiliation network by examining the degree of connectivity of entities in one mode, based on connections defined by the second mode.
單純複合體可用於研究隸屬網路所隱含的單模網路的內部結構，方法是基於第二模態定義的連接，檢查一種模態中實體的連通性程度。

The two-mode sociomatrix, the bipartite graph, and the hypergraph are alternative representations of an affiliation network.
雙模社會矩陣、二分圖和超圖是隸屬網路的替代表示。

The sociomatrix is an efficient way to present the information and is most useful for data analytic purposes.
社會矩陣是呈現資訊的有效方式，對於數據分析目的最有用。

Representing the affiliation network as a bipartite graph highlights the connectivity in the network, and makes the indirect chains of connection more apparent.
將隸屬關係網路表示為二分圖可突出顯示網路中的連通性，並使間接連接鏈更加明顯。

Since there is no loss or gain of information in one or another
由於一個或另一個沒有信息丟失或獲得

One-mode
單模式
Networks representation, the researcher's goals should guide selection of the best representation.
網路表示，研究人員的目標應指導選擇最佳表示。

As an example of an affiliation network, the authors will use the data collected by Galaskiewicz on chief executive officers (CEOs) and their memberships in civic clubs and corporate boards.
作為隸屬關係網路的一個例子，作者將使用Galaskiewicz收集的有關首席執行官（CEO）及其在公民俱樂部和公司董事會中的成員資格的數據。

The relationship between the sociomatrix for the comembership relation, Xx, whose entries indicate the number of events jointly attended by each actor, and the affiliation matrix, A, that indicates which events each actor is affiliated with, can be expressed concisely in matrix notation.
共隸關係的社會矩陣 XX 與隸屬矩陣 A 之間的關係可以用矩陣符號簡潔地表示，前者的條目表示每個參與者共同參加的事件的數量，後者表示每個參與者所屬的事件。

The matrix X.k" records the co-membership relation for actors
矩陣 X.k“ 記錄了參與者的共同成員關係

It is a symmetric, valued sociomatrix, indicating the number of events jointly attended by each pair of actors.
它是一個對稱的、有價值的社會矩陣，表示每對參與者共同參加的事件數量。

Each value of x{f is the product of the corresponding columns in A: The authors can define an h x h sociomatrix, X..II = {xt,}, that records the number of actors each pair of events has in common.
x{f 的每個值都是 A 中相應列的乘積： 作者可以定義一個 h x h 社會矩陣 X.。II = {xt，}，記錄每對事件共有的參與者數量。

The matrix X..II is a one-mode, symmetric, valued sociomatrix indicating the number of actors that each pair of events shares.
矩陣 X..II 是一個單模、對稱、有價值的社會矩陣，表示每對事件共享的參與者數量。

The authors illustrate the actor co-membership matrix, Xx, and the event overlap matrix, X..II, using both the hypothetical example of six children and three birthday parties and Galaskiewicz's data on CEOs and their membership in clubs and corporate boards.
作者說明了參與者共隸矩陣 Xx 和事件重疊矩陣 X。II，使用六個孩子和三個生日派對的假設例子，以及Galaskiewicz關於首席執行官及其在俱樂部和公司董事會中的成員資格的數據。

The authors discuss properties of affiliation networks, including properties of the one-mode networks of actors and of events, and of the two-mode affiliation network
作者討論了從屬網路的性質，包括參與者和事件的單模式網路以及雙模式隸屬網路的性質

Properties of Affiliation Networks
隸屬關係網路的屬性
The authors define and describe several properties of affiliation networks and show how these properties can be calculated from the affiliation matrix, A, or from the one-mode sociomatrices, XA' and X..H. The authors first consider properties of individual actors or events and discuss properties of networks of actors and/or of events
作者定義並描述了隸屬關係網路的幾個屬性，並展示了如何從隸屬關係矩陣 A 或單模社會矩陣 XA' 和 X 計算這些屬性。H.作者首先考慮了單個參與者或事件的屬性，並討論了參與者和/或事件網路的屬性

Properties 0/ Actors and Events
屬性 0/ Actor 和事件
Some simple properties of actors and events can be calculated directly from the affiliation matrix or from the one-mode sociomatrices.
參與者和事件的一些簡單屬性可以直接從隸屬矩陣或單模社會矩陣中計算出來。

On average each club has a membership of 6.533 CEOs from this sample
平均而言，每個俱樂部有 6.533 名來自此樣本的 CEO 會員

These measures of the rates of participation for actors or the size of events are appropriate for describing affiliation networks when the authors assume that all actors and events of interest are included in the data set.
當作者假設所有感興趣的參與者和事件都包含在數據集中時，這些對參與者的參與者或事件規模的測量適合於描述隸屬關係網路。

Let them consider the density of ties in the one-mode networks of actor co-memberships and event overlaps.
讓他們考慮一下參與者共同成員和事件重疊的單模式網路中的聯繫密度。

The value of d(.¥) for the co-membership relation can be interpreted as the mean number of events to which pairs of actors belong.
共同成員關係的 d（.¥） 值可以解釋為成對參與者所屬的事件的平均數。

The value of drAt) for the overlap relation can be interpreted as the mean number of actors who belong to each pair of events.
重疊關係的DrAt值可以解釋為屬於每對事件的參與者的平均數。

The authors will use Galaskiewicz's data on CEOs and their memberships in clubs and boards to illustrate the density of ties among actors and among events
作者將使用Galaskiewicz關於首席執行官及其在俱樂部和董事會中的成員資格的數據來說明參與者之間和活動之間的聯繫密度

The authors will use both the valued and dichotomous relations of actor co-memberships and event overlaps.
作者將同時使用演員共同成員和事件重疊的價值關係和二分關係。

The density of this valued relation is Ll(.K) = 1.412
此值關係的密度為 Ll（.K） = 1.412

This means that on average, pairs of CEOs share memberships in 1.412 clubs.
這意味著平均而言，一對CEO在1.412個俱樂部中共用會員資格。

A useful way to study reachability in an affiliation network is to consider the bipartite graph, with both actors and events represented as nodes.
研究隸屬網路中可達性的一個有用方法是考慮二分圖，參與者和事件都表示為節點。

Properties afAffiliation Networks
屬性afAffiliation Networks
4, 14, 15, 17, 20, 14, 15, 20, 23, I, 13, 19 7,14,20 14, 15, 25 14, 17,26 15, 16, 23.
4、14、15、17、20、14、15、20、23、I、13、19、7、14、20、14、15、25、14、17、26、15、16、23。

The authors have used cliques to study the co-membership and overlap relations, one could use other cohesive subgroup ideas, such as n· cliques or k-plexes for valued graphs, to study these relations.
作者使用集團來研究共隸關係和重疊關係，可以使用其他有凝聚力的子集團思想，例如n·取值圖的 cliques 或 k-plexes，以研究這些關係。

Recall that the value of a path in a valued graph as the smallest value of any line included in the path
回想一下，值圖中路徑的值是路徑中包含的任何行的最小值

The authors can use this idea to study cohesive subgroups based on levels of reachability either among actors in the co-membership relation or among events in the overlap relation.
作者可以使用這個想法來研究基於共同成員關係中參與者之間或重疊關係中事件之間的可達性水準的內聚子群。

The authors focus on the one-mode valued relation of co-membership for actors or the one-mode valued relation of overlap for events.
作者重點研究了行為者共同成員的單模價值關係或事件重疊的單模價值關係。

One can use the definitions for the value of a path to define connectedness for pairs of actors in the valued graph.
可以使用路徑值的定義來定義值圖中成對的參與者的連通性。

But using the idea of simplicial complexes, Doreian defines a set of actors connected at level q as a subset such that all pairs of actors in the path were co-members of at least q + 1 events.
但是，使用簡單複合體的概念，Doreian將在水準q上連接的一組參與者定義為一個子集，使得路徑中的所有參與者對都是至少q + 1事件的共同成員。

ComputationalIy, finding pairs of actors who are q-connected is equivalent to finding paths of level q in the valued graph (Doreian 1969).
在計算中，找到 q 連接的 actor 對等價於在值圖中查找 q 級的路徑 （Doreian 1969）。

A q-analysis consists of finding subsets of actors all of whom are connected at level q
q 分析包括查找參與者的子集，所有這些參與者都在 q 水平上連接

Taking Account 0/ Subgroup Size
考慮 0/ 子組大小
An important issue to consider when analyzing the one-mode networks that are derived from an affiliation network is that both the co-membership relation for actors and the overlap relation for events are valued relations based on frequency counts.
在分析從隸屬網路派生的單模網路時要考慮的一個重要問題是，參與者的共隸關係和事件的重疊關係都是基於頻率計數的值關係。

These four children form a maximal complete subgraph in the co-membership relation.
這四個子項在共同成員關係中構成了一個最大完整的子圖。

If the authors return to the affiliation matrix, A, the authors see that Allison and Eliot attended Party 3, Allison and Sarah attended Party 1, and Eliot and Sarah attended Party 2, and Ross was at all parties, Allison, Eliot, Ross, and Sarah were never all four present at any party
如果作者回到隸屬關係矩陣 A，作者會看到 Allison 和 Eliot 參加了第 3 次聚會，Allison 和 Sarah 參加了第 1 次聚會，Eliot 和 Sarah 參加了第 2 次聚會，Ross 參加了所有聚會，Allison、Eliot、Ross 和 Sarah 從未參加過任何聚會

In his application of hypergraphs to social networks, Seidman (1981a) uses the term "pseudo-event" to refer to a subset of actors that form a "clique" in the one-mode comembership relation but are not together in any event in the affiliation network.
Seidman（1981a）在將超圖應用於社交網路時，使用術語“偽事件”來指代在單一模式共成員關係中形成“集團”但在隸屬關係中無論如何都不在一起的參與者子集。

The authors can represent the colIection of subsets of events defined by the actors' memberships, along with the null set, the universal set, and the relation s; as a lattice.
作者可以表示由參與者的成員身份定義的事件子集的集合，以及零集、通用集和關係 s;作為格子。

The advantages of a Galois lattice for representing an affiliation network are the focus on subsets, and the complementary relationships between the actors and the events that are displayed in the diagram.
伽羅瓦格子表示隸屬網路的優點是關注子集，以及參與者與圖中顯示的事件之間的互補關係。

Let them reiterate some of the important features of affiliation networks that make them distinctive from the one-mode networks that the authors have discussed prior to this chapter, and briefly review some of the
讓他們重申隸屬關係網路的一些重要特徵，這些特徵使它們與作者在本章之前討論的單模式網路不同，並簡要回顧一些

Summary important issues to consider when analyzing affiliation networks
總結 分析隸屬關係網路時要考慮的重要問題
Affiliation networks are two-mode networks that focus on the affiliation of a set of actors with a set of events.
隸屬關係網路是雙模式網路，側重於一組參與者與一組事件的隸屬關係。

Since each event consists of a subset of actors, and each actor is affiliated with a subset of events, affiliation network data cannot be studied completely by looking at pairs of actors and/or pairs of events.
由於每個事件都由參與者的子集組成，並且每個參與者都隸屬於事件的子集，因此無法通過查看參與者對和/或事件對來完全研究隸屬網路數據。

Affiliation networks are two-mode networks, and the most comprehensive analyses would study both actors and events simultaneously, it is possible to study the one-mode networks, of actors or of events.
隸屬網路是雙模網路，最全面的分析將同時研究行為者和事件，可以研究行為者或事件的單模式網路。

Since affiliation networks are defined on subsets of actors and events there is loss of information and potential for misinterpretation when studying only the one-mode networks.
由於隸屬網路是在參與者和事件的子集上定義的，因此在僅研究單模網路時，存在資訊丟失和誤解的可能性。

For the most part the analyses that the authors have described assume that one has a complete affiliation network.
在大多數情況下，作者所描述的分析假設一個人有一個完整的隸屬關係網路。

That all actors and all events constituting the network are included.
構成網路的所有參與者和所有事件都包括在內。

If, on the other hand, the actors in % are a sample of actors from a larger popUlation, or if the events in .It are a sample from a larger popUlation of events, one must consider issues of sampling and estimation of the relevant network quantities.
另一方面，如果 % 中的 actor 是來自較大 popUlation 的 actor 樣本，或者如果 .它是從更大的事件群中抽取的樣本，必須考慮相關網路數量的抽樣和估計問題。

McPherson (1982) discusses how to estimate key network affiliation measures
McPherson（1982）討論了如何估計關鍵的網路隸屬關係測量

Part IV
第四部分
Many methods for the description of network structural properties are concerned with the dual notions of social position and social role.
許多描述網路結構特性的方法都涉及社會地位和社會角色的雙重概念。

In social network terms these translate into procedures for analyzing actors' structural similarities and patterns of relations in multirelational networks
在社會網路術語中，這些轉化為分析多重關係網路中行為者的結構相似性和關係模式的程式

These methods, which have been referred to as positional, role, or relational approaches, are the topic of Part IV.
這些方法被稱為位置方法、角色方法或關係方法，是第四部分的主題。

These methods are mathematically and formally diverse, they share a common goal of representing patterns in complex social network data in simplified form to reveal subsets of actors who are embedded in networks of relations and to describe the associations among relations in multirelational networks.
這些方法在數學上和形式上都是多樣化的，它們有一個共同的目標，即以簡化的形式表示複雜社交網路數據中的模式，以揭示嵌入關係網路中的參與者子集，並描述多關係網路中關係之間的關聯。

In Chapters and the authors take up more advanced approaches to the notions of role and position and explore alternative formal definitions of these concepts
在章節中，作者對角色和地位的概念採取了更高級的方法，並探索了這些概念的替代形式定義

These chapters are concerned with the algebraic analysis of role systems using relational algebras (Chapter 11) and more general definitions of equivalence (Chapter 12).
這些章節涉及使用關係代數（第 11 章）和更一般的等價定義（第 12 章）對角色系統進行代數分析。

The use of formal role and positional analysis to study social networks with a wider variety of relations started in the 1970's, with the publication of Lorrain and White's (1971) paper on structural equivalence.
1970年代，隨著Lorrain和White（1971）關於結構等價的論文的發表，開始使用正式角色和位置分析來研究具有更廣泛關係的社會網路。

I and j are structurally equivalent if i ~ k if and only
I 和 j 在結構上是等價的，如果 i ~ k 當且僅

Definition of Structural
結構的定義
For two actors to be structurally equivalent in a multirelational network, they must have identical ties to and from all other actors, on all relations.
要使兩個參與者在多關係網路中結構上等效，它們必須在所有關係上與所有其他參與者具有相同的聯繫。

Consider the valued relation of acquaintanceship in Freeman's EIES network
考慮弗里曼的EIES網路中熟人關係的價值

This quantity is measured as each person's reported friendship with each other member of the group on a five-point scale: 1) "unknown," 2) "person the author has heard of," 3) "person the author has met," 4) "friend," or 5) "close personal friend." In the strictest sense, two actors are structurally equivalent if they name and are named by exactly the same close personal friends, exactly the same friends, had met and been met by exactly the same others, and so on.
這個數量是以每個人報告的與小組其他成員的友誼來衡量的，分為五分制：1）“未知”，2）“作者聽說過的人”，3）“作者見過的人”，4）“朋友”，或5）“親密的私人朋友”。從最嚴格的意義上講，如果兩個演員的名字和名字完全一樣，由完全相同的私人朋友命名，完全相同的朋友，遇到和被完全相同的其他人認識，等等，那麼他們在結構上是等價的。

When a relation is reflexive (i ---+ i for all i) and self-ties are considered substantively meaningful, diagonal entries in the sociomatrix should be included in calculation of structural equivalence.
當關係是自反的（i ---+ i 表示所有 i）並且自聯繫被認為具有實質意義時，社會矩陣中的對角線條目應包含在結構等價的計算中。

The authors present a list of the steps that are required for a complete positional analysis
作者列出了完整位置分析所需的步驟

Simplification of Multirelational Networks
多關係網路的簡化
If all actors within each subset are structurally equivalent, when the rows and columns of the original sociomatrix are permuted so that actors who are assigned to the same equivalence class occupy rows and columns that are adjacent, the submatrices corresponding to the ties between and within positions are filled with either all O's or all l's.
如果每個子集中的所有參與者在結構上都是等價的，那麼當原始社會矩陣的行和列被置換，以便分配給同一等價類的參與者佔據相鄰的行和列時，對應於位置之間和位置內的聯繫的子矩陣將填充所有 O 或所有 l。

This is the definition of a graph homomorphism, which is important in the discussions of blockmodels and relational algebras
這是圖同態的定義，這在塊模型和關係代數的討論中很重要

This rule for constructing a reduced graph includes both a rule for assigning actors to positions and a rule for assigning ties between positions based on the presence or absence of ties between actors.
此用於構建簡化圖的規則既包括將參與者分配給位置的規則，也包括根據參與者之間是否存在聯繫來分配位置之間聯繫的規則。

This example illustrates some of the results of positional analysis methods: a partition of the actors into discrete subsets and a simplified description of the original social network data presenting the ties between positions rather than among individual actors.
這個例子說明瞭位置分析方法的一些結果：將參與者劃分為離散的子集，並簡化了原始社交網路數據的描述，呈現了位置之間的聯繫，而不是單個參與者之間的聯繫。

Structurally equivalent actors have identical entries in their corresponding rows and columns of the sociomatrix.
結構等效的參與者在社會矩陣的相應行和列中具有相同的條目。

Notice that no pairs of actors are structurally equivalent, since none of the off-diagonal distances is equal to O
請注意，沒有一對參與者在結構上是等價的，因為沒有一個對角線距離等於 O

Correlation as a Measure of Structural Equivalence
相關性作為結構等效性的度量
A second widely used measure of structural equivalence is the correlation coefficient.
第二個廣泛使用的結構等價度量是相關係數。

The authors denote the mean of the values in row i of the sociomatrix as Xi., and denote the mean of the values in column i as x.j, where the calculation excludes diagonal elements.
作者將社會矩陣第 i 行中值的平均值表示為 習，並將第 i 列中值的平均值表示為 x.j，其中計算不包括對角線元素。

Where all the sums are over k, and i =1= k, j =1= k
其中所有總和都超過 k，並且 i =1= k，j =1= k

These correlations are arranged in a g x g correlation matrix, which the authors denote by Cl. The (i,j)th element of Cl is the Pearson product-moment correlation coefficient, rij, between the ith row and column and the jth row and column of the sociomatrix.
這些相關性排列在 g x g 相關矩陣中，作者用 Cl 表示。Cl 的 （i，j） 元素是 Pearson 乘積矩相關係數 rij，位於社會矩陣的第 i 行和列與第 j 行和列之間。

Diagonal elements of the sociomatrix are excluded from calculation of the correlation.
社會矩陣的對角線元素被排除在相關性的計算之外。

If two actors are structurally equivalent, the correlation between their respective rows and columns of the sociomatrix will be equal to +1.
如果兩個參與者在結構上是等價的，則它們各自的社會矩陣的行和列之間的相關性將等於 +1。

Since the columns of the original matrix become the rows in its transpose, including the transposes in the calculation allows them to compare ties both to and from the actors.
由於原始矩陣的列成為其轉置中的行，因此在計算中包括轉置允許它們比較與參與者的聯繫和與參與者的聯繫。

This is the same conclusion that the authors reached using Euclidean distance as a measure of structural equivalence
這與作者使用歐幾里得距離作為結構等價度量得出的結論相同

Some Considerations in Measuring Structural Equivalence
測量結構等效性時的一些考慮因素
The authors turn to some considerations in the measurement of structural equivalence. The authors' comments focus on selecting a good measure for a given relation and a comparison of the two measures (Euclidean distance, and Pearson product-moment correlation coefficient).
作者轉向了結構等價性測量中的一些考慮因素。作者的評論側重於為給定關係選擇一個好的度量，並比較兩個度量（歐幾里得距離和皮爾遜積矩相關係數）。

This method was first used for analyzing social networks by H.
這種方法最初被H用於分析社交網路。

One way to display the results of a series of partitions from CONCOR is to construct a tree-diagram or a dendrogram indicating the degree of structural equivalence among the positions and identifying their members.
顯示 CONCOR 一系列分區結果的一種方法是構建樹形圖或樹狀圖，指示位置之間的結構等效程度並識別其成員。

Extensions, Generalizing CONCOR to multirelational networks and to valued relations is straightforward once the authors realize that the primary matrix that CONCOR analyzes is the correlation matrix, Cl, containing the Pearson product-moment correlation coefficients as measures of similarity among pairs of actors, After the first step of computing Cl the procedure of iterating correlations is identical regardless of the types of relations that were included in the calculation of Cl. Some Comments.
擴展，將 CONCOR 推廣到多關係網路和價值關係是很簡單的，一旦作者意識到 CONCOR 分析的主要矩陣是相關矩陣 Cl，其中包含 Pearson 乘積矩相關係數作為參與者對之間相似性的度量，在計算 Cl 的第一步之後，反覆運算相關性的過程是相同的，無論 Cl 的計算中包含何種關係類型。評論。

Another way to study the figure is to compare the proximity of actors in the multidimensional scaling figure with the partition that resulted from CONCOR or from hierarchical clustering of either the Pearson product-moment correlations or of the Euclidean distances
研究該圖的另一種方法是將多維縮放圖中參與者的鄰近性與 CONCOR 或 Pearson 積矩相關性或歐幾里得距離的分層聚類產生的分區進行比較

Recall that these three methods gave the same partition at the level of two positions.
回想一下，這三種方法在兩個位置的級別上給出了相同的分區。

Three arcs, 2 reflexive
三弧，2反身
The advice image is transitive at the level of the positions.
建議圖像在職位級別是可傳遞的。

The pattern for friendship is not as clear.
友誼的模式並不那麼清晰。

Interpreting blockmodels with multiple relations can be tedious.
解釋具有多個關係的塊模型可能很乏味。

One possible way to interpret multirelational blockmodels is to study pairs of image matrices to see whether they exhibit common kinds of multirelational patterns, such as mUltiplexity or exchange.
解釋多重關係塊模型的一種可能方法是研究圖像矩陣對，看看它們是否表現出常見類型的多重關係模式，例如mUltiplexity或交換。

Multiplexity of relations is the tendency for two or more relations to occur together.
關係的多重性是兩個或多個關係同時發生的趨勢。

"is a friend of" and "spends time with" are two relations that might tend to occur together if people are free to choose the people they spend time with.
“是朋友”和“花時間在一起”是兩種關係，如果人們可以自由選擇與他們共度時光的人，它們可能會同時發生。

Multiplexity in a blockmodel would be apparent if two or more image matrices were identical.
如果兩個或多個圖像矩陣相同，則塊模型中的多重性將很明顯。

Exchange occurs when one relation "flows" one way, and the second relation "flows" back.
當一個關係以一種方式「流動」 而第二個關係「流回」 時，就會發生交換。

"pays money to" and "delivers goods to" are two relations that form an exchange in an economic transaction.
“向”和“向”是經濟交易中形成交換的兩種關係。

Transitivity [li~!]m[!!~n
傳遞性 [li~！]m[!!~n
The researcher should be able to lIse the several approaches described to arrive at a consistent and theoretically meaningful statement about the positions in a network, the characteristics of actors in the positions, and how the positions are related to each other.
研究人員應該能夠理解所描述的幾種方法，以得出關於網路中位置、位置中參與者的特徵以及位置如何相互關聯的一致且理論上有意義的陳述。

White (1963) draws the analogy between a formal organization and certain kinship structures, and lays the groundwork for a program of research on formal role analysis
White（1963）將正式組織與某些親屬結構進行了類比，併為正式角色分析的研究計劃奠定了基礎

He notes "Primary roles can be cumulated into chains defining compound roles", pointing toward a focus on associations among relations.
他指出，“主要角色可以累積成定義複合角色的鏈條”，並指出了對關係之間關聯的關注。

Formalizing the ideas of interrelatedness, interlocking, or bundles of relations is one goal of formal network role analysis
將相互關聯性、互鎖性或關係束的概念形式化是正式網路角色分析的目標之一

These methods are different from methods for network positions that focus on properties of subsets of actors.
這些方法不同於關注參與者子集屬性的網路位置方法。

Network methods for social roles focus on relations and on the associations among these relations, rather than on network properties of actors or subsets of actors.
社會角色的網路方法側重於關係和這些關係之間的關聯，而不是行為者或行為者子集的網路屬性。

The the authors present formal network definitions for the notion of social role and show how this concept applies to the different levels of analysis of networks.
作者提出了社會角色概念的正式網路定義，並展示了這一概念如何應用於不同層次的網路分析。

In order to present methods and models for multiple relations, it is useful to employ algebraic notation rather than the sociometric and graph theoretic notations.
為了呈現多重關係的方法和模型，使用代數符號而不是社會計量和圖論符號是有用的。

The authors describe some properties of the operation of composition of relations
作者描述了關係組合操作的一些特性

11.2.2 Properties of Composition and Compound Relations
11.2.2 組成的性質和複合關係
There are several important things to note about the operation of composition of relations.
關於關係組合的運作，有幾件重要的事情需要注意。

Examples of compound relations of long strings that might be socially meaningful include relations such as "a friend of a friend of a friend," or "a boss of a friend of a friend," or "mother's mother's mother." Each string of relations that forms a compound relation is referred to as a word, and the length of a word is the number of primitive relations in it.
可能具有社會意義的長字串複合關係的例子包括諸如「朋友的朋友的朋友」或「朋友的朋友的老闆」或「母親的母親的母親」等關係。形成複合關係的每個關係串都稱為一個詞，一個詞的長度是其中原始關係的數量。

These long compound relations are computed through a series of operations of composition of two relations, for example, (U 0 U) 0 U, or through a series of Boolean matrix products.
這些長複合關係是通過兩個關係的組合的一系列運算來計算的，例如，（U 0 U） 0 U，或者通過一系列布爾矩陣乘積。

The first collection contains all words that could be constructed from a given set of primitive relations, regardless of whether the words give rise to equivalent images.
第一個集合包含所有可以從一組給定的原始關係構造的單詞，而不管這些單詞是否產生等效的圖像。

11.3.1 Multiplication Tables and Relational Structures
11.3.1 乘法表和關係結構
One way to display the result of composition of relations for a given network is in a multiplication table.
顯示給定網路關係組合結果的一種方法是在乘法表中。

Composition graph table for a hypothetical network the relation "oversees the work of" in a three-level corporate hierarchy, or as the relation "is the parent of" in a three-generation family tree.
假設網路的組成圖表，該關係在三級公司層次結構中“監督”其工作，或作為三代家譜中“是其父級”的關係。

In this example there are Rs = 5 distinct images: the primitive relations Hand L, and three additional distinct compound relations H L, H H, and 0.
在此示例中，有 Rs = 5 個不同的圖像：原始關係 Hand L，以及另外三個不同的複合關係 H L、H H 和 0。

This compound relation could be interpreted as "parent of a parent" or "boss of a boss." The entry in the second row and second column of the body of the table shows the compollnd relation LL
這種複合關係可以解釋為「父母的父母」或「老闆的老闆」。表正文的第二行和第二列中的條目顯示了 compollnd 關係 LL

Notice that this is equivalent to the relation L. and so adds no new image to the table, or new entry to 1/.
請注意，這等價於關係 L.，因此不會向表中添加新圖像，也不會向 1/ 添加新條目。

The information in the multiplication table describes the set of equivalences among the relations; that is, the results of composition tell them which primitive or compound relations produce identical images.
乘法表中的資訊描述了關係之間的等價集;也就是說，構圖的結果告訴他們哪些原始關係或複合關係產生相同的圖像。

A F AF FA FF
FA FF 的 A F
Let them look at the role structure generated by the image matrices for the relations of advice and friendship for Krackhardt's high-tech managers.
讓他們看看圖像矩陣為克拉克哈特的高科技經理提供建議和友誼關係所產生的角色結構。

This multiplication table was adapted from an analysis using UC/NET IV (Borgatti, Everett, and Freeman 1991).
該乘法表改編自使用UC/NET IV（Borgatti， Everett， and Freeman 1991） 的分析。

These equivalence classes are given, along with some of the words that generate the same image
給出了這些等價類，以及一些生成相同圖像的單詞

This set of equations expresses the fact that the role structure describes a partition of the set of all possible words that could be constructed from the primitive relations.
這組方程表達了這樣一個事實，即角色結構描述了可以從原始關係構造的所有可能單詞的集合的分區。

This shows that there are two subsets of images that operate when they are the first element in a compound relation.
這表明，當圖像是複合關係中的第一個元素時，有兩個圖像子集可以運行。

The second strategy is based on simplifying the multiplication table that expresses the composition of relations
第二種策略基於簡化表示關係組成的乘法表

11.4.1 Simplification by Comparing Images
11.4.1 通過比較圖像進行簡化
One goal of simplification and reduction of a multiplication table is to add further equations among pairs or collections of images to reduce the total number of distinct elements in the table.
簡化和減少乘法表的一個目標是在圖像對或圖像集合之間添加進一步的方程，以減少表中不同元素的總數。

For the example of advice and friendship, if the authors focus on similarities among columns in the table, the authors are led to a different simplification than if the authors focus on rows
以建議和友誼為例，如果作者關注表中各列之間的相似性，則作者會被引導到與作者關注行不同的簡化

This second set of equations, which is a homomorphic reduction, is: The authors present the permuted and partitioned multiplication table in Fig-.
第二組方程是同態約簡，是： 作者在圖-中給出了置換和分區的乘法表。

Pattison (1993) has developed a complementary approach to homomorphic reduction of relational algebras that preserves the property of inclusions among images in the network
Pattison（1993）開發了一種互補方法，用於關係代數的同態約簡，該方法保留了網路中圖像之間的包含屬性

Both of these approaches have the goal of representing the essential features that are shared between role structures.
這兩種方法的目標都是表示角色結構之間共用的基本特徵。

Consider imposing the equations among images that hold for Krackhardfs high-tech managers on the role table for the Bank Wiring room
考慮將 Krackhardfs 高科技經理的形象強加於銀行佈線室的角色表上

The authors will denote these new classes by Q{NT, Q~NT, ...
作者將用 Q{NT， Q~NT， ...

Examining this figure, the authors see that within each submatrix, the labels for images are all in the same equivalence class, and this simplification preserves the operation of composition
通過檢查該圖，作者發現在每個子矩陣中，圖像的標籤都處於相同的等價類中，並且這種簡化保留了構圖的操作

This reduction of the role structure for the Bank Wiring room is a homomorphic reduction.
組布線室角色結構的這種簡化是同態簡化。

If the authors consider the equations among compound relations that are expressed in the set of equivalence classes of images for the joint homomorphic reduction of these two role structures, the authors see that there are five kinds of compound relations operating in these groups.
如果作者考慮在圖像的等價類集合中表示的複合關係之間的方程，用於這兩個角色結構的聯合同態還原，作者會發現在這些組中有五種複合關係在起作用。

Let them consider some issues that arise in defining regular equivalence for nondirectional relations
讓他們考慮在定義非方向關係的正則等價時出現的一些問題

12.4.2 Regular Equivalence for Nondirectional Relations
12.4.2 非方向關係的正則等價
As many authors have noted, in a graph in which there are no isolates, the maximal regular equivalence consists of a single equivalence class containing all nodes (Faust 1985; Doreian 1987, 1988b; Borgatti 1988).
正如許多作者所指出的，在沒有分離物的圖中，最大正則等價由包含所有節點的單個等價類組成（Faust 1985;Doreian 1987， 1988b;Borgatti 1988）。

For a nondirectional relation with no isolates, all actors in the single maximal regular equivalence class are adjacent to some other actor, who is in the equivalence class.
對於沒有隔離的非方向關係，單個最大正則等價類中的所有參與者都與等價類中的其他參與者相鄰。

A partition consisting of a single equivalence class is trivial, and probably uninteresting.
由單個等價類組成的分區是微不足道的，而且可能無趣。

A nondirectional relation may contain other regular equivalence partitions.
非方向關係可以包含其他正則等價分區。

The maximal regular equivalence partition for this graph is {1, 2,3, 4}.
此圖的最大正則等價分區為 {1， 2,3， 4}。

One useful approach for studying regular equivalence in graphs is the graph theoretic concept of neighborhood (Everett, Boyd, and Borgatti 1990).
研究圖中正則等價的一種有用方法是鄰域的圖論概念（Everett， Boyd， and Borgatti 1990）。

Since the neighborhood of a node consists of all nodes adjacent to that node, nodes that are regularly equivalent must have the same equivalence classes of nodes in their neighborhoods across all relations.
由於節點的鄰域由與該節點相鄰的所有節點組成，因此，在所有關係中，常規等效節點的鄰域中必須具有相同的等價節點類。

In order to be regularly equivalent, actors must be adjacent to the same kinds of other actors
為了在規則上等價，參與者必須與相同類型的其他參與者相鄰

This approach to defining regular equivalence is especially useful for studying regular equivalence in nondirectional relations.
這種定義正則等價的方法對於研究非方向關係中的正則等價特別有用。

Before the authors discuss measures of regular equivalence, let them consider how to represent regular equivalence partitions using a regular equivalence blockmodel
在作者討論正則等價的度量之前，讓他們考慮一下如何使用正則等價塊模型來表示正則等價分區

12.4.3 Regular Equivalence Blockmodels
12.4.3 正則等價塊模型
Recall that a blockmodel consists of a mapping of actors into equivalence classes according to the particular equivalence definition, and for each pair of positions, a statement of whether or not there is a tie present from one position to another position.
回想一下，塊模型由根據特定的等價定義將參與者映射到等價類中組成，並且對於每對位置，都包含從一個位置到另一個位置是否存在平局的陳述。

Let them consider two definitions of equivalence that focus on the types of ties in which each actor is involved
讓他們考慮兩個等價的定義，這兩個定義側重於每個行為者所參與的聯繫類型

These two approaches, Winship and Mandel's local role equivalence and Breiger and Pattison's ego algebras, consider associations among relations from the perspectives of individual actors
這兩種方法，Winship和Mandel的局部角色等價以及Breiger和Pattison的自我代數，從個體行為者的角度考慮了關係之間的關聯

The authors will consider the associations among relations from the perspectives of individual actors
作者將從個體行為者的角度考慮關係之間的關聯

To describe these approaches it will be useful to return to Merton's (1957) ideas of role relation and role set, which the authors discussed at the beginning of this chapter.
為了描述這些方法，回到Merton（1957）的角色關係和角色集思想將是有用的，作者在本章開頭討論了這一點。

For this example the authors will consider the set of all distinct primitive and compound relations.
在這個例子中，作者將考慮所有不同的原始關係和複合關係的集合。

In the two sections the authors present two different definitions and measures of equivalence for individual roles
在這兩節中，作者提出了兩種不同的定義和個別角色的等效性衡量標準

These two methods, local role equivalence (Winship and Mandel 1983, and Mandel 1983) and ego algebras (Breiger and Pattison 1986) focus on sets of primitive and compound relations, but they differ in terms of which relations are included in the set, how individual roles are defined, and how similarity of individual roles is calculated.
這兩種方法，局部角色等價（Winship and Mandel 1983，Mandel 1983）和自我代數（Breiger和Pattison 1986）側重於原始關係和複合關係的集合，但它們在集合中包含哪些關係，如何定義單個角色以及如何計算單個角色的相似性方面有所不同。

Notice that actors 2 and 3 have identical equations among relations and identical right multiplication tables, as do actors 4, 5, and 6
請注意，參與者 2 和 3 在關係之間具有相同的方程和相同的右乘法表，參與者 4、5 和 6 也是如此

H r-H1 2-1L
Two actors have identical ego algebras, and are ego-algebraically equivalent (EA), if the equivalences among relations and the composition of relations are the same from each actor's perspective.
兩個參與者具有相同的自我代數，並且從每個參與者的角度來看，如果關係之間的等價和關係的組成相同，則它們是自我代數等價的 （EA）。

Actors i and j are equivalent, i ~ j, if Y j , the partition of Y for actor i, is identical to !/j, the partition of [/ for actor j, and their right multiplication tables are identical.
Actor i 和 j 是等價的，i ~ j，如果 Y j 是 actor i 的 Y 分區，則與 Actor j 的 [/ 的分區 ！/j 相同，它們的右乘法表相同。

The authors turn to measuring the similarity of ego algebras
作者轉向測量自我代數的相似性

11.7.3 Measming Ego Algebra Similarity
11.7.3 測量自我代數相似性
To measure the similarity of ego algebras the authors use the same approach that the authors used to compare the role algebras for two groups.
為了測量自我代數的相似性，作者使用了與作者用來比較兩組角色代數相同的方法。

Breiger and Pattison (1986) compare ego algebras by the joint right homomorphism of two ego algebras.
Breiger和Pattison（1986）通過兩個自我代數的聯合右同態來比較自我代數。

The authors will denote the joint right homomorphic reduction of the ego algebras for actors i and j by !lfJ-IT.
作者將用 ！lfJ-IT 表示參與者 i 和 j 的自我代數的聯合右同態約簡。

The joint right homomorphic reduction is a coarser partition of the set f/, since it equates relations that are identical from the perspective of either individual actor.
聯合右同態約簡是集合 f/ 的較粗略的劃分，因為它等價從任何一個個體參與者的角度來看相同的關係。

A measure of the degree of equivalence of two ego algebras is a measure of how much "coarser" the partition described by their joint right homomorphic reduction is, compared to the partitions of the two ego algebras.
兩個自我代數的等價程度的度量是衡量它們的聯合右同態約簡所描述的分區與兩個自我代數的分區相比有多“粗糙”。

The authors can measure the distance between two ego algebras by slimming the distance each is from their joint right homomorphic reduction.
作者可以通過縮小兩個自我代數與它們的關節右同態約簡的距離來測量兩個自我代數之間的距離。

In both examples the authors used the routine JNTHOM in the program ROLE (Breiger 1986).
在這兩個例子中，作者在程式ROLE（Breiger，1986）中使用了例程JNTHOM。

As Wc noted above, for this example there are three subsets of actors who arc ego-algebraically equivalent (EA)
如上所述，在這個例子中，有三個參與者的子集，它們在自我代數等價 （EA） 上是弧形的

These subsets are: consider the relations of advice and friendship for Krackhardl's high-tech managers.
這些子集是：考慮 Krackhardl 高科技經理的建議和友誼關係。

There are fewer examples of applications of these methods to substantive problems
將這些方法應用於實質性問題的例子較少

Part V
第五部分
The first question that an analyst must answer is: "What is the stochastic nature of the random variables?" In other words: "What distribution do the author's random variables follow?" These distributions allow a researcher to test hypotheses about various properties of a directed graph under study, such as the number of mutual dyads.
分析師必須回答的第一個問題是：「隨機變數的隨機性質是什麼？換句話說：「作者的隨機變數遵循什麼分佈？這些分佈允許研究人員測試有關所研究的有向圖的各種屬性的假設，例如相互二元組的數量。

These properties will be described at length .
這些屬性將詳細描述。

A statistical dyadic analysis is only possible if the authors allow the counts of the dyad census to be random variables; that is, if the authors consider the sociomatrix under study to represent a random directed graph.
只有當作者允許二元普查的計數是隨機變數時，才有可能進行統計二元分析;也就是說，如果作者認為所研究的社會矩陣代表隨機有向圖。

Katz and Powell (1955) proposed an index, which the authors will label PKP, to measure the tendency for actors in a group to reciprocate choices more frequently than would occur by chance
Katz和Powell（1955）提出了一個指數，作者將其命名為PKP，以衡量群體中行為者比偶然發生更頻繁地回報選擇的傾向

Such an index refines the examination of the counts in the dyad census, since the index can be used to compare groups and relations with unequal numbers of actors.
這種指數細化了對二元人口普查中計數的檢查，因為該指數可用於比較具有不等數量行為者的群體和關係。

The simplest conditional uniform random digraph distribution conditions on the graph property
圖屬性上最簡單的條件均勻隨機二圖分佈條件

Such a distribution is useful when studying the randomness of choices made by each individual actor
在研究每個參與者所做選擇的隨機性時，這種分佈是有用的
One could assume that the sociomatrix was distributed as a uniform random matrix, conditional on the outdegrees and number of mutuals
人們可以假設社會矩陣分佈為一個均勻的隨機矩陣，條件是相互的出度和數量

This allows all statistical inferences to be made only among those sociomatrices with the same outdegrees and M value as observed in the data set.
這允許所有統計推斷僅在那些與數據集中觀察到的具有相同度數和 M 值的社會矩陣中進行。

Each of these random digraphs is likely under the uniform distribution, conditional on a fixed set of indegrees
這些隨機二分法中的每一個都可能處於均勻分佈之下，以一組固定的內度為條件

This number is the size of the sample space of digraphs under this distribution.
這個數位是這個分佈下二合體的樣本空間的大小。

The distribution discussed in these paragraphs is one of the most frequently used random directed graph distributions in social network analysis
這些段落中討論的分佈是社交網路分析中最常用的隨機有向圖分佈之一

It is a uniform distribution which conditions on the numbers of mutual, asymmetric, and null dyads in the digraph - that is, the dyad census itself.
它是一種均勻分佈，它以二元圖中互對、不對稱和零二元對稱的數量為條件，即二元普查本身。

It is quite useful in social network analysis, when working with a specific set of graph properties to fix or condition on all "lower-level" graph properties
在社交網路分析中，當使用一組特定的圖形屬性來修復或調節所有“較低級別”的圖形屬性時，它非常有用

This is exactly the approach that the authors take, where the UIMAN distribution will be used extensively when studying triads - that is, the authors fix the dyad census to study triadic frequencies.
這正是作者所採用的方法，在研究三元組時，UIMAN分佈將被廣泛使用——也就是說，作者將二元組普查固定為研究三元組頻率。

Such hypotheses are usually tested by examining not the entire triad census and its expectation, but linear combinations of it, which the authors discuss
這種假設通常不是通過檢查整個三合會人口普查及其期望來檢驗的，而是通過檢查作者討論的線性組合來檢驗的

14.3.4 Mean and Variance of Linear Combinations of a Triad Census
14.3.4 三元組普查線性組合的均值和方差
As the authors have mentioned and demonstrated, linear combinations of the triad census, defined as Lu tu Tu where the lu are the coefficients of the.
正如作者所提及和證明的，三元組人口普查的線性組合，定義為 Lu tu Tu，其中 lu 是 的係數。

The first step in the testing process is to consider how these hypotheses can be "operationalized" in terms of triads; that is, what predictions these theories make about the various triadic configurations that occur in a data set.
測試過程的第一步是考慮如何根據三元組「操作化」這些假設;也就是說，這些理論對數據集中出現的各種三元配置做出了哪些預測。

By "chance," the authors mean the expected numbers of these configurations that would arise as given by a random directed graph distribution, assuming that the hypothesis is true
作者所說的“偶然”是指這些配置的預期數量，這些配置由隨機有向圖分佈給出，假設假設為真

Note that this comparison strategy is identical to the standard approach to significance testing in statistics: let the data give the empirical frequencies or value of the relevant statistic, and compare the empirical value(s) with the value(s) to be expected based on some null model.
請注意，這種比較策略與統計學中顯著性檢驗的標準方法相同：讓數據給出相關統計量的經驗頻率或值，並將經驗值與基於某些零模型的預期值進行比較。

Under one of the random directed graph distributions, the authors can calculate the expected value and covariance matrix of T, and the expected number for this configuration (equation (14.22) and its variance (equation (14.23))
在其中一個隨機有向圖分佈下，作者可以計算 T 的期望值和協方差矩陣，以及該配置的期望數（方程 （14.22） 及其方差（方程 （14.23））

This expected number is l'PT, and the standard error is -/1'''£Ti, where PT, the mean triad census vector, is given by the components of equation (14.19).
這個預期數位是 l'PT，標準誤差是 -/1'''£Ti，其中 PT，平均三元組普查向量，由方程 （14.19） 的分量給出。

The configurations associated with the theory yield a set of weighting vectors, to be applied to the counts of the triad census, since the triad types contain the various predicted configurations.
與該理論相關的配置產生一組加權向量，用於三元組人口普查的計數，因為三元組類型包含各種預測配置。

A complete social network analysis begins by using methods from Parts Ill, IV, and V of this book
完整的社交網路分析從使用本書第 Ill、IV 和 V 部分的方法開始

Part VI
第六部分
Statistical Analysis of Single Relational Networks by Dawn Iacobucci. The authors turn the attention to stochastic models for social network data.
Dawn Iacobucci 對單一關係網路的統計分析。作者將注意力轉向社交網路數據的隨機模型。

Parameters that quantify the "structural effects" present in a network, such as reciprocity and tendencies toward differential indegrees, can be estimated simultaneously; for example, the authors can model actor expansiveness while controlling for differential actor popularity.
可以同時估計量化網路中存在的「結構效應」的參數，例如互惠性和差異度的趨勢;例如，作者可以對演員的擴張性進行建模，同時控制不同的演員受歡迎程度。

The authors begin this chapter by presenting models for a network with measurements on a single, directional relation for one set of actors.
作者在本章的開頭介紹了一個網路模型，該模型對一組參與者的單個方向關係進行了測量。

The model the authors present for a single relation includes parameters to measure the probabilistic tendencies of all of these substantive effects: expansiveness, popularity, and reciprocity.
作者為單一關係提出的模型包括用於衡量所有這些實質性影響的概率趨勢的參數：擴展性、受歡迎程度和互惠性。

The authors estimate these parameters using log linear modeling techniques.
作者使用對數線性建模技術估計這些參數。

Referring to network data, if the authors include P's in the model, and if actor 1 is chosen by four of the eight actors, fitted probabilities that nl is chosen must sum to (4/8) = 0.50
參考網路數據，如果作者在模型中包括 P，並且 8 個參與者中的 4 個選擇了參與者 1，則選擇 nl 的擬合概率之和必須為 （4/8） = 0.50

This equating is the critical computation that will produce maximum likelihood estimates of cell expected values, and parameters.
這種等價是關鍵計算，它將產生單元格期望值和參數的最大似然估計值。

The authors' models for single relational network data Can be fit by following the theory discussed above; one focuses on the model parameters, and their sufficient statistics, which are margins of the Y-array.
作者的單關係網路數據模型可以通過遵循上述理論進行擬合;一個關注模型參數及其充分的統計數據，即 Y 陣列的邊緣。

More manageable sizes (2 x 2 x 2 x 2 if the authors use one attribute variable or 4 x 4 x 2 x 2 if the authors use two attribute variables) than their corresponding y-arrays (21 x 21 x 2 x 2 for both of the relations)
比其相應的 y 陣列（兩個關係均為 21 x 21 x 2 x 2）更易於管理的大小（如果作者使用一個屬性變數，則為 2 x 2 x 2 x 2 x 2，如果作者使用兩個屬性變數，則為 4 x 4 x 2 x 2）

15.2.3 The Basic Model with Auribute Variables
15.2.3 具有 Auribute 變數的基本模型
The model the authors fit to this new contingency table (the W-array defined in equation (15.8)) is a special case of the basic model (15.3), subject to the constraints placed on the parameters, which arise through the use of the actor attribute variables and the assumption of stochastic equivalence.
作者擬合到這個新的列聯表的模型（等式（15.8）中定義的W陣列）是基本模型（15.3）的一個特例，受制於對參數的約束，這些約束是通過使用參與者屬性變數和隨機等價假設產生的。

This version of model (15.3) follows: 10gP(Yijk/ = 1) =.
此版本的模型 （15.3） 如下：10gP（Yijk/ = 1） =。

If the authors aggregate the actors into four age and tenure subsets (S = 4), and model the resulting W-array, the authors estimate only (4 - 1)(2 - 1) = 3 a's and 3 p's, 1, and 1 fJ, for a total of just 8 parameters
如果作者將參與者聚合為四個年齡和任期子集 （S = 4），並對生成的 W 陣列進行建模，則作者僅估計 （4 - 1）（2 - 1） = 3 個 a 和 3 個 p、1 和 1 fJ，總共只有 8 個參數

This simplification is due to the stochastic equivalence assumption.
這種簡化是由於隨機等價假設。

The adjustment that is needed takes the parameter estimates for a model with attribute variables (which are based on S(S -1)/2 pairs of subsets), and calculates fitted values for all g(g - 1)/2 pairs of actors, compares these fitted values to the relational data contained in the original sociomatrix.
所需的調整採用具有屬性變數的模型的參數估計值（基於 S（S -1）/2 對子集），並計算所有 g（g - 1）/2 對參與者的擬合值，將這些擬合值與原始社會矩陣中包含的關係數據進行比較。

Not small 1 Not small
不小 1 不小
The younger and older actors have different numbers of friends
年輕和年長的演員有不同數量的朋友

Their friendship expansiveness differs, as can be seen from the parameter estimates from model (15.9), fit to the friendship relation, using just age as the actor attribute variable.
從模型（15.9）的參數估計值可以看出，他們的友誼擴展性是不同的，擬合到友誼關係，僅使用年齡作為演員屬性變數。

The models that the authors have presented allow researchers to study patterns of ties for a single relational variable among individual actors, or among subsets.
作者提出的模型允許研究人員研究單個參與者之間或子集之間單個關係變數的聯繫模式。

Models postulated for such data focus on the relations alone, without consideration of who sent or received the ties, or of any attributes of these actors
為此類數據假設的模型僅關注關係，而不考慮誰發送或接收了關係，或這些參與者的任何屬性

These models assume that all expansiveness parameters are constant across all actors or all subsets of actors.
這些模型假定所有參與者或所有參與者子集的所有擴展性參數都是恆定的。

When considering whether to go from the W- to the V-array, the authors are further assuming all subsets, or all actors, are homogeneous with respect to the dyadic interactions on the relational variable under study.
在考慮是否從 W 陣列轉到 V 陣列時，作者進一步假設所有子集或所有參與者在所研究的關係變數上的二元相互作用方面都是同質的。

No parameters appear in the model that depend on the actors, i or j, or the subsets, s(i) or s(j), because the authors have aggregated over all these possibilities, and formed a table that cross-classifies only the levels of the relational variables.
模型中沒有出現依賴於參與者 i 或 j 或子集 s（i） 或 s（j） 的參數，因為作者已經聚合了所有這些可能性，並形成了一個僅對關係變數級別進行交叉分類的表。

15.4 ONondirectionai Relations
15.4 ONondirectionai 關係
Y-array, designed to reflect the dyadic states that are possible with a nondirectional relation: Yijk.
Y 陣列，旨在反映非方向關係中可能的二元狀態：Yijk。

To study whether the families differ with respect to business or marriage, the authors fit the special case of the basic model without the ')The author parameters.
為了研究家庭在商業或婚姻方面是否不同，作者擬合了基本模型的特殊情況，沒有“）作者參數。

Comparing the basic model to the model without these actor-level parameters, the authors found that the ')The author effects are large for both marital ties (LlG2 = 108.13 - 87.97 = 20.16).
將基本模型與沒有這些行為者水平參數的模型進行比較，作者發現，對於兩種婚姻關係來說，“）作者效應都很大（LlG2 = 108.13 - 87.97 = 20.16）。

The authors can conclude that the families are different with respect to the volume and patterns of their marital and business ties to others.
作者可以得出結論，這些家庭在與他人的婚姻和商業聯繫的數量和模式方面是不同的。

Five families have no business ties with the others, and have -00 parameter estimates.
五個家庭與其他家庭沒有業務聯繫，參數估計值為 -00。

The authors modeled these relations using wealth as an attribute of each actor.
作者使用財富作為每個參與者的屬性來類比這些關係。

For the analyses of the w's, the )'i's for marriage are not large (LlG2 = 1.21 with Lldf = 1) but they are for business (LlG2 = 7.59 again with Lldf = 1)
對於w的分析，婚姻的）'i並不大（LlG2 = 1.21，Lldf = 1），但它們是商業的（LlG2 = 7.59，Lldf = 1）

These results suggest that wealth· is quite important in distinguishing families who have business ties, but not for marital arrangements.
這些結果表明，財富·對於區分有商業關係的家庭非常重要，但對於婚姻安排則不然。

Wealthy families enter into business relationships at different rates than less wealthy families, but wealth is not an important influence on marital ties.
富裕家庭與不富裕家庭建立商業關係的速度不同，但財富對婚姻關係的影響並不大。

These include Bayesian estimation of PI parameters as described in Wong (1987), and the pseudo-likelihood estimation described in Strauss and Ikeda (1988) designed for the Markov random graphs of Frank and Strauss (1986)
其中包括Wong（1987）描述的PI參數的貝葉斯估計，以及Strauss和Ikeda（1988）為Frank和Strauss（1986）的瑪律可夫隨機圖設計的偽似然估計

15.5 Q9Recent Generalizations of Pi
15.5 Q9圓周率的最新概括
The Bayesian ideas offered by Wong (1987) allow a priori information about the
Wong （1987） 提供的貝葉斯思想允許關於

Prior information might change or even improve parameter estimates in other network data sets
先前的資訊可能會改變甚至改進其他網路數據集中的參數估計

Another development to note is the work reported in Strauss and Ikeda (1990).
另一個值得注意的發展是施特勞斯和池田（1990）報告的工作。

Strauss and Ikeda compared the performance of standard maximum likelihood estimates to their maximum pseudo-likelihood (MP) estimates both in a simulation study, and by analyzing the "like" relation measured on the monks in the monastery studied by Sampson (1967)
Strauss 和 Ikeda 在類比研究中比較了標準最大似然估計值與最大偽似然 （MP） 估計值的性能，並通過分析 Sampson （1967） 研究的寺院僧侶的“相似”關係

In the simulations, they looked at the performance of the estimates in five replicated networks containing fifteen, twenty, or thirty actors.
在類比中，他們研究了包含15個、20個或30個參與者的五個複製網路中的估計性能。

15.6 ®Single Relations and Two Sets of Actors various log-linear models
15.6 ® 單關係和兩組參與者各種對數線性模型

In this new situation, the authors create not a fourdimensional array, but a three-dimensional contingency table of size gxhxC, defined as follows: 1 if the ordered pair < ni, mj > takes on the value Xij = k o otherwise.
在這種新情況下，作者創建的不是四維數位列，而是大小為 gxhxC 的三維列聯表，定義如下：1 如果有序對< ni，則 mj > 取值 Xij = k o，否則。

Aggregating over all actors gives a very simple one-dimensional V-array: j describing only the relational data, not distinguishing among the sending or receiving actors
聚合所有參與者會得到一個非常簡單的一維 V 陣列：j 僅描述關係數據，不區分發送或接收參與者

Models fit to this array would contain only A and () parameters.
擬合到此陣列的模型將僅包含 A 和 （） 參數。

To fit models to a y-array, use the following program file: TITLE 'MY ANALYSIS'
要將模型擬合到 y 陣列，請使用以下程式檔： 標題“我的分析”

BY L K BY L
To fit models to a w-array, use these commands: TITLE 'MY WANALYSIS'. The commands "TITLE," "FILE HANDLE," and "DATA LIST" initiate SPSSx and read in the data file.
要將模型擬合到 w 陣列，請使用以下命令：TITLE 'MY WANALYSIS'。命令「TITLE」、“FILE HANDLE”和“DATA LIST”啟動 SPSSx 並讀取數據檔。

The standard mathematical representation of a positional analysis frequently uses blockmodels to describe and study the equivalence classes determined by a set of measured relations.
位置分析的標準數學表示經常使用塊模型來描述和研究由一組測量關係確定的等價類。

It is not proper to use relational data to find a blockmode representation, and test this same representation on that data set.) If statistical tests are desired in a network analysis, the authors recommend the use of statistical methods from the beginning of the analysis
使用關係數據查找塊模式表示，並在該數據集上測試相同的表示形式是不合適的。如果在網路分析中需要統計檢驗，作者建議從分析開始就使用統計方法

Such methods, as described can be used to find partitions of actors, and lead to proper statistical tests and measures of goodness-of-fit.
如上所述，這些方法可用於查找參與者的分區，並導致適當的統計檢驗和擬合優度的測量。

As described can be used to find partitions of actors, and lead to proper statistical tests and measures of goodness-of-fit
如上所述，可用於查找參與者的分區，並導致適當的統計檢驗和擬合優度的測量

Another approach centers on the evaluation of a particular positional analysis technique using standard data sets.
另一種方法是使用標準數據集評估特定位置分析技術。

The second approach, as mentioned, is based on statistical theory for social network data
如前所述，第二種方法基於社交網路數據的統計理論

This idea uses a statistical or stochastic blockmodel to represent mathematically the equivalence classes defined on the actors.
這個想法使用統計或隨機塊模型在數學上表示在參與者上定義的等價類。

The authors note that one can generate a permutation distribution for this index by considering all possible permutations of the actors to positions, and calculating R2 for each permutation
作者指出，通過考慮參與者對位置的所有可能排列，並計算每個排列的R2，可以為該指數生成排列分佈

This leads to a valid, nonparametric statistical test for the goodness of an observed fit.
這導致了對觀察到的擬合優度的有效非參數統計檢驗。

There is no parametric statistical theory for this index
該指數沒有參數統計理論

16.1.2 Structurally Based Blockmodels and Permutation Tests
16.1.2 基於結構的塊模型和排列測試
It should not be surprising to find that ties predicted by a blockmodel are extremely similar to the observed ties.
發現塊模型預測的領帶與觀察到的領帶極為相似也就不足為奇了。

The authors turn to a discussion of goodness-of-fit indices which are based on specific, parametric statistical models
作者轉而討論基於特定參數統計模型的擬合優度指數

This is a rather different approach to assessing the fit of a network data set to a particular partition of actors to positions.
這是一種完全不同的方法，用於評估網路數據集與特定參與者位置分區的擬合度。

This statistical approach, based on the statistical models described, has associated with it a natural goodness-of-fit index that follows directly from the models under consideration.
這種統計方法基於所描述的統計模型，與它相關的是直接來自所考慮的模型的自然擬合優度指數。

Attribute variables can greatly reduce the number of parameters in a model through the modeling of a W-, rather than a V-array
屬性變數可以通過對 W 陣列而不是 V 陣列進行建模來大大減少模型中的參數數量

These stochastic blockmodels will be discussed for valued and muItirelational data sets, and will be illustrated by using the countries trade network.
這些隨機塊模型將針對價值數據集和多關係數據集進行討論，並將使用國家貿易網路進行說明。

These posterior partitions are more difficult to find and evaluate statistically, but are highly desirable because of their similarity to relational analysis, which is based on posterior partitions
這些後部分區更難在統計上找到和評估，但由於它們與基於後部分區的關係分析相似，因此非常可取

These a posteriori stochastic blockmodels are very similar to the positional analyses of Chapters 9 and 10, since they use the relational data to obtain the positions; this "data dredging" does not allow for proper, significance tests of the fit of actors to the derived positions.
這些後驗隨機塊模型與第 9 章和第 10 章的位置分析非常相似，因為它們使用關係數據來獲取位置;這種「數據疏通」不允許對參與者與派生位置的擬合度進行適當的、顯著性測試。

Of primary interest to them are the likelihood-ratio statistics, which, as the authors discuss shortly, can be used to evaluate the goodness-of-fit of a stochastic blockmodel
他們最感興趣的是似然比統計，正如作者稍後討論的那樣，它可以用來評估隨機塊模型的擬合優度

16.2.4 Goodness-oJ-Fit Indices for Stochastic Blockmodels
16.2.4 隨機塊模型的優度-oJ-擬合指數
As discussed earlier there is a large literature on indices designed to measure how well a blockmodel fits a given network data set
如前所述，有大量關於索引的文獻，旨在衡量塊模型與給定網路數據集的擬合程度

Most of these measures are lacking because they are not based on statistical models, and they do not have convenient and well-known distributions.
這些度量大多是缺乏的，因為它們不是基於統計模型的，而且它們沒有方便和眾所周知的分佈。

One solution to this problem, discussed by Wasserman and Anderson (1987), begins with the assumption that one has a stochastic blockmodel, consisting of a p(x) and a mapping of actors to B positions.
Wasserman 和 Anderson （1987） 討論過這個問題的一個解決方案，首先假設有一個隨機塊模型，由 p（x） 和參與者到 B 位置的映射組成。

If one desires to compare two stochastic blockmodels with differing number of positions (and one of the p(x)'s is not a special case of the other), one can compare G2,s normalized by their degrees of freedom: G21dj
如果想要比較兩個位置數不同的隨機塊模型（其中一個 p（x） 不是另一個的特例），則可以比較按其自由度歸一化的 G2，s：G21dj

This normalization is commonly used in categorical data analysis, and its evaluation is equivalent to that of a statistic divided by its mean.
這種歸一化通常用於分類數據分析，其評估等價於統計量除以其均值。

16.3.1 Statistical Analysis of Multiple Relational Networks
16.3.1 多關係網路的統計分析
There is a wide variety of models for network data consisting of measurements on two or more relations.
網路數據有各種各樣的模型，包括對兩個或多個關係的測量。

Most of the models can be fit using standard categorical data analysis techniques, especially those found in the computer package GLIM (Baker and Nelder 1978; Payne 1985; and the appendix to Wasserman and Iacobucci 1986)
大多數模型都可以使用標準的分類數據分析技術進行擬合，尤其是那些在計算機軟體包GLIM中發現的模型（Baker and Nelder 1978;佩恩 1985;以及 Wasserman 和 Iacobucci 1986 的附錄）

These techniques are identical to those illustrated in the last chapter on simpler network data sets involving just one relation.
這些技術與上一章中介紹的那些技術相同，這些技術僅涉及一個關係的簡單網路數據集。

The first extension of these dyadic interaction models to mUltiple relations came in Fienberg and Wasserman (1980) and Fienberg, Meyer, and Wasserman (1981)
這些二元交互模型首次擴展到 mUltiple 關係是在 Fienberg 和 Wasserman （1980） 以及 Fienberg， Meyer 和 Wasserman （1981） 中

Their models extend Holland and Leinhardt's PI by focusing on the associations among the relations rather than on the similarities and differences among individual actor attributes.
他們的模型擴展了Holland和Leinhardt的PI，關注關係之間的關聯，而不是單個參與者屬性之間的異同。

Fienberg, Meyer, and Wasserman (1985) presented models that could include both actor and subset parameters, as we]] as interactions that measure the interrelatedness of the different relations.
Fienberg， Meyer， and Wasserman （1985） 提出了可以同時包含參與者和子集參數的模型，因為我們]] 作為測量不同關係相互關聯性的交互作用。

Novel applications of these models can be found in Wasserman (1987), Iacobucci and Wasserman (1987, 1988), and Wasserman and Iacobucci (1988, 1989).
這些模型的新應用可以在Wasserman（1987），Iacobucci和Wasserman（1987,1988）以及Wasserman和Iacobucci（1988,1989）中找到。

Research on the diffusion of innovations among the actors in a small, closed set has frequently utilized stochastic models to study how such innovations percolate through network structures. Rogers (1979) gives a thorough overview of such models and studies. Rapoport (1953) and Coleman, Katz, and Menzel (1957) have made important contributions to such modeling, and the authors refer the interested reader to reviews of this research in Kemeny and Snell (1962), Bartholomew (1967), and Coleman (1964)
關於創新在小型封閉集合中參與者之間的擴散的研究經常使用隨機模型來研究這種創新如何滲透到網路結構中。Rogers（1979）對此類模型和研究進行了全面概述。Rapoport （1953） 和 Coleman， Katz， and Menzel （1957） 對這種建模做出了重要貢獻，作者向感興趣的讀者推薦了 Kemeny 和 Snell （1962）、Bartholomew （1967） 和 Coleman （1964） 對這項研究的評論

Epilogue
結語
The authors conclude this book by speculating a bit about the future of social network methodology.
作者在本書的最後對社交網路方法論的未來進行了一些推測。

The equivalences and inclusions among a set of relations measured on a specific network is one of the most important issues in multirelational studies, and a statistical approach to this problem should be quite welcome
在特定網路上測量的一組關係之間的等價性和包含性是多關係研究中最重要的問題之一，對這個問題的統計方法應該非常受歡迎

Such statistical approaches should be developed, and should become an integral part of any social network analysis.
這種統計方法應該得到發展，並應成為任何社會網路分析的一個組成部分。

Social network methods have been developed to study one-mode networks with a single, usually dichotomous and nondirectional relation
已經開發了社交網路方法來研究具有單一，通常是二分和非方向關係的單模式網路

Methods designed for these limited data can be generalized to directional, valued, or multirelational networks, and less frequently to two-mode networks.
為這些有限數據設計的方法可以推廣到定向網路、價值網路或多關係網路，而較少推廣到雙模式網路。

An active area of current research in social network methodology is development of methods for measuring and analyzing properties of local or ego-centered networks.
目前社會網路方法論研究的一個活躍領域是開發測量和分析局部或以自我為中心的網路屬性的方法。

One area of network analysis that needs more work is development of general propositions about the structure of social networks based on replication across a large number of networks.
網路分析的一個需要更多工作的領域是開發關於基於大量網路複製的社交網路結構的一般命題。

17.6 Computer Technology corporating replication across a number of independent networks include Bernard, KiIlworth and Sailer's research on informant accuracy and Freeman's work of appropriate models of the notion of social group (Freeman 1992a)
17.6 計算機技術在許多獨立網路中包括Bernard、KiIlworth和Sailer對線人準確性的研究，以及Freeman對社會群體概念的適當模型的工作（Freeman 1992a）