Hyo-Shin Kwon & Yi-Yi Chen

Hyo-Shin Kwon & Yi-Yi Chen

Wasserman and Fraust (1994)

Two important features of affiliation networks The focus on subsets (a subset of actors and of events) the duality of the relationship between actors and events.

Application and Rationale First, individuals affiliations with events provide direct linkage between the actors and/or between the events. (Most common!) Second, contact among individuals who participate in the same social events provides conditions under which pairwise ties among individuals become more likely. Third, the interaction between actors and events as a social system that is important to study as a whole.

Density Reachability, Connectedness, and Diameter Affiliations create connections both between actors through membership in events, and between events through shared members. A useful way to study reachability in an affiliation network is to consider the bipartite graph, with both actors and events represented as nodes(see slide 8). If all pairs of nodes (both actors and events) are reachable, the affiliation network is connected. And, the diameter of an affiliation network is the length of the longest path between any pair of actors and/or events. Cohesive Subsets of Actors or Events clique at level c

Fraust (1997)

The inherent properties of affiliation networks The presence of two modes The duality of actors and events The non-dyadic affiliation relation

Degree Eigenvector Closeness Betweenness Flow betweenness

Give centrality indices for both actors and events Be extendable to subsets of actors and events Focus on the linkages between actors and events through overlapping memberships Capture subset-superset inclusion relations between actors and events

Degree centralities The degree centrality of an actor is equal to the sum of the sizes of its event. The degree centrality of an event is equal to the sum of the number of memberships of its actors.

Eigenvector centrality One-mode network othe strength of the actor's ties to other network members othe centrality of these other actors Two-mode network othe duality between actor and event centralities

Closeness centrality Closeness centrality is the length of any shortest path between the nodes and is not applicable to valued relations. The closeness centrality of an actor is a function of the minimum distances to its events. The closeness centrality of an event is a function of the minimum distances to its actors. Since a two-mode network consists of valued relations, closeness centrality deserves further investigation.

Betweenness centrality Betweenness centrality for a one-mode dyadic network focuses on the extent to which actors sit on geodesic paths between other pairs of actors. Flow betweenness centrality An extension of betweenness centrality that is applicable to valued relations For a pair of actors, the value of the relation might be "their amount of interaction, the time they spend in one another's company, the range of different social settings in which they interact..." (Freeman et al., 1991, p. 145). This concept, however, still deserves further investigation.

It is important to have centrality scores for both actors and events in an affiliation network. Affiliation relation is non-dyadic, and thus focuses on subsets. In an affiliation network, actors create linkages between events, and events create linkages between actors Subset-superset relationships between actors' affiliations and events' memberships capture the distinction between 'primary' and 'secondary' actors and events. One is not necessarily justified in drawing conclusions about collections larger than pairs from the one-mode relations.

Laumann and Knoke (1987)

Two assumptions underlie the research design. First, corporate entities are the key policy-domain actors. Second, it is assumed that supraindividual structural arrangements among these corporate entities must be taken into account in formulating an adequate explanation of policy domain event participation.

Policy Domains Domain Membership Structural Relations The Policy Process Problem Recognition Option Generation Agenda Placement Events and Scenarios

The essence of the conceptual framework for the analysis of policy decision making is a structural complex that connects consequential organizational actors with a set of temporally arrayed policy events. How is the structure of events interfaced with the structure of actors in their policy participation (cell d)? To understand how national policy unfolds, one must take into account how organizations perceive and respond to an opportunity structure for effecting policy outcomes that is created by the temporal sequence of policy-relevant events (Matrix B).

Yi-Yi Chen 3.22.2011

Bearden & Minz (1987) Identifying sources of business cohesion from status and institution Series of categorical comparisons -- Affiliation data without social network analysis tools Breiger (1974) Duality & transitivity: converting interpersonal, intergroup, and person-togroup networks Simplifying intergroup data to identify subgroups of women Biggest group is not so crucial!

Borgatti & Everett (1997) ways of applying traditional network analytic techniques to 2-mode data based on bipartite graphs correspondence analysis, MILEN Defining density, centrality, centralization, and subgroups in 2-mode data Bonacich (1991) The duality of group and individuals Ways to control size influence in centrality

Bearden, James and Beth Mintz. (1987).

Known: Interlocking directorates, particularly bank board membership, are the source of cohesion in business community. What is the source of influence? 1. Elite status: club membership & policy making group 2. Institution/ organization: interlocking present and retired executive What circumstances which one is important? What are the different role structures of each? In what level (nation/local) does it work best?

1. Corporation ties created by shared directors (interlock) to see the characteristics of individuals uniting business community 2. Director ties created by shared board memberships (are in the same two or more boards) to examine the characteristics of individuals having more than one position in the biggest firms

90% companies are directly tied to at least one other firms. Connectivity is only 3%. 3, 2% #of shared directors >=4, 1% 0, 10% 2, 13% 1, 84%

Businessmen with secondary affiliations are the most crucial in unity. affiliated to large corporations- institution retired executives- class

Smaller businessmen are more likely to sit on the bank boards and help system unity.

component = grouping of directors 13 components identified They are clustered by region! The national and semi-national components play the bridging role among nations. Two-stage process of cohesion formation and labor division: (1)a local consensuses-forming process (2) tied into a national network of class relations.

Directors who are members of important policy making organizations and sit on two corporate boards are more likely to maintain the types of interlock patterns.

Those with 2 positions and no club membership are more likely to be component members, and 74.2% of these directors are either active or retired executives. membership.

Although big linkers are typically elite, the non-elite with 4 or more positions (mostly active executives) are more likely to be grouped in components.

Individuals with two positions are either policy group members, club members from the smaller business, or non-club members who are active or retired executives.

Bank board members with past or present employment in largest corporations are likely to participate in the interlock relationships.

Bank directors are more likely to maintain interlock patterning which creates components.

Nation level is where the social and political segments of corporate directors most closely overlap.

Routes to participate in the highest American capitalism leadership of one of the largest companies social elite combined with leadership of a smaller firm In the corporate network, important individuals are socially elite directors (representing class!), from the smaller business community as well as retired executives of sampled companies (institution!). In the director network, the ties maintained by directors through banks where class and institution interests meet.

Breiger, Ronald L. (1974)

Membership matrix Reflexibility: A person who belong to any group relates to himself by that fact.

Equation 3. P= A(A T ) Equation 4. G= (A T )A The matrix A T of group-to-group ties is equivalent to A except its rows are interchanged or transposed with its columns. That is, A T is of dimensions g times P and A T ij= A ji for any i and j. <application> distinct matrixes P and G are derivable from A

Social bonds individuals as autonomous actor (Goffman) (1) collectivities through membership (2) other individuals through social relations individuals as link in the organic society (Breiger) (3) common membership (4) collectivities through social relationships

person-to-event data collected Convert to person-to-person(connectivity 91%) Simplify data to identify subgroups Make a G matrix listing events and # of their common participants like Figure 3a Delete events without 0 Make a new P matrix based on less events (connectivity 30%) Identify subgroups in the new P matrix

Sometimes asymmetric ties are interesting. When person C s primary affiliation is the secondary affiliation of person D, C is possibly affected by D in the group. Equations of symmetric data can be extended/reasoning as analogous in the asymmetric case.

Borgatti Stephen P & Martin Everett (1997).

social sciences network analysis Analysis unit INDIVIDUALS PAIRS of INDIVIDUALS Variable Data matrix monadic attributes dyadic attributes Person-by-attribute Person-by-person Cases are persons Pairs of actors Variables are attribute Dyadic attribute or relation Way-mode 2 way 2 mode 2 way 1 mode Example Faculty and their demographics Relationships between faculty

Way #1. Correspondence analysis Kind of factor analysis of two-mode data: looking for latent factor (correlation pattern) behind the ties within and between actor; making distances between the points are meaningful. Meaning of distance in graph: The women are placed close together if the women attended mostly the same events. Social events are placed near each other if they were attended by mostly the same women. Women-points are placed near event-points if those women attended those events.

Way #1. Correspondence analysis

Way#2 bipartite graph actor and event as nodes.

Way#3. Combining correspondence analysis and classical graph presentation

Way#4. Computing geodesic distance between in bipartite graph and add to ordinary multidimensional scaling

Way #5. MINLEN, best readability

Normalization formula of network indexes is different for 2-mode data because the denominators (often are the maximum number of ties) are usually smaller and often not linear (different for woman nodes and event nodes)

1-mode data 2-mode data in bipartite graph Density n(n-1) n i n 0 in undirected case 2n i n 0 in directed case Centrality degree k-1 n i or n 0 for women, it s the # of events for events, it s the # of women Closeness n-1 ni+2n0-2 The minimum distance of a node to another mode in the same vertex set is 2. Betweenness Total # of equivalent paths between the 2 nodes For n 0 > n i, 2(n 0-1)(n i -1) For n 0 <n i, n i (n i -1)/2 + (n 0-1)(n 0-2)/2 + (n 0-1)(n i -1) Eigenvector. generally improve centrality scores centrality (1)

One way is to change the denominators in formula. The other way is to use single mode centralization, measuring the extent to which nodes in one vertex are central relative only to other nodes in the same vertex set. suggested!

The subgroup toolkit in UCINET doesn t fit because all nodes of the same type are necessarily two links distant. Special routines for 2 mode data have been developed. biclique: maximal complete bipartite subgraph of a given bipartite graph.

Biplex: maximual bipartite graph with vertex sets V1 and V2 of sizes p and q, where every member of V1 is connected to (q-m) vertices in V2 and very member of V2 is connected to (p-1) members of V1. --a biclique is a (0, 0) biplex.

FACTIONS: block modeling & traditional clustering

Bonacich, Phillip (1991)

Centrality involves the duality of groups and individuals. the measure of centrality takes into account the indirect as well as the direct paths connecting positions. The centrality of each board is a linear function of the centralities of the groups with which it overlaps and the amount of overlap.

Equation (3): Ag=λp Group centralities are a function of the centralities of their members. Equation (4): A t p=λg Individual centralities are a function of the centralities of the groups to which they belong. Equation (5): AA t =λp, A t Ag=λ 2 g The centrality scores are the positive eigenvectors corresponding to this largest eigenvalue.

Centrality measures are strongly affected by the sheer size of the group and the number of groups. Ways to control the effect of size: 1. number of common members adjusted for sizes of groups and individuals modify A t A before using Equation (5), Equation (7) 2. Relative centrality: C - C 1, Equation (9) & Appendix B 3. Correspondence analysis

People whose Centrality increases most with this measure attended few events that attended by many others.

The three largest groups decline in relative centrality because they are larger than they are central.

In correspondence analysis the highest scores for individuals, positive or negative, go to those whose attendance patterns were most characteristic of their cliques. The highest magnitudes for groups, positive or negative, go to those that had the purest pattern of attendance by one clique or another.