Cohesion. Relational and Group. -0- Copyright 2006 Steve Borgatti. All rights reserved.

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3 Ways to Approach This Many ways to define or theoretically conceive of cohesion cohesion outcome What is the mechanism that would relate cohesion to the outcome of interest? Define cohesion consistent with this mechanism For each way, we can then devise an operational measurement Don t confuse the measure with the construct 2 Copyright 26 Steve Borgatti. All rights reserved.

4 Adjacency & Strength of Tie Raw dyadic data Positive ties Guttman scale of social closeness or obligation Valued relations Frequency of interaction Duration of relation Intensity 3 Copyright 26 Steve Borgatti. All rights reserved.

5 Multiplexity Multiplexity is often what is meant by relational embeddedness As in economic ties being embedded in social ties Combination of (the right set of) ties can be seen as yielding greater closeness than just one tie 4 Copyright 26 Steve Borgatti. All rights reserved.

6 Embedded Ties A tie (u,v) is structurally embedded if there exists node p (possibly several nodes) such that (u,p) œ E and (v,p) œ E I.e, then endpoints u and v have friends in common 5 Copyright 26 Steve Borgatti. All rights reserved.

7 Simmelian Ties Krackhardt s definition: A dyad has a simmelian tie if it is reciprocal ties to each other and to third parties The value of a simmelian tie is the number of third parties they have in common Ideally, it is the number of cliques they have in common 6 Copyright 26 Steve Borgatti. All rights reserved.

8 Reachability If there exists a path from u to v of any length, then v is said to be reachable from u The reachability matrix R in which rij = if I can reach j records the relational cohesions in the graph Is a weak form of cohesion minimal in fact Can define a weak form of Simmelian ties on the reachability graph 7 Copyright 26 Steve Borgatti. All rights reserved.

9 Geodesic Distance Adjacency Geodesic Distance a b c d e f g h i j a b c d e f g h i j a a b b c c d d e e f f g g h h i i j j More nuance in the representation of nonconnection 8 Copyright 26 Steve Borgatti. All rights reserved.

12 Independent Paths A set of paths is nodeindependent if they share no nodes (except beginning and end) They are lineindependent if they share no lines S T 2 nodeindependent paths from S to T 3 lineindependent paths from S to T Copyright 26 Steve Borgatti. All rights reserved.

13 Connectivity Line connectivity λ(s,t) is the minimum number of lines that must be removed to disconnect s from t Node connectivity κ(s,t) is minimum number of nodes that must be removed to disconnect s from t S T 2 Copyright 26 Steve Borgatti. All rights reserved.

14 Menger s Theorem Menger proved that the number of line independent paths between s and t equals the line connectivity λ(s,t) And the number of nodeindependent paths between s and t equals the node connectivity κ(u,v) 3 Copyright 26 Steve Borgatti. All rights reserved.

15 Maximum Flow If ties are pipes with capacity of unit of flow, what is the maximum # of units that can flow from s to t? Ford & Fulkerson show this was equal to the number of lineindependent paths S T 4 Copyright 26 Steve Borgatti. All rights reserved.

17 Measures of Group Cohesion Density & Average degree Average Distance and Diameter Number of components Fragmentation Distanceweighted Fragmentation Cliques per node Connectivity Centralization Core/Peripheriness Transitivity (clustering coefficient) 6 Copyright 26 Steve Borgatti. All rights reserved.

18 Density Number of ties, expressed as percentage of the number of ordered/unordered pairs Low Density (25%) Avg. Dist. = 2.27 High Density (39%) Avg. Dist. =.76 7 Copyright 26 Steve Borgatti. All rights reserved.

21 Average Degree Average number of links per person Is same as density*(n), where n is size of network Density is just normalized avg degree divide by max possible Often more intuitive than density Density.47 Avg Deg Copyright 26 Steve Borgatti. All rights reserved. 23 Density.4 Avg Deg 4

26 F Measure of Fragmentation Proportion of pairs of nodes that are unreachable from each other F = 2 i> j n( n ) r ij = if node i can reach node j by a path of any length r ij = otherwise If all nodes reachable from all others (i.e., one component), then F = If graph is all isolates, then F = r ij 25 Copyright 26 Steve Borgatti. All rights reserved.

27 Computation Formula for F Measure No ties across components, and all reachable within components, hence can express in terms of size of components F = k s k ( s k n( n ) ) Sk = size of kth component 26 Copyright 26 Steve Borgatti. All rights reserved.

29 Heterogeneity/Concentration Sum of squared proportion of nodes falling in each component, where s k gives size of kth component: 2 s k H = k n Maximum value is /n Can be normalized by dividing by /n. If we do, we obtain the F measure = 28 Copyright 26 Steve Borgatti. All rights reserved. F k s k n( n ( s k ) )

31 Breadth DistanceWeighted Fragmentation Use average of the reciprocal of distance letting / = Bounds B = i, j n( n ) lower bound of when every pair is adjacent to every other (entire network is a clique) upper bound of when graph is all isolates d ij 3 Copyright 26 Steve Borgatti. All rights reserved.

32 Connectivity Line connectivity λ is the minimum number of lines that must be removed to disconnect network Node connectivity κ is minimum number of nodes that must be removed to disconnect network S T 3 Copyright 26 Steve Borgatti. All rights reserved.

33 Transitivity Proportion of triples with 3 ties as a proportion of triples with 2 or more ties Aka the clustering coefficient A B D C E T {C,T,E} is a transitive triple, but {B,C,D} is not. {A,D,T} is not counted at all. cc = 2/26 = 46.5% 32 Copyright 26 Steve Borgatti. All rights reserved.

35 Core/Periphery Structures Does the network consist of a single group (a core) together with hangerson (a periphery), or are there multiple subgroups, each with their own peripheries? C/P struct. Clique struct. 34 Copyright 26 Steve Borgatti. All rights reserved.

38 Blocked/Permuted Adjacency Matrix CORE PERIPHERY CORE PERIPHERY Corecore is block Coreperiphery are (imperfect) blocks Peripheryperiphery is block 37 Copyright 26 Steve Borgatti. All rights reserved.

39 Idealized Blockmodel CORE PERIPHERY CORE PERIPHERY c i = class (core or periphery) that node i is assigned to if c = C O R E o r c = C O R E δ ij i j = otherw ise 38 Copyright 26 Steve Borgatti. All rights reserved.

40 Partitioning a Data Matrix Given a graphmatrix, we can randomly assign nodes to either core or periphery Search for partition that resembles the ideal 39 Copyright 26 Steve Borgatti. All rights reserved.

41 Assessing Fit to Data a ij = cell in data matrix c i = class (core or periphery) that node i is assigned to if c = C O R E o r c = C O R E δ ij i j = otherw ise ρ = a ij δ ij i, j A Pearson correlation coefficient r(a,d) is b tt 4 Copyright 26 Steve Borgatti. All rights reserved.

44 Continuous Model Xij~ CiCj Strength or probability of tie between node i and node j is function of product of coreness of each Central players are connected to each other Peripheral players are connected only to core 43 Copyright 26 Steve Borgatti. All rights reserved.

46 CP Structures & Morale Core/Peripheryness Study by Jeff Johnson of a South Pole scientific team over 8 months C/P structure seems to affect morale 3 25 Group Morale Month 45 Copyright 26 Steve Borgatti. All rights reserved.

Centralization Degree to which network revolves around a single node

NODE CONNECTIVITY AND CONDITIONAL DENSITY D. White and F. Harary. Definitions of Cohesive Sets in Social Networks

NODE CONNECTIVITY AND CONDITIONAL DENSITY D. White and F. Harary Definitions of Cohesive Sets in Social Networks The connectivity K (G) kappa of a connected graph G is the minimum number of nodes needed