PAD 637 Summary Week 6: Cohesive Subgroups and Core-Periphery Structures

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1 PAD 637 Summary Week 6: Cohesive Subgroups and Core-Periphery Structures Xiao Liang,Hyo-Shin Kwon Wasserman and Faust, chapter7 Cohesive Subgroups: In this chapter, the authors discuss methods for finding cohesive subgroups of actors within one-mode networks with a single set of actors and a single relation. Pp249 Definition: Cohesive subgroups are subsets of actors among whom there are relatively strong, direct, intense, frequent, or positive tires. Pp249 Structure of the chapter: 1. Theoretical motivation for studying cohesive subgroups and different types of subgroups and their properties. 2. How to assess the cohesiveness of network subgroups 3. Directional relations and valued relations 4. Interpretation of Cohesive subgroups 5. Alternative approaches for studying cohesiveness in networks (multidimentional scaling and factor analysis) Required knowledge: Graph theory (Chapter4) 1. Theoretical motivation for studying cohesive subgroups and different types of subgroups and their properties. 1.1 Theoretical motivation - Studying the emergence of consensus among members of a group: direct and indirect positive relationships and the emergence of consensus - One expects greater homogeneity among persons who have relatively frequent face-to-face contact or who are connected through intermediaries, and less homogeneity among persons who have less frequent contact. pp250

2 - Clique: isolated and tightly connected groups make up a clique; within such highly cohesive groups, individuals tend to have very homogeneous beliefs. (quated from Collins 1998, pp250) Social forces operation through direct contact among subgroup members, through indirect conduct transmitted via intermediaries, or through the relative cohesion within as compared to outside the subgroup. Pp251 Key definitions: social group, subgroup, clique 1.2 Types of subgroups Define cohesive subgroups! How? Key: the ties! There are at least two different aspects to the concept of a cohesive subgroup: 1) the concentration of ties within the subgroup 2) and a comparison of strength or frequency of ties within the subgroup to the strength or frequency of ties outside the subgroup. pp267 (from Alba 1973) 1) the concentration of ties within the subgroup - subgroups based on complete mutuality Definition of a clique: A clique in a graph is a maximal complete subgraph of three or more nodes. It consists of a subset of nodes, all of which are adjacent to each other, and there are no other nodes that are also adjacent to all of the members of the clique (luce and Perry 1949; Harary, Norman, and Cartwright 1965 pp254, restriction: the clique contains at least three nodes Considerations: pp256 very strict: the size of the cliques will be limited by the degree of the nodes (example: ask the respondents to list their three best friends) : Actual data rarely contain interesting cliques, since the absence of a single tie among subgroup members prevents the subgroup from meeting the clique definition. There is no internal differentiation among actors within a clique. Since a clique is complete, within the clique all members are graph theoretically identical --- lack of the ability to discover interesting internal structure For large dataset, there may be numerous, but largely overlapping, cliques in the group, and they might not be very informative. Way of weaken the notion of clique: 1) reahcability, path distance and diameter; 2) nodal degree - subgroups based on reachability and diameter

3 Subgroup members might not be adjacent, but if they are not adjacent, then the path connecting them should be relatively short. n-cliques: An n-clique is a maximal subgraph in which the largest geodesic distance between any two nodes is no greater than n. d(i,j)n for all n i, n j N s d(i,j) is geodesic distance for node i and node j. n i is node i, n j is node j, N s is a node set. 2 is often a useful cutoff value, 2-cliques are subgraphs in which all members need not be adjacent, but all members are reachable through at most one intermediary. Considerations: Because it does not require that the paths connecting the nodes are all within the subgroup, there are problems: 1) a subgraph may have a diameter greater than n, 2) an n-clique might be disconnected. N-clans and n-clubs:ann-clan is an n-clique in which the geodesic distance, d(i,j), between all nodes in the subgraph is no greater than n for paths within the subgraph. An n-club is defined as a maximal subgraph of diameter. That is, an n-club is a subgraph in which the distance between all nodes within the subgraph is less than or equal to n; further, no nodes can be added that also have geodesic distance n or less from all members of the subgraph. Sprenger and Stokman (1989 ), Hardly anybody has used n-class and n-clubs, and more research is needed on theses cohesive subgroup ideas. The n-clans in a social network are relatively easy to find. The n-clubs are difficult to find. The social theory behind the reachability of cohesive subgroups: 1) Erickson (1988) The cohesive subgroup definitions based on reachability are important for understanding processes that operate through intermediaries, such as the diffusion of clear cut and widely salient information. 2) Hubbell (1965)ties between actors are channels for the transmission of influence. Pp263 - subgroups based on nodal degree These definitions are based on the adjacency of subgroup members. These approaches are based on restrictions on the minimum number of actors adjacent to each actor in a subgroup. pp263 (in another word, these definitions require that all subgroup members be adjacent to some minimu number of other subgroup members) Motivation of the definition: 1) understanding processes that operate primarily through direct contacts among subgroup members. 2) vulnerability of n-cliques. Pp264 K-plexes: a k-plex is a maximal subgraph containing g s nodes in which each node

4 is adjacent to no fewer than g s -k nodes in the subgraph. In other words, each node in the subgraph may be lacking ties to no more than k subgraph members. (pp265). An important property of a k-plex is that the diameter of a k-plex is constrained by the value of a k. Consideration of k-plexes: researcher should restrict the size of a k-plex so that it is not too small relative to the number of ties that are allowed to be missing. Pp266 K-cores: k-core is a subgraph in which each node is adjacent to at least a minimum number, k, of the other nodes in the subggraph. (Although k-cores themselves are not necessarily interesting cohesive subgroups, they are areas of a graph in which other interesting cohesive subgroups will be found.)pp267 2) a comparison of strength or frequency of ties within the subgroup to the strength or frequency of ties outside the subgroup. - The idea is that cohesive subgroups should be relatively cohesive within compared to outside pp267 - An ideal subgroup: complete component (or strong alliance by Freeman): consists of ties between all pairs of members within the subgroup, and no ties from subgroup members to actors not in the subgroup. - LS sets: An LS set is a subgroup definition that compares ties within the subgroup to ties outside the subgroup by focusing on the greater frequency of ties among subgroup members compared to the ties from subgroup members to outsiders.consider the subgraph Gs with the node set Ns, and the subsets of nodes that can be taken from Ns. We will define a subset of nodes taken from Ns as L, so that L Ns. The set of nodes, Ns, is an LS set if any proper subset L Ns has more lines to the nodes in Ns- L (other nodes in the subset) than to N-Ns (nodes outside the subset). Pp269 Summary of LS sets: 1) relatively robust, and do not contain splinter groups 2) - Lambda Sets: in a given graph, any two LS sets either are disjoint or one LS set contains the other. 3) within a graph there is a hierarchical series of LS sets. The idea is that a cohesive subset should be relatively robust in terms of its connectivity. (Borgatti, Everett, and Shirey 1990)pp269 A cohesive subset should be hard to disconnect by the removal of lines from the subgraph. Pp269. Edge connectivity (line connectivity): the extent to which a pair of nodes remains connected by some path, even when lines are deleted from the graph. (pp269) Notion: The line connectivity of nodes i and j, denoted (i,j), is equal to the minimum number of lines that must be removed from the graph in order to leave no path between the two nodes.

5 Lambda Sets: A lambda set is a subset of nodes, Ns N, such that for all i,j,k Ns, and ln-ns, (i,j) >(k,l) Summary of Lambda Sets:1) no over lapping only containing 2) more general than LS sets. Any LS set in a network will be contained within a lambda set. 3) A given network is more likely to contain lambda sets than it is to contain LS sets. Pp270 4) no need for members of a lambda set to be adjacent, they may be quite distant from one another in the graph. 2. How to assess the cohesiveness of network subgroups 2.1 Bock and Husain (1950) :iteratively to construct subgroups so that the ratio of the strength of ties within the subgroup to ties between subgroups does not decrease appreciably with the addition of new members. If there are g members in the whole network, and g s members in a subgroup Ns, then: in s g s in s s g s jns 1 jns g g g s x x ij ij The numerator of this ratio is the average strength of ties within the subgroup and the denominator is the average strength of the ties that are from subgroup members to outsiders. The ratio =1, >1 or <1. For a dichotomous relation the numerator is the density of the subgroup. For a valued relation the numerator is the average strength of ties within the subgroup. 2.2 Probability of obtaining the density of a subgroup equal to or greater than the observed density, given the density of the graph. H0: There is no difference between the density of the subgraph and the density of the graph as a whole. The probability of observing q or more lines in a subgraph of size g s form a graph with L lines is:

6 P L s q gg ( 1) L L 2 k g ( g 1) K 2 gg ( 1) 2 gs( gs 1) 2 gs gs1 min( L, ) s s 2 kq If the probability in the equation is small, then the observed frequency of lines within the subgraph is greater than would be expected by chance, given the frequency of lines in the graph as a whole. Thus, this probability can be interpreted as a p-value for the null hypothesis that there is no difference between the density of the subgraph and the density of the graph as a whole. 3. Directional relations and valued relations - Cliques based on reciprocated ties Redefine the directional graph: x min ij x 1 if x ij = x ji =1, min ji 0 otherwise, The relation X min can then be analyzed using methods for finding cliques or other cohesive subgroups in a nondirectional relation. (Note: if there are few reciprocated ties the resulting symmetric relation will be quite sparse, and might not yield many cohesive subgroups.) - N-clliques in directional relations Connectivity in directional relations: Use definition of semipaths and connectivity for directed graphs. Pp274 Semipath: A semipath from node I to node j 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. In a semipath the direction of the arcs is irrelevant. The length of a semipath is the number of

7 arcs in it. Four types of connectivity: A pair of nodes, i, j, is : (i) Weakly n-connected if they are joined by a semipath of length n or less. (ii) Unilaterally n-connected if they are joined by a path of length n or less from i to j, or a path of length n or less from j to i. (iii) strongly n-connected if there is a path of length n or less from i to j, and a path of length n or less from j to I; the path from i to j may contain different nodes and arcs than the path from j to i. (iv) Recursively n-connected if they are strongly n-connected, and the path from i to j uses the same nodes and arcs as the path from j to I, in reverse order. Four different kinds of cohesive subgroups based on the four types of connectivity.pp267 (i) A weakly connected n-clique is a subgraph in which all nodes are weakly n-connected, and there are no additional nodes that are also weakly n-connected, to all nodes in the subgraph. A unilaterally connected n-clique is a subgraph in which all nodes are unilaterally n-connected, and there are no additional nodes that are also unilaterally n-connected to all nodes in the sub-graph. A strongly connected n-clique is a subgraph in which all nodes are strongly n-connected, and there are no additional nodes that are also strongly n-connected to all nodes in the subgraph. A recursively connected n-clique is a subgraph in which all nodes are recursively n-connected, and there aer no additional nodes that are also recursively n-connected to all nodes in the subgraph. Note: Finding weakly connected n-cliques and recursively connected n-cliques requires symmetrizing the relation using the appropriate rule, and then using a standard n-clique algorithm. - Valued Relations In general, a cohesive subgroup of actors in a valued network is a subset of actors among whom ties have high values. Pp278 Cliques, n-cliques, and k-pleses: Clique at level c: a subgraph G s with actor set N s is a chique at level c if for all actors i, j N s, xij c, and there is no actor k, such that kij cfor all i N s (peay 1975a) A path of level c: A path at level c in a valued graph is a path in which all lines have values of c or greater.

8 A n-clique at level c requires that geodesics between subgroup members all of whom are reachable at level c by a pth of length n or less. (Peay 1975a) pp279 AK-plex at level c in a valued relation requires that all g s members of the k-plex have ties with values of c or greater to no fewer than g s k of the other k-plex members. Finding Cliques, n-cliques, and k-plexes: 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. Define ( c ) xij 1 if xij c 0 otherwise 4. Interpretation of Cohesive subgroups (i) Individual actor Members and non-members of cohesive subset. What are the differences between members and non-members. And non-members can also occupy critical locations between groups. (ii) Susbset of actors If the network data set contains information on attributes of the actors, then one can use these attributes to describe the subsets. (iii) Whole group 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. Pp Alternative approaches for studying cohesiveness in networks (multidimentional scaling and factor analysis) - Matrix permutation approache The idea is that if actors who choose each other occupy rows and columns that are close in the sociomatrix, then ties that are present will be concentrated on the main diagonal of the sociomatrix, and ties that are absent will be concentrated far from the main diagonal of the sociomatrix. The goal is to permute the rows (and simultaneously the columns) of the sociomatrix to concentrate choices along the main diagonal (Katz 1947) pp285 For an entire matrix a summary measure of how close large values of xij are to the main diagonal is given by:

9 g g x 2 ij i j for i j i1 j1 If the value of equation is small, then 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. On the other hand, if the value of this quantity is relatively large, then the ordering of rows and columns in the sociomatrix probably is not the best possible ordering for revealing cohesive subgroups of actors. Pp2 Summary: 1) A matrix permutation analysis does not locate discrete subgroups. 2) It can be quite informative to present the sociomatrix with rows and columns permuted to suggest the subgroup structure. - Multidimentional scaling Fore revealing which actors are close to each other, and for presenting possible cleavages between subgroups. Multidimenational scaling: 1) It is a vary general data analysis technique. 2) It seeks to represent proximities among a set of entities in low-demensional space so that entities that are more proximate to each other in the input data are closer in the space, and entities that are less proximate to each other are farther apart in the space. - Factor analysis pp290. Direct factor analysis. Factor analysis of a correlation or covariance matrix derived from the rows of a sociomatrix have been used to reveal aspects of network structure.. Bonacich (1972b) shows that if a group contains non-overlapping subsets of actors in which actors within each subset are connected by either adjacency or paths, then a factor analysis of the sociomatrix will reveal this subgroup structure. One should be quite cautious about using factor analysis on dichotomous data, since results can be quite unstable. Borgatti, S.P. & Everett, M.G. (1999). Models of Core/Periphery structures. Social Networks, 21(4), Abstract:A common but informal notion in social network analysis and other fields is the concept of acore-periphery structure. The intuitive conception entails a dense, cohesive core and a sparse, unconnectedperiphery. This paper seeks to formalize the intuitive notion of a core-periphery structure and suggestsalgorithms for detecting this structure, along with statistical tests for

10 testing a priori hypotheses. Differentmodels are presented for different kinds of graphs_directed and undirected, valued and non-valued.inaddition, the close relation of the continuous models developed to certain centrality measures is discussed. Intuitive concept: 1) A group or networkcannot be subdivided into exclusive cohesive subgroups or factions, although someactors may be much better connected than others. 2) In the terminology of blockmodeling, the core isseen as a 1-block, and the periphery is seen as a 0-block. This is the sense in whichbreiger_1981.uses the terms. The blocks representing ties between the core andperiphery can be either 1-blocks or 0-blocks. 3) 3) Given a map of the space, such asprovided by multidimensional scaling, nodes that occur near the center of the picture arethose that are proximate not only to each other but to all nodes in the network, whilenodes that are on the outskirts are relatively close only to the center. Two models: discrete model and continuous model 1) Discrete model - Definition: In this section we explore the idea that the core periphery model consists of twoclasses of nodes, namely a cohesive subgraph_the core.in which actors are connected toeach other in some maximal sense and a class of actors that are more loosely connectedto the cohesive subgraph but lack any maximal cohesion with the core. - Pattern matrix (ideal matrix): - A simple measure of how well the real structure approximates the idealmodel is

11 In the equations, a indicates the presence or absence of a tie in the observed data, c i refers to the class_core (or periphery) that actor i is assigned to, and (subsequently called the pattern matrix) indicates the presence or absence of a tie in the ideal image.for a fixed distribution of values, the measure achieves its maximum value when andonly when A(the matrix of a) and Δthe matrix of. are identical, which occurs when A has a perfect core/periphery structure. Thus, a structure is a core/peripherystructure to the extent that ρ is large. - Testing a priori partitions If we obtain a partition of nodes into core and periphery blocks a priori, we can useeq. 1. as the basis for a statistical test for the presence of a core/periphery structure.this is precisely the QAP test described by Mantel1967.and HubertandSchultz, 1976; Hubert and Baker, The test is a permutation test for theindependence of two proximity matrices. - Additional pattern matrices 2) Continuous model - Definition: Each node is assigned a measure of coreness. In aeuclidean representation, this would correspond to distance from the centroid of a singlepoint cloud. In this way, we assume the network data consist of continuous values representing strengths or capacities of relationships, an obvious approach is to continue using strengths or capacities of relationships, and obvious approach is to continue using correlation to

12 evaluate fit, but define the structure matrix as follows: Where C is a vector of nonnegative values indicating the degree of coreness of each node. - Estimating coreness empirically The objective is to obtain values of C so as to maximize the correlation between the dta matrix and the pattern matrix associated with Eq: - Coreness and centrality All coreness ar centrality measures, but the converse is not necessarily true. Caldeira, Grogory A. (1998). Legal Precedent: Structures of Communication between State Supreme Courts. Social Networks, 10(1), Abstract: In making and justifying choices, state supreme courts rely on many sources of authority,including the precedents of other courts, To the extent that appellate judges borrow, reject, andreview each other's decisions, the several state supreme courts form a network for the communicationof political information. In this paper I focus on identifying, describing, and explaining thebases of discrete networks among the state supreme courts. More specifically, using all interstatereferences for all states and thedistrict of Columbia in the calendar year 1975, I rely on clusteringtechniques to uncover coherent and consistent "networks," and discriminant analysis to ferret outthe essential bases of the groupings of state supreme courts. Do interpretable blocks of statesupreme courts emerge? If so, what binds these sets of appellate courts? Do constellations ofleaders and followers develop?do networks go beyond particular regions? Data, Methods, and Model - Data: the data come from a count of all references each state supreme court as well as the Court of Appeals for the District of /Columbia made to every other state court of last resort during Methods: Network Analysis: Social cohesive Structural equivalence: a is structurally equivalent to b if a relates to every object x of C in exactly the same way as b does. Discriminant Analysis Where do the ties from (originate)? It is better look at the individual characteristics of the actors. In the paper, the author use discriminanat analysis to find the linear combination of characteristics of courts that best separates the state supreme courts. (membership in particular cliques and positions)

13 Findings: - Social cohesion: Clustering analysis:pp41, software SOCK Multidimentional plot of distances among the state supreme courts Discriminant functions for cliques in a social network

14 The author use eight independent variables: policy innovaqtion, legal capital, prestige, caseload, legal professionalism, population, wealth, and progressivism of a state s public policy to discriminate among the groupings of state supreme courts. - Structure equivalence Position Cluster (by STRUCTURE) The author identify 7 positions, here is the admixture of regionalism, prestige, and other influences in the membership of the seven positions. The basic Interpositional relationships in the network

15 This table is about which positions interact with each other and which sets of courts lead or follow others. Table A is about the densities of communication within and between the blocks or social positions. Table B is the author replace each entry with either a 0 or a 1 according to the density of connections within the cell in table a. The author choose 2.0, the average density of each connection in the original matrix as the cut-off point to make dichotomize the matrix.

16 Everett & Borgatti: Peripheries of Cohesive Subgroups 1 Main Topics Exploring ways of identifying actors that are not members of a given cohesive subgroups, but who are sufficiently well tied to the group to be considered peripheral members Using the information of peripheral members to explore the structure of the network as a whole Key Concepts A network has a core/periphery structure if the network can be partitioned into two sets: a core and a periphery. A core s members are densely tied to each other and a periphery s members have more ties to core members than to each other. In this paper, the conception of a core is equal to a cohesive subgroup, and in particular as a clique. N-cliques: an n-clique is a maximal subgraph in which the largest geodesic distance between any two nodes is no greater than n. An Example: Figure 1 (pp. 402) - The single 1-clique is {c,d,e}. K-plexes: a k-plex is a maximal subgraph containing g s nodes in which each node is adjacent to no fewer than g s k nodes in the subgraph. K-cores: a k-core is a subgraph in which each node is adjacent to at least a minimum number, k, of the other nodes in the subgraph. An Example: Figure 2 (pp. 401) 1 Everett, M.G. and Borgatti, S.P. (1999). Peripheries of Cohesive Subgroups. Social Networks, 21(4):

17 - The set {1,2,3,4,5} is a 2-plex, as each node is connected to at least 5-2 = 3 others in the set. To add node 6 to the set would require adding two ties from node 6 to the set, so that it would have 6 2 = 4 ties to other group members. - The set {1,2,3,4,5} is a 3-core, and the set {1,2,3,4,5,6} is a 2-core. Definition of Peripheries Two extremes of a continuum of possible definitions of peripheries B A - A: Periphery as actors clearly associated with the core - B: Periphery as simply all outsiders A general definition of periphery should be able to encompass all of these possibilities. First Definition : Let G be any graph, and let C be a cohesive subgroups, called a core of G. Then the periphery P is G-C. If ʋ P then we say is in the k-periphery if is a distance less than or equal to k from C. - The measure of distance between a node and a group C is left deliberately undefined, so as to keep the definition completely general. - Both the method of detecting the cohesive subgroup and the distance measure can be determined by the researcher and can therefore be related to the type of data to be analyzed. (ex. shortest distance (nearest neighbor), maximum distance (farthest neighbor), average distance, medium distance, and so on) - It is not necessary that distance to the core should be defined in terms of graph-theoretic distance at all. For instance, a metric based on the extent to which the node is structurally equivalent to the members of the core would do just as well. Second Definition : For ʋ V, if ʋ C then CP (ʋ) = 1, otherwise let q be the minimum number of edges incident with ʋ that are required to make ʋ part of C and let r be the number of those edges that are already incident with ʋ. Then CP (ʋ) = r. We call q CP (ʋ) the coreness of vertex ʋ. - The CP measure varies from 0 to 1. All core actors have a value of 1. - While K-peripheries are defined in terms of distance, the CP measure is defined in terms of volume of ties.

18 P D : Peripheral Degree Index meaning the density of periphery-to-periphery interaction P D (ʋ) = An Example: Figure 3 (pp. 399) Number of peripheral actors connected to ʋ Total number of peripheral actors - Definition 1: Consider the network in Figure 1, taking the clique (1,2,3,4) as the core. All other points in the graph comprise the 1-periphery, even though some of these nodes are better connected to the core than others. - Definition 2: Given that we are using cliques as the basis for identifying the core in Figure 1, to enlarge the core we much supply a vertex that is adjacent to all four of the existing core members. This would mean that q = 4. It follows that in Figure 1 the vertices with degree 1 (6,7,9,10) have CP (ʋ) = 0.25, the vertices with degree 2 (5,8) have CP (ʋ) = 0.5 and all other vertices (those in the core) have CP (ʋ) = 1. - Hence, it is very appropriate that we use distance to define the 1-periphery, and then use density of ties to subdivide the 1-periphery. - In Figure 1, all peripheral actors have P D (ʋ) = 0. If some peripheral actors have higher values than others in a given dataset, they may not view the core as a group to get more involved with. Concerning the density of the peripheral actors as a group, a low peripheral density for a 1-periphery could indicate an elite structure in which the periphery members only wish to have contacts with the elite core. An Alternative Approach One of the problems with the 1-periphery is that it can be quite large. We can use the values of CP and P D to restrict the size of the periphery, but these values do not give any insight or restriction on the structure of the periphery. An Alternative Approach 1: The generalization of cliques that contain parameters that can be adjusted so as to relax the conditions of membership An Conceptual Example: Figure 4 (pp. 402)

19 - The single 1-clique is {c,d,e}. - The two 2-cliques that include {c,d,e} are {b,c,d,e} and {c,d,e,f}. To reduce the size of the corresponding peripheries, we consider the 2-clique periphery to be {b,f}, which are the two nodes in the inclusive 2-cliques that are not in the 1-clique. A Practical Example: Taro data reported in Hage and Harary (1983) Table 1 Taro cores and peripheries (pp. 403) 2-plexes 4-plex-periphery 1-periphery This dataset looks at the relation of gift-giving among 22 households in a Papuan village. - Table 1 lists the 2-plexes with four or more actors. We shall use these as the cores. Each core is simply considered in turn and is the focus of an individual analysis. - For each core we find the associated 4-plexes (i.e., the 4-plexes which contains the 2-plex as a subset) and list the vertices contained in the 4-plexes which are not in the 2-plex. This we label as the 4-plex periphery. - The third column gives the 1-periphery of the core as defined previously. We can see immediately by looking at the second two columns that this method has the desired effect of reducing the size of the corresponding peripheries. This is principally because we are only identifying those actors that are close to becoming core members, whereas the 1-periphery includes highly peripheral actors. Note that the elements in the 4-plex periphery could not have been identified by using the CP or PD values as filters. This is because these measures are calculated independently for each actor. An Alternative Approach 2: The idea of k-cores - Definition: a k-core is a connected maximal induced subgraph which has minimum degree greater than or equal to k. (Seidman, 1983)

20 - They may not be cohesive themselves, but any cohesive structures within the network must be contained within them. - Seidman describes them as seedbeds for cliques or other cohesive structures. An Empirical Example: Zachary data. - The largest value of k that has a k-core containing the 2-plex is 4, the 4-core is {1,2,3,4,8,9,14,31,33,34} which yields a periphery consisting of {9,31,33,34}. - This immediately demonstrates the problem of using k-cores, since this group is actually another clique that is located close to the core 2-plex. - Clearly the k-core model is useful in identifying peripheries, but care must be taken as the k-cores contain both cores and peripheries and it may be difficult to separate out individual actors into appropriate groups. Completing the picture An individual actor can be in more than one core and more than one periphery. Clearly cores that overlap sufficiently could be merged into single cores. These observations suggest that a 2-mode analysis of the data based upon membership in the cores and peripheries may help provide a better picture of the overall pattern of connections in a social network. CP matrix: a 2-mode matrix in which the rows are actors, the columns are cores, and the cells indicate the relationship the coreness of the row actor with respect to the column core. Table 2 CP matrix for cliques of Taro data (pp. 407) Actor C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 R R R R R R R R R R R R R R R R R

21 R R R R R A clique analysis of the Taro data yields 10 cliques each containing three actors. - Since each clique contains exactly three actors the CP scores for each actor with relation to each clique are either 0, 0.33, 0.66 or 1.0, as shown in Table 2. The CP matrix was then submitted to the correspondence analysis procedure in UCINET 5 for Windows. The resulting plot is given in Figure 5 (pp.405). Figure 5. Correspondence analysis of Taro CP matrix. - The points labeled with an R correspond to the actors, the points labeled with a C correspond to the core/periphery structures. - Three or four basic groups are visible, suggesting that some of the core/periphery structures are essentially the same. - The map clearly indicates which actors are associated with which aggregated core/periphery complexes.

22 Directed graphs Although we have developed these methods with undirected graphs in mind, the concepts can be extended fairly easily to the case of directed graphs. We can think of a periphery as the union of two peripheries: an out-periphery and an in-periphery. The out-periphery consists of all the actors that receive arcs from the core while the in-periphery consists of all the actors that send arcs to the core. With directed data we can also define the concept of a strong periphery, which is the intersection of the in-periphery and out-periphery. Similarly, concepts such as peripheral density can be extended to in-periphery density and out-periphery density. Frank: Identifying Cohesive Subgroups2 Main Topics This article applies recent advances in the statistical modeling of social network data to the task of identifying cohesive subgroups from social network data. Through simulated data, the author describes a process for obtaining the probability that a given sample of data could have been obtained from a network in which actors were no more likely to engage in interaction with subgroup members than with members of other subgroups. By obtaining the probability for a specific data set, through further simulations, the article develops a model which can be applied to future data sets. Through simulated data, the author characterizes the extent to which a simple hill-climbing algorithm recovers known subgroup memberships. The author applies the algorithm to data indicating the extent of professional discussion among teachers in a high school, and he shows the relationship between membership in cohesive subgroups and teachers' orientations towards teaching. Introduction Cohesive subgroups have been a crucial link between individuals and organizations; individuals are most strongly influenced by the members of their primary groups; and primary groups are integral to understanding people within the contexts of their communities. 2 Frank, Kenneth A. (1995). Identifying Cohesive Subgroups. Social Networks, 17(1):

23 Since actors are directly influenced by their interpersonal interactions, subgroups based on the pattern of interaction are more likely to be related to the sentiments and actions of actors than subgroups based on formal positions. Methodologists have developed and employed various techniques for identifying cohesive subgroups from data indicating the extent of interactions between each pair of actors. The author contends that probably the most promising approaches for identifying non-overlapping cohesive subgroups are those which utilize goodness of fit, or statistical criteria, associated with the fitting of subgroups to social network data. Cohesive Subgroups and Stochastic Blockmodels The statistical models developed for sociometric data (or directed graphs) have been applied to the task of identifying blocks of stochastically equivalent actors. This approach is important because it can be applied to identification of cohesive subgroups (Wasserman and Anderson, 1987). This application can start by the p 1 model (Holland and Leinhardt, 1981). For n actors and n (n 1) possible pairs of actors i and i, define: X ii = 1 if actor i chooses (in a sociometric sense), or initiates an exchange with, i 0 otherwise. Exchange is referred as a form of interaction which is captured by social network data. Holland and Leinhardt (1981) define X to be the matrix of all X ii, and describe it as a random variable. Now define as the realization of one of the possible matrices of. The model is then defined as p 1 = P(X = x) = K exp* pm + θx ++ + i α i x i+ + i β i x +i + (1) where x ++, x i+, and x +i are computed from the marginals of. The number of reciprocated or symmetric or mutual relationships also is computed from the marginals of x and is represented by m. The parameter p represents the extent of reciprocity in the network, θ the density of the network, α is the expansiveness of the actors (associated with out-degree, x i+ ) and β is the attractiveness of the actors (associated with in-degree, x +i ). Assuming that actors are partitioned into G subgroups B 1, B 2,, B G. The p 1 models then can be expanded to include parameters, λ gh, representing exchanges within and between subgroups.

24 p 1 (x λ) = K exp* pm + θx ++ + i α i x i+ + i β i x +i + g,h λ gh x ++ (B g B h )+ (2) where denotes the vector of block parameters λ g,h, and x ++ (B g B h ) is the observed count of exchanges in the B g B h block. When presenting a single subset of blocks, B g B g, and a single parameter, λ c ( c for cohesion), associated with the sum of within-subgroup exchanges, λ c x ++ (B g B g ) p 1 (x λ) = K exp* pm + θx ++ + i α i x i+ + i β i x +i + λ c x ++ (B g B g )+ (3) The number of exchanges in the subgroup is large relative to the subgroup size. Therefore, the author replace the αs, βs, and p with a term associated with the subgroup sizes in terms of the number of pairs which are contained by subgroup boundaries (B g B g ). The new model becomes p 1 (x λ) = K exp* θx ++ + γ(b g B g ) + λ c x ++ (B g B g )} (4) This log-linear model includes a main effect for the overall extent of exchange, associated withx ++, a main effect for the number of pairs contained by subgroup boundaries, associated with (B g B g ), and an effect based on the interaction of x ++ (B g B g ). Therefore, the model can be restated at the level of the pair of actors i and i' in terms of the logit model (Agresti 1984): log ( P,X ii c =1 X ii - ) = θ P,X ii =0 X c 0 + θ 1 samegroup ii - ii (5) where samegroup ii = 1 if actors i and i are members of the same subgroup, 0 otherwise. Model (5) can be equated with p 1 model by defining four new quantities. - The first is X + ii which is the full sociomatrix when X ii is forced to be 1. - The second is X ii which is the full sociomatrix when X ii is forced to be 0. c - The third is X ii which is the complement of X ii, and represents the entire matrix X, except for the cell (i, i') - Last, z(x)is defined as a vector of network statistics. The p 1 models can be defined as log ( P,X ii c =1 X ii - ) = P,X ii =0 X c ii - Θ,z(X + ii ) z(x ii )- (6)

25 By multiplying Θ,z(X + ii ) z(x ii )-, the model (6) can be re-expressed as log ( P,X ii c =1 X ii - ) = θ P,X ii =0 X c 0 + θ 1 (if i and i' are in the same subgroup) (7) ii - This right-hand side is precisely the same as in (5). Strauss and Ikeda (1990) show that maximizing the likelihood for model (7) is equivalent to maximizing the pseudo-likelihood function PL(θ) = Pr (X ii = 1 X c ii ) X ii Pr (X ii = 1 X ii c ) (1 X ii ) i =i (8) which expresses the likelihood conditioning on X c ii and therefore does not assume independence of pairs in X. But one can maximize the likelihood in model (5) to obtain estimates of θ 0 and θ 1 without assuming independence of pairs. Clustering to Maximize θ 1 When defining G 2 to be the likelihood ratio statistic which measures the adequacy of the fit of a given log-linear or logit model to sociometric data, one natural approach to identifying cohesive subgroups is to maximize the change in G 2 between a model which includes θ 1 (e.g. model 5) and an identical model except for the removal of the term involving θ 1, such as log ( P,X ii =1-1 P,X ii =1- ) = θ 0 (9) Alternatively, since the focus is on θ 1, one can identify cohesive subgroups by assigning actors so as to maximize θ 1. Unlike G 2, θ 1 is invariant with respect to network size, has a range from - to and has a readily understandable interpretation. Typically, the subgroup memberships are not known. When defining to be the lower triangular portion of an n n matrix, with ω ii = 1 if actors i and i' are in the same subgroup, 0 otherwise, and the restriction that if ω ij = 1 and ω jk = 1 then ω ik = 1. In effect, is a block-diagonal 'target' matrix indicating subgroup membership, and ω ii is considered as entities which a clustering algorithm estimates. By estimating Ω, latent subgroups can be identified so as to maximize θ 1 Ω., X. In this sense, the subgroup memberships are 'estimated' and so is θ 1, which is referred to as θ 1, where θ 1 is half the log-odds of the cells in Table 1 based on X and Ω. The hill-climbing algorithm

26 - Given a criterion for evaluating the extent to which subgroups are cohesive, θ 1, a simple hill-climbing, or iterative partitioning, algorithm have been employed to identify cohesive subgroups. - Everitt (1986) showed that this type of algorithm will converge on a local maximum in the criterion, and, in Appendix A, the author restates this procedure in terms of the E-M (expectation-maximization) algorithm, which would ensure convergence to maximum likelihood estimates of θ 1 and. Testing Whether There Are Underlying Subgroup Processes: Evaluating the Internal Validity of θ 1 Cohesive subgroups could be salient even if there is reciprocity beyond that within subgroup boundaries. Therefore, instead of testing whether θ 1 is different from zero or whether there is reciprocity not captured by subgroup boundaries, it may be more sensible to test whether θ 1 is greater than θ 1base ; where θ 1base is the value of θ 1 associated with subgroups identified by the algorithm when applied to data which are generated without regard for subgroup memberships. This can be tested through the change in G 2 between the following two models compared to a χ 2 distribution on one degree of freedom: log ( P,X ii =1-1 P,X ii =1- ) = θ 0 + θ 1base samegroup ii (10) and log ( P,X ii =1-1 P,X ii =1- ) = θ 0 + θ 1base samegroup ii + θ 1 subgroup processes samegroup ii (11) The null hypothesis is that θ 1 subgroup processes = 0 Specific case: Simulation customized to each new data set - The author estimates θ 1base and tests whether θ 1 subgroup processes = 0 by simulating a data set in terms of teachers. General case: Predicting θ 1base - The author has simulated data according to the experimental design which allows him to predict θ 1base based on characteristics of the network. In so doing, he estimates the value which could be used as the basis for testing whether θ 1 subgroup processes is different from zero. Performance of the Algorithm: Evaluating Ω The iterative partitioning algorithm proceeds linearly until it identifies a local maximum in θ 1. But there are more sophisticated algorithms which do not

27 proceed linearly as they seek to identify a superior local, or even the global, maximum. The iterative partitioning algorithm could be extended through optimal annealing procedures by introducing the possibility of random re-assignment of actors as θ 1 approaches a local maximum (Hajek 1988). One can also identify several local maxima by initiating the algorithm based on random subgroup assignments, and then, upon convergence, initiating a second application of the algorithm based on a different set of random subgroup assignments. In order to address whether or not the algorithm recovers the true subgroup memberships (the set of assignments associated with the global maximum), the author simulates sociometric data sets (X) based on varying values of the concentration of exchanges within subgroups (θ 1 subgroup processes ), network size ( ), maximum subgroup size (Max (n g )), and maximum number of exchanges any actor initiated (C max ). - First, he describes experimental design for estimating linear effects of network size, subgroup sizes, maximal number of exchanges initiated, and concentration of exchanges within subgroups on recovery of subgroup memberships. - Then, X and have been generated, and an outcome measure has been identified. - Results: a simple hill-climbing algorithm performs quite well in recovering known subgroup memberships in simulated data. Interpretation of Subgroup Memberships The author has applies the methods of identifying the cohesive subgroups to a dataset of teachers. By going this, he has confirmed that teachers engage in exchanges within identifiable subgroup boundaries, and that the boundaries are likely to be similar to those associated with the global maximum in θ. In section 6, the author interprets the subgroup memberships with respect to the context of the school and the characteristics of the teachers. See the below table 1. Table 1 Characteristics of teachers by subgroup Teacher Race Gender First subject field Moral agency Subgroups = A/1 2 Black Female English Black Female Business Black Female Business Black Female Social Science Subgroups = B/2 15 Black Male Maths

28 14 White Male Music White Male Physical Black Male Physical ** White Male Physical Subgroups = C/3 13 White Female Other White Female English White Male Other White Male Maths Subgroups = D/4 17 Other Male English White Female Foreign White Female Special Education White Female Science Other Female Special Education Subgroups = E/5 3 White Male Social Science White Male Science White Male Maths White Male Science White Male Maths White Male Other In Table I, it is clearly observed that teachers are more likely to be members of a subgroup with others of the same race (a χ 2 test of independence and Fisher's exact test produced p < 0.01). Teachers also are aligned according to gender (again, a χ 2 test of independence and Fisher's exacts test produced p < 0.01). Moral agency indicates each teacher s tendency toward being a moral agent. This measure was one of four based on each teacher's extent of agreement (1 = strongly disagree to 4 = strongly agree) with items referring to the teacher's sentiments and orientation towards teaching. - The teachers do not uniformly emphasize the moral agency orientation. - A Kruskal-Wallis test revealed that the probability of achieving the differences in moral agency among the subgroups by chance alone was Using Tukey's honestly significant differences, the members of subgroup A were significantly higher on moral agency than were the members of subgroup C. - Note that this difference cannot be attributed to subject field, since both subgroups contain teachers of various fields, including English. - The black female teachers of subgroup A, one of whom was married to the principal, took the principal's sense of mission more seriously than did other

29 teachers in the school (especially those in subgroup C), many of whom discounted the principal as weak and equivocating. - This is reflected in the fact that, when interviewed, the teachers of Subgroup A perceived their role to be more that of 'saving' students, than did the teachers in subgroup C. While from these data we cannot tell whether the teachers formed their subgroups based on common sentiments and actions, or whether the sentiments and actions emerged as a result of within-subgroup discussions, this type of cohesion-based subgroup analysis is important to understanding the sentiments and actions of teachers within the context of their school organization. Moore: The Structure of a National Elite Network3 Main Topics This paper addresses a long-disputed issue: the degree of integrating among political elites in the United States. This issue is examined through an investigation of the structure of an elite interaction network as revealed by recently developed procedures for network analysis. The data, taken from the American Leadership Study conducted by the Bureau of Applied Social Research in 1971 and 1972, consist of interviews with 545 leaders of major political, economic and social institutions. The study s wide institutional representation, sociometric data, and focus on major issues of the early 1970s make it virtually unique for examining elite integration. Introduction Ruling class and power elite theorists such as Mills and Domhoff find a considerable amount of integration, with various bases, in the national power structure. Pluralists find little integration among elites in diverse sectors. Elites are seen as fragmented rather than integrated since each is involved primarily with its own relatively narrow concerns and constituencies. This study assesses the extent of integration in a network of political elites in the United States. The concept of political elite integration has several dimensions, at least, including social homogeneity, value consensus and personal interaction. 3 Moore, Gwen. (1979). The Structure of a National Elite Network. American Sociological Review, 44: