Generalized Blockmodeling

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Overview

After establishing its mathematical foundations, this integrated study of blockmodeling, the most frequently used technique in social network analysis, generalizes blockmodeling for the examination of many network structures. It also includes a broad introduction to cluster analysis. The authors propose direct optimizational approaches to blockmodeling which yield blockmodels that best fit the data, and create the potential for many generalizations and a deductive use of blockmodeling.
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'This is a clearly presented and insightful book that provides an excellent mix of mathematical rigor and practical application. I would unhesitatingly recommend the book to anyone interested in social network analysis or discrete clustering methods.' Journal of Classification
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Meet the Author

Patrick Doreian is a Professor of Sociology and Statistics at the University of Pittsburgh and is chair of the Department of Sociology. He has edited the Journal of Mathematical Sociology since 1982 and has been a member of the editorial board for Social Networks since 2003. He was a Centennial Professor at The London School of Economics during 2002. He has been a Visiting Professor at the University of California-Irvine and the University of Ljubljana. His interests include social networks, mathematical sociology, interorganizational networks, environmental sociology and social movements.

Vladimir Batagelj is a Professor of Discrete and Computational Mathematics at the University of Ljubljana and is chair of the Department of Theoretical Computer Science at IMFM, Ljubljana. He is a member of editorial boards of Informatica and Journal of Social Structure. He was visiting professor at University of Pittsburgh in 1990 to 1991 and at University of Konstanz (Germany) in 2002. His main research interests are in graph theory, algorithms on graphs and networks, combinatorial optimization, data analysis and applications of information technology in education. He is coauthor (with Andrej Mrvar) of Pajek - a program for analysis and visualization of large networks.

Anu∫ka Ferligoj is a Professor of Statistics at the University of Ljubljana and is dean of the Faculty of Social Sciences. She is editor of the series Metodoloski zvezki since 1987 and is a member of the editorial boards of the Journal of Mathematical Sociology, Journal of Classification, Social Networks, and Statistics in Transition. She was a Fulbright scholar in 1990 and Visiting Professor at the University of Pittsburgh. She was awarded the title of Ambassador of Science of the Republic of Slovenia in 1997. Her interests include multivariate analysis (constrained and multicriteria clustering), social networks (measurement quality and blockmodeling), and survey methodology (reliability and validity of measurement).

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Generalized Blockmodeling
Cambridge University Press
0521840856 - Generalized Blockmodeling - by Patrick Doreian, Vladimir Batagelj, and Anuška Ferligoj
Excerpt



1

SOCIAL NETWORKS AND BLOCKMODELS


These are exciting times for social network analysts. As Hummon and Carley (1993) observed, their area has emerged as an integrated social scientific specialty. Some (e.g., Berkowitz 1982:150 and Rogers 1987:308) have declared that social network analysis is revolutionary for the social sciences. Doreian (1995) argued that this is a premature judgment. It was not clear then - nor is it clear now - that there is a network paradigm in the sense of "a set of shared methods, standards, modes of explanation, or theories or a body of shared knowledge" (Cohen 1985:26) adhered to by all network analysts. In part, this can be attributed to the field's having its historical origins in a wide variety of disciplines. Fields such as anthropology, business administration, communication, history, mathematics, political science, and sociology have scholars whose research, at least in part, includes network analytic ideas. Even though some network analysts view social networks as their field, they do not share all of the features of a specialty listed by Cohen. However, network analysts do agree that social networks are important - even crucial - and that network-based explanations of social phenomena have a distinctive character. Wellman (1988) is most persuasive in arguing that network accounts of social phenomena, in addition to being distinctive, are also more potent. Even so, network analysts differ on some of the specifics. In our view, this is a positive feature of the specialty given a commitment to using network tools of some sort.

Certainly the trappings of a coherent social science specialty are in place: The International Association of Social Network Analysts (INSNA) was formed in 1976. Members of INSNA have received Connections, a professional newsletter - now available online - that has linked their invisible college since 1977, and a specialty journal (Social Networks) was created in 1978 specifically for the emerging field. There is an annual (International Sunbelt Social Network) conference. Within this institutionalized forum, network analysts have pursued a wide variety of topics with regard to substance and the development of network analytic tools.

The so-called knowledge generated by scientists working within a particular discipline or sets of disciplines is conditioned by the technology they use. For our purposes here, we view technology (both hardware and software) as the means by which sets of tools can be fashioned. Even though it is far too early to know if social network analysis is in the vanguard of a scientific revolution for the social sciences, it is useful to examine the first stage of a scientific revolution. Cohen (1985) characterized it as an intellectual revolution that "occurs whenever a scientist (or a group of scientists) devises a radical solution to some major problem or problems, finds a new method of using information . . . sets forth a new framework for knowledge into which existing information can be put in a wholly new way . . . [and] introduces a set of concepts that change the character of existing knowledge or proposes a revolutionary new theory" (1985:28). We believe that social network analysis in general, and blockmodeling in particular, has this character.

Lorrain and White's (1971) paper on structural equivalence was dramatically different and changed1 the way many social network analysts viewed their field. Work on blockmodeling and positional analysis blossomed. In Hummon and Carley's citation study of the articles in the first 12 volumes of Social Networks, positional analysis was identified as one of the dominant interests.

There are two broad rationales for considering blockmodeling as an important set of network analytic tools. One is substantive and concerns the delineation of role structures. In general, there are well-defined places in social structures. We call these places positions and provide an intuitive description of this term later in this chapter and a much fuller characterization in Chapter 6. A social role is a set of expectations that are coupled to the positions. Parent is a position, and there are expectations as to what parents should do. Child is another position with (age-graded) expectations of what children should do. These positions and roles form a coupled system in which parents and children have expectations of each other.2 Chief Executive Officer (CEO) is another position defined in a corporation, as is the position of Vice President (VP). There are coupled expectations of how CEOs and VPs should behave with regard to each other and to other actors inside (and outside) the corporation. These roles include and, more importantly, generate social relations with observed network behavior, especially interaction, providing indicators of the fundamental social structure in which social actors are located (see Figure 1.3 and Table 1.8 in the paragraphs that follow). Actors in those roles will share some common features. As Schwartz and Sprinzen (1984) put it, "the occupants of a common position will exhibit a common pattern of relations, across multiple relations, consistently tending to have certain relations with occupants of particular other positions and to not have the same type of relation with occupants of yet other positions" (1984:104). Using the network in Figure 1.3 (shown in the paragraphs that follow) as an example, we find that informal social interactions are more likely to occur at each level of the hierarchy rather than to involve people from different levels. In this example of a network, the idea of positions and roles is built into the organizational structure. Blockmodeling tools provide ways of describing these structures in a theoretically informed way.

The second rationale is more pragmatic in the sense of discerning the fundamental (basic) structure of social networks. Most, if not all, social networks have a nonrandom structure. Put differently, they have systematic features that characterize their social structures. For small networks, be they formal roles systems or generated empirical networks, these structures are relatively straightforward to discern. However, once networks are large or complex, identifying their basic structure becomes a difficult task. We note that the structure of the small Kansas Search and Rescue network that is described in Section 2.2.4 with just 20 organizations is such an example, for it has a structure that is nonobvious upon visual inspection. On the basis of our experience of fitting many blockmodels, we also note that a perceived obvious structure need not be fundamental and that blockmodeling can suggest alternative structures to analysts. Blockmodeling provides a coherent approach to identifying fundamental structures of social networks, and generalized blockmodeling provides a much expanded set of blockmodeling tools.

The following treatment of generalized blockmodeling is offered in the hope that such an integrated presentation will secure its intellectual foundations and establish a fruitful methodological basis for the analysis of a wide range of relational structures and network processes. In short, our goal here is to present an integrated and generalized treatment of blockmodeling. We do not present a chronological history of blockmodeling. Readers who are familiar with network analysis and with conventional blockmodeling can skip ahead to Section 1.6 for a concise statement of generalized blockmodeling and some of the ways in which it can be done.

We use the rest of this chapter to provide an intuitive statement of social network analytic ideas (Section 1.1), a description of blocks as parts of networks (Section 1.2), a listing of some block types (Section 1.3), a discussion of specifying blockmodels with some concrete examples (Section 1.4), a preliminary characterization of blockmodeling that starts with conventional blockmodeling (Section 1.5), and a general statement about generalized blockmodeling, including a description of how it generalizes and greatly extends conventional blockmodeling (Section 1.6). In section 1.7 we provide an outline map of the book that is represented in Figure 1.5. Sections marked with ⊙ contain more techanical materials and can be skipped on a first reading.

1.1 AN INTUITIVE STATEMENT OF NETWORK IDEAS

We start with an intuitive statement of some social network issues, problems, and ideas. Our intent is to provide an overview of some key network analytic terms as they relate to blockmodeling. These ideas help provide motivation for the formal tools we begin to develop in Chapter 3 and use throughout the subsequent chapters.

Social networks consist of social actors with one or more social relations defined over those actors. Social actors and social relations are everywhere - so much so that is impossible to think of social life without also thinking of social relations.

Most people, at many points in their lives, belong to small groups. There have been an extraordinary number of definitions of this concept. Because we do not want to get into a protracted terminological debate, we consider a group to be two or more individuals who are interdependent through sustained interaction.3 Groups can be found in workplaces, in religious organizations, in dormitories, and so on. Terms such as support groups and friendship groups acknowledge the widespread existence of relatively small collections of individuals sharing some kind of salient identity. People's relations with each other in their groups are crucial for understanding their actions within those groups. Indeed, there is a large literature devoted to the study of the structure of social relations in groups.4 We note its existence here, in passing, but we return to it in Chapters 6-12. These relations help determine the ways in which human groups evolve through time. See, for example, some of the contributions to Doreian and Stokman's (1997) edited volume on social network evolution.5 Affective relations or sentiments (Homans 1950) having positive and negative values (such as like or dislike) are particularly relevant. We consider signed relations in detail in Chapter 10.

Network analytic techniques are not confined to small-group applications, and, as a first step away from a small-group focus, we note that there can be social relations between groups. Two bowling teams can have a friendly rivalry, or two support groups can provide mutual support. For these two examples, it is unlikely that a group constrains the activities of its members with regard to between-group behavior. However, intergroup relations can be negative. Some people belong to gangs, and there are important social relations between gangs, notably being allied with, or an enemy of, other gangs. Relations between the gangs set both the permitted and proscribed behaviors for individuals belonging to them. So too for families: The drama of Shakespeare's Romeo and Juliet is driven by the enmity between the houses of Capulet and Montague.

Many people work in organizations in which there are social relations such as "has authority over" or "receives orders from" as well as some of the interpersonal relations that are created when people work and necessarily interact with each other. Homans (1950) made a distinction between the external and internal systems of small human groups. The external system is part of the wider organizational environment, including the physical layout of work areas such as the Bank Wiring Room (BWR; Roethlisberger and Dickson, 1939). We describe some of the classic BWR data in Chapter 2 and report analyses of them in Chapters 6, 7, and 10. There are relations between different departments of an organization. In an organization manufacturing some durable product, minimally, raw resources and partially finished goods move between different organizational units. There can be relations between organizations also. Organizations shipping goods back and forth have social relations such as "transports to" and "communicates with." The sets of agencies providing social services to a population in some geographical location are linked by a rich set of social relations that include referrals (of clients) between agencies, the provision of services for other agencies, and money flows.

Cities are linked by a variety of infrastructures - roads, railways, optical fibers, and satellite links - that permit people to travel between places, make telephone calls, and fax messages to one another. Economic goods can be moved between areas even if they are far apart. We can think of countries as being linked by many of the social relations that also link cities. Certainly, they have trading ties, can recognize each other diplomatically, form alliances, and go to war - all of which are social relations.

The foregoing discussion makes it clear that we do not confine the term social actor to individuals. Any human collectivity with a clear identity is a potential social actor: groups, organizations, cities, and nations have the potential to act. At a minimum, interaction involves outputs from actors that are coupled to inputs from other actors. According to Olsen (1968:32), social interaction "occurs whenever one social actor affects the thoughts or actions of another social actor in some fashion" (emphasis added). Furthermore, we regard a social relationship as enduring social interaction.6 In this sense, then, we can talk of relationships between organizations or any well-defined collectivities. We recognize that the words organizations interact can be expressed in terms of the behavior of specific individuals within these organizations. We can take the statement about organizations interacting as a shorthand expression for the actions of all of the human actors involved in that transaction. However, we want to use the term interact in a much more general sense. The key concept is to identify social actors as socially defined units. Transactions can occur between any social collectivities that are identified as meaningful, well-defined social units.

1.1.1 Fundamental types of social relations

In each of the examples described thus far, the social actors are individuals, groups, organizations, and the like. They are social and they can act. We use social actor as a generic term. If the focus is on one type of social actor, people for instance, then in terms of our narrative thus far, attention will be focused on the relations between people. If the focus is on organizations, attention is confined to relations between these organizations.7 A specific network analysis can have multiple types of social actors. For a particular discussion, the relevant social actors can also be referred to as the units (of analysis) consistent with both the idea of socially defined units and most discussions of research methods.

Figure 1.1. Transatlantic industries friendship ties. Image not available in HTML version

One-mode social relations. For this type of relation, a one-mode social network is defined as the set of social actors and the relations defined only between them. In most discussions of social networks, this is the meaning of the term network, and we also adopt this usage of the term. The units are all of a single type. For a small group, this will be the people and their social ties - for example, "likes" or "is my best friend." As an example, consider the diagram8 in Figure 1.1 for a Little League baseball team (Transatlantic Industries) with 13 boys (Fine 1987). Each boy was asked to name his three best friends.9 The social actors are the boys and the relation is "my best friend." The lines represent the presence of the social ties from one boy to another. The line from Tom to Ron shows Tom's choice of Ron as someone he likes well enough to be among his top three choices. Ron does not view Tom in the same way. In a similar fashion, Tim and Boyd have a tie in one direction but not in the other. Ron and Frank have a reciprocal tie whereas Darrin and Tim have no tie between them. The rest of the figure can be read in the same fashion. All of the information concerning this social relation is contained in the diagram. Such figures were called sociograms by Moreno (1934), which is a term still in use today.

1. Acciaiuoli 9. Medici
2. Albizzi 10. Pazzi
3. Barbadori 11. Peruzzi
4. Bischeri 12. Pucci
5. Castellani 13. Ridolfi
6. Ginori 14. Salviati
7. Guadagni 15. Strozzi
8. Lamberteschi 16. Tornabuoni

Figure 1.2. Elite families in Florence in the 15th century. Image not available in HTML version

Figure 1.1 illustrates the concept of a graph in a pictorial representation of the set of social actors and the relation defined over them. The relation is the set of pairs of boys (as actors) that are linked by the social ties. The graph is the display of the boys and the pairs of boys where one is chosen by the other. This and related concepts are defined in Chapter 4. When our attention is on a graph as a representation of a social network, we will use the term vertex for a represented social actor. For the ties among the social actors, we will use the term lines (in a graph).

As a second example, consider the elite Florentine families in the 15th century as shown in Figure 1.2. Here the social relation is "linked by marriage" with the lines showing the existence of a marriage tie between families. This graph can be read in the same way as Figure 1.1. For example, the Medici family has marriage ties to six other families. The Bischeri family is not among those linked directly by marriage to the Medici family. However, it is linked with the Peruzzi family by marriage. Again, all of the information concerning the incidence of marriage linkages is contained in the graph.

We note one difference between the relations depicted in Figures 1.1 and 1.2. For the boys in the Little League baseball team the social relation has a direction (from the chooser to the chosen), while the relation for the Florentine families does not. If we preserved the distinction of men marrying women and women marrying men, then the marriage tie between families would be directed also.10 Two types of lines in the graph are distinguished: edges (undirected lines) and arcs (directed lines). Generically, we use the term lines, and the context will make clear the specific type of tie.

Using a visual version of a graph is one way of representing a social relation. An alternative form for representing the same information about the presence of a social relation is a square array or matrix. Table 1.1 provides the same information that is contained in Figure 1.1. Here the numeral 1 represents the presence of a tie between two actors, and the numeral 0 represents the absence of a tie between actors.

Table 1.1. Matrix with the Ties of the Transatlantic Industries Team
1 2 3 4 5 6 7 8 9 10 11 12 13
Ron 1 0 0 1 1 1 0 0 0 0 0 0 0 0
Tom 2 1 0 1 0 0 0 0 0 0 0 1 0 0
Frank 3 1 0 0 1 0 0 0 0 0 0 1 0 0
Boyd 4 1 1 1 0 0 0 0 0 0 0 0 0 0
Tim 5 1 0 1 1 0 0 0 0 0 0 0 0 0
John 6 0 0 0 0 1 0 0 0 0 0 0 1 1
Jeff 7 0 1 0 0 0 0 0 1 1 0 0 0 0
Jay 8 0 1 0 0 0 0 1 0 1 0 0 0 0
Sandy 9 0 1 0 0 0 0 1 1 0 0 0 0 0
Jerry 10 1 0 0 0 0 1 0 0 0 0 0 0 0
Darrin 11 1 1 0 0 0 0 0 0 0 1 0 0 0
Ben 12 1 0 0 0 0 1 0 0 0 1 0 0 0
Arnie 13 0 0 0 0 1 1 0 0 0 0 0 0 0

Table 1.2 shows exactly the same marriage data contained in Figure 1.2. Both the rows and columns are coordinated by the family names, and the presence of a 1 indicates a marriage tie between the two families and 0 represents the absence of such a tie. The matrix representation of ties is known also as a sociomatrix. It is possible, and often desirable, to examine multiple relations defined over the same collection of social actors.

It should be clear that any social relation, as defined above, can be represented either by a graph or an array (a matrix). These representations take the same abstract forms, regardless of the nature of the units and the relations between them. We make extensive use of both throughout this volume. While pictures do have a visual immediacy and have value because of this, matrix representations have greater relevance for blockmodeling. In essence, the technical core of what we propose throughout this book is a sophisticated form of row and column permutation to reveal structure. Of course, once we have the fundamenal form of a network, graphical methods are again important for displaying this form.

Table 1.2. Padgett’s Florentine Families Marriage Data
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Acciaiuoli 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
Albizzi 2 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0
Barbadori 3 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0
Bischeri 4 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0
Castellani 5 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0
Ginori 6 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Guadagni 7 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1
Lamberteschi 8 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
Medici 9 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1
Pazzi 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
Peruzzi 11 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0
Pucci 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Ridolfi 13 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1
Salviati 14 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0
Strozzi 15 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0
Tornabuoni 16 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0

Membership social relations. The second fundamental network relation is membership or affiliation. Although it was not described explicitly in our discussion thus far, it was acknowledged implicitly. Gang members belong to gangs, people are employed by organizations, and individuals belong to families. For membership ties, there are two types of units - and the data for such a relation are often referred to as two-mode data. (See Wasserman and Faust 1994:29-30 and 291-298, for a discussion of two-mode data.) One of the classic membership data sets is shown in Table 1.3 (Davis, Gardner, and Gardner 1941).

The data represent the attendance patterns of 18 women for 14 events. Typically, such two-mode data are represented in rectangular arrays. In this case, the array is coordinated by women for the rows and by events for the columns. It is straightforward to construct Table 1.4, which contains counts (off the main diagonal) of the number of times each pair of women was at the same event. For example, Eleanor and Brenda jointly attended events E2, E4, E8, and E12. The values in the diagonal contain counts of the number of events each woman attended. Eleanor attended just four events while Brenda attended seven events. In addition to the events attended by Eleanor, Brenda was also present at events E7, E10, and E13. These two-mode data will be analyzed with our blockmodeling tools for two-mode data in Chapter 8.

Table 1.3. Membership Relation for Women and Events
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14
Eleanor 0 1 0 1 0 0 0 1 0 0 0 1 0 0
Brenda 0 1 0 1 0 0 1 1 0 1 0 1 1 0
Dorothy 0 0 0 0 0 1 0 0 0 0 0 1 0 0
Verne 0 0 0 1 1 1 0 0 0 0 0 1 0 0
Flora 1 0 0 0 0 1 0 0 0 0 0 0 0 0
Olivia 1 0 0 0 0 1 0 0 0 0 0 0 0 0
Laura 0 1 1 1 0 0 1 1 0 1 0 1 0 0
Evelyn 0 1 1 0 0 1 1 1 0 1 0 1 1 0
Pearl 0 0 0 0 0 1 0 1 0 0 0 1 0 0
Ruth 0 1 0 1 0 1 0 0 0 0 0 1 0 0
Sylvia 0 0 0 1 1 1 0 0 1 0 1 1 0 1
Katherine 0 0 0 0 1 1 0 0 1 0 1 1 0 1
Myrna 0 0 0 0 1 1 0 0 1 0 0 1 0 0
Theresa 0 1 1 1 0 1 1 1 0 0 0 1 1 0
Charlotte 0 1 0 1 0 0 1 0 0 0 0 0 1 0
Frances 0 1 0 0 0 0 1 1 0 0 0 1 0 0
Helen 1 0 0 1 1 0 0 0 1 0 0 1 0 0
Nora 1 0 0 1 1 1 0 1 1 0 1 0 0 1

In a specific network study, both one-mode and two-mode relations can be observed. It may well be the case that one of the relations has some importance for the other. If we look at people belonging to gangs, their memberships will constrain their choices of friends. Although employees work for organizations, they do this in distinct organizational units, and their locations in these units constrain their interactions with other organizational members. Although the linkage of these relations will not be perfect, we will say that one relation constrains the other. We expand these ideas formally in Chapter 3 (Section 3.2.2).

Table 1.4. Woman-by-Woman Joint Attendance of Events Relation
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Eleanor 1 4
Brenda 2 4 7
Dorothy 3 1 1 2
Verne 4 2 2 2 4
Flora 5 0 0 1 1 2
Olivia 6 0 0 1 1 2 2
Laura 7 4 6 1 2 0 0 7
Evelyn 8 3 6 2 2 1 1 6 8
Pearl 9 2 2 2 2 1 1 2 3 3
Ruth 10 3 3 2 3 1 1 3 3 2 4
Sylvia 11 2 2 2 4 1 1 2 2 2 3 7
Katherine 12 1 1 2 3 1 1 1 2 2 2 6 6
Myrna 13 1 1 2 3 1 1 1 2 2 2 4 4 4
Theresa 14 4 6 2 3 1 1 6 7 3 4 3 2 2 8
Charlotte 15 2 4 0 1 0 0 3 3 0 1 1 0 0 4 4
Frances 16 3 4 1 1 0 0 4 4 2 2 1 1 1 4 2 4
Helen 17 2 2 1 3 1 1 2 1 1 2 4 3 3 2 1 1 5
Nora 18 2 2 1 3 2 2 2 2 2 2 6 5 3 3 1 1 4 8




© Cambridge University Press
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Table of Contents

Preface; 1. Social networks and blockmodels; 2. Network data sets; 3. Mathematical prelude; 4. Relations and graphs for network analysis; 5. Clustering approaches; 6. An optimizational approach to conventional blockmodeling; 7. Foundations for generalized blockmodeling; 8. Blockmodeling two-mode network data; 9. Semirings and lattices; 10. Balance theory and blockmodeling signed networks; 11. Symmetric-acyclic blockmodels; 12. Extending generalized blockmodeling; Bibliography; Author index; Subject index.
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