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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

8.6 (j9Analysis of Actors and Events
8.6 (j9參與者和事件分析


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.

The text discusses social groups, role structures, network analysis, statistical approaches, and methods for studying social networks.

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.

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.

The text discusses various applications of affiliation networks, methods for gathering social network data, and techniques for analyzing social networks.

It covers topics such as hypergraphs, data collection techniques, nodal degree, graph density, cohesive subgroups, two-mode networks, and role structures.

The text also mentions the use of graph theory in social network analysis and the evaluation of methods for positional analysis.

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.

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.

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.

It also covers the importance of actor attributes, Bayesian estimation, and the use of sociometric and graph theoretic notations.

The text emphasizes the mathematical definitions provided by social network analysis and the significance of network data in understanding social relationships.

The text discusses various concepts related to social network analysis, including equivalence, relations, graphs, and structural properties of actors and groups.

It also covers the use of Bayesian ideas, maximum likelihood estimates, and matrix permutation analysis in network analysis.

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.

Additionally, it mentions the application of these methods to real-world problems and the concept of blockmodels in network analysis.

It covers methods for analyzing social networks, such as composition of relations, regular equivalence, and measures of structural equivalence.

The text also touches on the importance of network theory and measurement in understanding social structures.

Additionally, it mentions the use of graph theory and formal definitions to translate theoretical concepts into formal measures for network analysis.

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.

It also covers topics such as affiliation networks, transitivity, graph theory, matrix operations, and random directed graph distributions.

The text emphasizes the importance of understanding network properties and measures for social and behavioral science research.

Key points

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

We have described how to compare role structures

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

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

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


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

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

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.

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.

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

Theoretical Motivations

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.

Balanced relations were quite common in empirical work; theorists were quick to pose theories about why such things occurred so frequently

This concept led to a very active thirty-year period of empirical, theoretical, and quantitative research on triples of individuals.

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

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.

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.

Such theories argue that units are not acting independently from one another, but rather influence each other

Focusing on such structural variables opens up a different range of possibilities for, and constraints on, data analysis and model building.

The historical examination of empirical, theoretical, and mathematical developments in network research should convince the reader that social

Fundamental Concepts in Network

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.

The methods of network analysis provide explicit formal statements and measures of social structural properties that might otherwise be defined only in metaphorical terms.

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.

There are several key concepts at the heart of network analysis that are fundamental to the discussion of social networks

These concepts are: actor, relational tie, dyad, triad, subgroup, group, relation, and network.

Many important social network methods and models focus on the triad; a subset of three actors and the tie(s) among them.

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.

Having defined actor, group, and relation the authors can give a more explicit definition of social network.


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

Distinctive Features of Network Theory and Measurement

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.

Such features provide the necessary motivation for the topics discussed in this book.

Many network analysis methods provide formal definitions and descriptions of structural properties of actors, subgroups of actors, or groups

These methods translate core concepts in social and behavioral theories into formal definitions expressed in relational terms.

Social network data require measurements on ties among social units; attributes of the actors may be collected

Such data sets need social network methods for analysis.

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.

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.


Social network data consist of at least one structural variable measured on a set of actors.

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.

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.

Friendships among people are most likely studied using questionnaires or interviews, rather than using archival or historical records.

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.

The nature of the structural variables determines which analytic methods are appropriate for their study.

It is crucial to understand the nature of these variables.

The data collection techniques described here determine, to some degree, the characteristics of the relations

Structural and Composition Variables

There are two types of variables that can be included in a network data set: structural and composition.

Structural variables are measured on pairs of actors and are the cornerstone of social network data sets.

Structural variables can measure business transactions between corporations, friendships between people, or trade between nations.

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.

There are types of structural variables that are measured on two sets of entities.

A two-mode network data set contains measurements on which actors from one of the sets have ties to actors in the other set.

The second mode in an affiliation network is a set of events to which the actors belong.

In affiliation network data the two modes are the actors and the events.

In such data, the events are defined not on pairs of actors, but on subsets of actors.

Boundary Specification and

Sampling identify the population to be studied, and if sampling is necessary, worry about how to sample actors and relations.

The second example comes from the study of community leaders by Laumann and Pappi (1973)

They asked community leaders to define the boundary by identifying the elite actors in the community of Altneustadt.

Name all persons [who] are in general very influential in Altneustadt

From these lists, each of which can be considered a sample of the relevant actors in the elite network, the actor set was enumerated.

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.

Methodology for the latter situation is considerably different from methods for the former.

One example is data arising from an ego-centered network design

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.

The authors turn to a discussion of one-mode, two-mode, and affiliational, and egocentric and special networks

One-Mode Networks

Suppose the network under study is one-mode, and involves measurements on just a single set of actors.

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.

In addition to relational information, social network data sets can contain measurements on the characteristics of the actors.

Such measurements of actor attribute variables constitute the composition of the social network.

Relations in a two-mode network measure ties between the actors in one set and actors in a second set.

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.

Social network data consist of one relations measured among a set of actors.

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


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.

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: ...

Observing interactions among actors is another way to collect network data

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.

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.

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

Measurement Validity, Reliability, Accuracy, Error

As Freeman and Romney (1987) note, "social structure refers to a relatively prolonged and stable pattern of interpersonal relations" (1987, pages 330-331)

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.

Important concerns in social network measurement are the validity, reliability, and measurement error in these data.

Little work has been done on the issues of validity, reliability, and measurement error in social network data.

They found that what people report about their interactions is related to the long-range social structure, rather than to particular instances

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.

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.

Kraekhardt's High-teeh Managers

This is a one-mode network, with three relations measured on a set of people

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.

This is a one-mode network with two relations measured among a set of families

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.

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.

Social network data consist of measurements on a variety of relations for one or more sets of actors.

OMultiple Relations
OMultiple 關係

Graph theoretic notation can be generalized to multirelatiorial networks, which could include both directional and nondirectional relations.

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").

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.

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

Sociometric notation is general enough to handle valued relations

Sociometric Notation

Sociometry is the study of positive and negative affective relations, such as liking/disliking and friends/enemies, among a set of people.

A social network data set consisting of people and measured affective relations between people is often referred to as sociometric.

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

This notation can be viewed as complementary to graph theoretic notation described .

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

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.

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.

Friendship at Beginning of Year

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

Consider how this single tie is coded with sociometric notation.

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.

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.

Assuming the relation for the heterogeneous pairs is directional, the originating actor must belong to a different set than the receiving actor.

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.

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

Note how this array codes the information in the directional relation between two sets of actors.

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

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.

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

Why 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

Most of these examples will consist of hypothetical data created to demonstrate specific properties of graphs.

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

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.

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.

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

The authors turn to an example to demonstrate nodal degree and graph density


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

These properties are important in themselves, but are building blocks for later properties.

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.

The two components in this graph are the subgraphs generated by the subsets: let them consider the paths between a pair of nodes.

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.

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.

The authors will focus on the most important directed graph concepts including the nodal degrees, walks, paths, reachability, and connectivity

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.

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.

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.

The density of a directed graph is equal to the proportion of arcs present in the digraph.

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.

The length of a path is the number of arcs in it

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.

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.

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.

Reachability and Connectivity in Digraphs

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.

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.

The diameter of a weakly or unilaterally connected directed graph is undefined

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.

The arcs in the converse connect the same pairs of nodes as the arcs in the digraph, but all arcs are reversed in direction.

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.

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.

Many relations are valued; that is, the ties indicate the strength or intensity of the tie between each pair of actors.

The graph for a valued relation must convey more information by representing the strength of an arc or a line.

Signed Graphs and Signed Directed Graphs

Relations are measured in which the ties can be interpreted as being either positive or negative in affect, evaluation, or meaning.

A signed graph is a graph whose lines carry the additional information of a valence: a positive or negative sign.

A complete signed graph is a signed graph in which all unordered pairs of nodes are included in the set of lines.

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 "-".

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.

In a signed directed graph the most general cycles are usually referred to as semicycles.

Signed graphs and signed directed graphs generalize graphs and directed graphs by allowing the lines or arcs to have valences.

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

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.

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

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.

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.

In order to define these concepts for valued graphs, the authors must consider the values attached to each of the lines in a path.

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.

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.

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.

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.

Sometimes the patterns of ties between actors is not clear until the authors permute both the rows and the columns of the matrix.

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.

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

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.

The authors show how they can be studied by examining matrices, relations, and graphs.

In the discussion of graphs the authors have focused on simple graphs, which, by definition, exclude loops.

A simple graph is irreflexive, since no.

If all loops are present, the graph represents a reflexive relation.

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.

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.

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.

Actors in a network are represented as nodes of a graph.

Nondirectional ties between actors are represented as lines between the nodes of a graph.

Directed ties between actors are represented as arcs between the nodes in a digraph.

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.

The excellent text by Frank (1971) is more mathematically advanced and focuses on social networks.

Roberts (1976, 1978) and Hage and Harary (1983) provide very readable, elementary introductions to graph theory, with many concepts illustrated on anthropological network data.

A more mathematical discussion of tournaments can be found in Moon (1968). Berge (1989) discusses hypergraphs in detail

Part III

One of the primary uses of graph theory in social network analysis is the identification of the "most important" actors in a social network.

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

These definitions yield actor indices which attempt to quantify the prominence of an individual actor embedded in a network.

Measures such as outdegree and indegree are quite likely to be different, and prestigious actors are usually those with large indegrees, or "choices" received

Both centrality and prestige indices are examples of measures of the prominence or importance of the actors in a social network.

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 .

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

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

Comparisons and Extensions

Several authors have compared the performance of the many centrality and prestige indices discussed either on real or simulated data, or both

Earlier researchers, such as Stogdill (1951), concentrated on different measures of actor degrees, focusing attention on only one centrality index.

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.

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.

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.

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.

The authors will return to clusterability as a generalization of structural balance later

Signed Directional Relations

Suppose that the relation under investigation is directional, so that the relevant representation is a signed digraph.

To generalize balance to such structures requires some care, since there are a number of ways to examine cycles in directed graphs.

Consider the triple shown, which has one negative arc, and two positive arcs

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.

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.

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

A single unbalanced cycle or semicycle insures that the graph or digraph is not balanced.

It is natural to consider how many cycles or semicycles in a graph or digraph do not have positive signs.

From this consideration, one can develop graph-level indices measuring the amount of unbalance in a structure.

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.

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.

An index such as this is usually referred to as a cycle index for balance.

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

Clusterability about social structure from the writings of these theorists

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

Some ties, especially those that makecycles have negative signs, may be reinforced by a wide variety of institutional, economic, and political constraints.

The most important aspect of structural balance is that the nodes in a balanced graph can be partitioned into two subsets or clusters

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.

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.

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.

These two theorems give the conditions under which a signed graph has a clustering; that is, under what conditions on the

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.

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.

If there is a triple of actors in a cycle containing three negative lines, these three actors can be partitioned into three different clusters.

The following theorem extends clusterability to complete signed graphs; its last condition is very important

It comes directly from 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.

Through these theorems, the properties of the triples of nodes in a graph tell them whether theoretically important structural properties are present.

The clusters of actors appeared to be ranked, or hierarchical in nature, with the actors "on the bottom" choosing those "at the top"

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

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.

This ranked clusterability model was quite elegant, but little used.

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

The authors take the idea of ranked clusters for complete signed digraphs, and do not consider arcs with negative signs.

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.

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.


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.

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.

This area of research is described .

The the authors discuss methods for finding cohesive sub~oups of actors within a social network.

Social Group and Subgroup

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.

The authors' discussion is divided into sections, each of which takes up methods that are primarily motivated by one of these ideas

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.

Subgroups Based on Complete

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.

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.

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.

Subgroups Based on Reachability and Diameter

Reachability is the motivation for the first cohesive subgroup ideas that extend the notion of a clique.

These alternative subgroup ideas are useful if the researcher hypothesizes that important social processes occur through intermediaries.

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.

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

Let them return to the example of marriage and business relations among Padgett's Florentine families to illustrate 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;.

Guadagni Medici Ridolfi Strozzi Tornabuoni; Medici Pazzi Salviati.

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;.

. 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


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

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

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.

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.

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

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.

Several researchers have proposed measures for the extent to which ties are concentrated within a subgroup, rather than between subgroups

Measures of Subgroup

The authors discuss extensions of cohesive subgroups to relations that are valued or directional

These extensiOIlN allow the cohesive subgroup ideas discussed to be applied to a much wider range of social network data.

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.

A more flexible way to extend cohesive subgroup ideas to directional relations uses definitions of semipaths and connectivity for directed graphs.

In a valued relation the authors can study cohesive subgroups that vary in the strength of ties among members.

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.

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

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.

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.

The result is the possible assignment of actors to one or more cohesive subgroups.

More exploratory, approach to cohesion in social networks seeks to represent the group structure in a network as a whole.

Collections of actors among whom there are relatively strong ties can become more visible by displaying functions or rearrangements of the graphs or sociomatrices.

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.

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.

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.

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

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.

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.

The authors presented methods for assessing the cohesiveness of subgroups.

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.

The authors will refer to these data as affiliation network data, or measurements on an affiliation variable.

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

Affiliation networks differ in several important ways from the types of social networks the authors have discussed so far.

Affiliation networks are two-mode networks, consisting of a set of actors and a set of events.

Affiliation networks describe collections of actors rather than ties between pairs of actors.

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 .

. Affiliation networks consist of subsets of actors, rather than pairs of actors;.

Affiliation networks allow one to study the dual perspectives of the actors and the events.

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.

The authors discuss how affiliation networks establish linkages among the entities in each of the modes.

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


The authors review some of the more influential theoretical and substantive contributions to the study of affiliation networks.

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.

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

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.

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.

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.

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.

The two-mode sociomatrix, the bipartite graph, and the hypergraph are alternative representations of an affiliation network.

The sociomatrix is an efficient way to present the information and is most useful for data analytic purposes.

Representing the affiliation network as a bipartite graph highlights the connectivity in the network, and makes the indirect chains of connection more apparent.

Since there is no loss or gain of information in one or another


Networks representation, the researcher's goals should guide selection of the best representation.

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.

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.

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

Properties of Affiliation Networks

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.

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.

Let them consider the density of ties in the one-mode networks of actor co-memberships and event overlaps.

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.

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

The authors will use both the valued and dichotomous relations of actor co-memberships and event overlaps.

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.

A useful way to study reachability in an affiliation network is to consider the bipartite graph, with both actors and events represented as nodes.

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.

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

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.

The authors focus on the one-mode valued relation of co-membership for actors or the one-mode valued relation of overlap for events.

One can use the definitions for the value of a path to define connectedness for pairs of actors in the valued graph.

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.

These four children form a maximal complete subgraph in the co-membership relation.

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.

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.

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

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.

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.

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.

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.

For the most part the analyses that the authors have described assume that one has a complete affiliation network.

That all actors and all events constituting the network are included.

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

Part IV

Many methods for the description of network structural properties are concerned with the dual notions of social position and social role.

In social network terms these translate into procedures for analyzing actors' structural similarities and patterns of relations in multirelational networks

These methods, which have been referred to as positional, role, or relational approaches, are the topic of Part IV.

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.

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

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.

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

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.

Consider the valued relation of acquaintanceship in Freeman's EIES network

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.

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

Simplification of Multirelational Networks

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

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.

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.

Structurally equivalent actors have identical entries in their corresponding rows and columns of the sociomatrix.

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

A second widely used measure of structural equivalence is the correlation coefficient.

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.

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.

This is the same conclusion that the authors reached using Euclidean distance as a measure of structural equivalence

Some Considerations in Measuring Structural Equivalence

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).

This method was first used for analyzing social networks by H.

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.

Three arcs, 2 reflexive

The advice image is transitive at the level of the positions.

The pattern for friendship is not as clear.

Interpreting blockmodels with multiple relations can be tedious.

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.

Multiplexity of relations is the tendency for two or more relations to occur together.

"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.

Multiplexity in a blockmodel would be apparent if two or more image matrices were identical.

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.

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.

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

He notes "Primary roles can be cumulated into chains defining compound roles", pointing toward a focus on associations among relations.

Formalizing the ideas of interrelatedness, interlocking, or bundles of relations is one goal of formal network role analysis

These methods are different from methods for network positions that focus on properties of subsets of actors.

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.

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.

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.

The authors describe some properties of the operation of composition of relations

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.

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.

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.

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.

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.

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.


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.

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

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.

This shows that there are two subsets of images that operate when they are the first element in a compound relation.

The second strategy is based on simplifying the multiplication table that expresses the composition of relations

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.

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

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

Both of these approaches have the goal of representing the essential features that are shared between role structures.

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

This reduction of the role structure for the Bank Wiring room is a homomorphic reduction.

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.

Let them consider some issues that arise in defining regular equivalence for nondirectional relations

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.

A partition consisting of a single equivalence class is trivial, and probably uninteresting.

A nondirectional relation may contain other regular equivalence partitions.

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.

In order to be regularly equivalent, actors must be adjacent to the same kinds of other actors

This approach to defining regular equivalence is especially useful for studying regular equivalence in nondirectional relations.

Before the authors discuss measures of regular equivalence, let them consider how to represent regular equivalence partitions using a regular equivalence blockmodel

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.

Let them consider two definitions of equivalence that focus on the types of ties in which each actor is involved

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

The authors will consider the associations among relations from the perspectives of individual actors

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.

For this example the authors will consider the set of all distinct primitive and compound relations.

In the two sections the authors present two different definitions and measures of equivalence for individual roles

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

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.

Breiger and Pattison (1986) compare ego algebras by the joint right homomorphism of two ego algebras.

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.

The authors can measure the distance between two ego algebras by slimming the distance each is from their joint right homomorphic reduction.

In both examples the authors used the routine JNTHOM in the program ROLE (Breiger 1986).

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

Part V

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.

These properties will be described at length .

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.

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

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.

The simplest conditional uniform random digraph distribution conditions on the graph property

Such a distribution is useful when studying the randomness of choices made by each individual actor

One could assume that the sociomatrix was distributed as a uniform random matrix, conditional on the outdegrees and number of mutuals

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

This number is the size of the sample space of digraphs under this distribution.

The distribution discussed in these paragraphs is one of the most frequently used random directed graph distributions in social network analysis

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.

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

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.

Such hypotheses are usually tested by examining not the entire triad census and its expectation, but linear combinations of it, which the authors discuss

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.

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

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.

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.

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

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.

The authors begin this chapter by presenting models for a network with measurements on a single, directional relation for one set of actors.

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.

The authors estimate these parameters using log linear modeling techniques.

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.

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.

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.

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

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.

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.

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

These models assume that all expansiveness parameters are constant across all actors or all subsets of actors.

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.

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.

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.

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.

Wealthy families enter into business relationships at different rates than less wealthy families, but wealth is not an important influence on marital ties.

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)

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

Another development to note is the work reported in Strauss and Ikeda (1990).

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.

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”

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.

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

Such methods, as described can be used to find partitions of actors, and lead to proper statistical tests and measures of goodness-of-fit.

As described can be used to find partitions of actors, and lead to proper statistical tests and measures of goodness-of-fit

Another approach centers on the evaluation of a particular positional analysis technique using standard data sets.

The second approach, as mentioned, is based on statistical theory for social network data

This idea uses a statistical or stochastic blockmodel to represent mathematically the equivalence classes defined on the actors.

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

This leads to a valid, nonparametric statistical test for the goodness of an observed fit.

There is no parametric statistical theory for this index

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.

The authors turn to a discussion of goodness-of-fit indices which are based on specific, parametric statistical models

This is a rather different approach to assessing the fit of a network data set to a particular partition of actors to positions.

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.

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.

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

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

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

Most of these measures are lacking because they are not based on statistical models, and they do not have convenient and well-known distributions.

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.

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.

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.

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.

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).

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


The authors conclude this book by speculating a bit about the future of social network methodology.

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

Such statistical approaches should be developed, and should become an integral part of any social network analysis.

Social network methods have been developed to study one-mode networks with a single, usually dichotomous and nondirectional relation

Methods designed for these limited data can be generalized to directional, valued, or multirelational networks, and less frequently to two-mode networks.

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.

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.

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)

These studies test general propositions about networks using the network as the unit of analysis.

One .should be able to display actor attributes and nodal or subgroup properties along with the graph
一個 .應該能夠顯示參與者屬性和節點或子組屬性以及圖形

17.7 Networks and Standard Social and Behavioral Science
17.7 網路與標準社會和行為科學

One area where a great deal of work remains is integrating network concepts and measures into more general social and behavioral science research.

Network is a catch phrase in many disciplines the precise use of network measures has not fully diffused to these areas.

In part the usual institutional and intellectual barriers between disciplines inhibit diffusion.

Theperception of the technical sophistication required to use network ideas may dissuade potential users.

The authors expect the greater availability of network analysis software, and greater ease of interface with standard statistical analysis software will make network ideas more exportable to the wider community.

Jf and when greater consensus develops among network researchers about key network properties and measures, it should be easier to communicate appropriate use of network methods to nonnetwork specialists.

The authors hope that this book will help in this regard.

The authors are excited about the future prospects for social network methods, and look forward to incorporating these advances into the second edition of this book

Graph Definition and Analysis Package (Sprenger and Stokman 1989) was developed through collaboration of researchers from the Universities of Amsterdam, Groningen, Nijmegen, and Twente (Sprenger and Stokman 1989).