Cohesion & Subgroups

Cohesive subgroups

Durkheim (social solidarity)1) Ideological: (Common Consciousness)2) Relational: (Structural Cohesion)

Groups that are held together wellGroups should have connectednessCohesion = a field of forces that keep people in the groupresistance of the group to disruptive forcessticking together

Social group and subgroupFriedkin (1984):(uniformity)

Social group and subgroupCollins (1988):: (1)(2) (clique)

Social group and subgroup(social forces)

Social group and subgroup(1) (mutuality of ties)(2) (closeness)(reachability)(3)(the frequency of ties)(4)(the relative frequency of ties among subgroup members compared to non-members)

Social group and subgroup(1) (mutuality of ties)(2) (closeness)(reachability)(3)(the frequency of ties)(4)

Connectivity & Cohesion1) Actors must be connected: a collection of isolates is not cohesive. Not cohesiveMinimally cohesive: a single path connects everyone

Connectivity & Cohesion1) Reachability is an essential element of relational cohesion. As more paths re-link actors in the group, the ability to hold together increases. Cohesion increases as # of paths connecting people increasesThe important feature is not the density of relations, but the pattern.

Connectivity & CohesionConsider the minimally cohesive group:Moving a line keeps density constant, but changes reachability.

Connectivity & CohesionWhat if density increases, but through a single person?

Connectivity & CohesionD = . 39More cohesiveD = . 39Minimal cohesion

Connectivity & CohesionSubstantive differences between networks connected through a single actor and those connected through many.Minimally CohesiveStrongly CohesivePower is centralizedPower is decentralizedInformation is concentratedInformation is distributedExpect actor inequalityActor equalityVulnerable to unilateral action Robust to unilateral actionSegmented structureEven structure

Def 2. A group is structurally cohesive to the extent that multiple independent relational paths among all pairs of members hold it together.

GN(graph) RL(adjacent)Gss(Ns)(subset)(subgraph)

(complete graph)(complete subgraph)

(reachable)(lines)(the length of a the shortest path)geodesic

Diametergeodesic

geodesic

(connected)(reachable)

maximal((maximal)(components)(maximal connected subgraph)(disconnected)

Subgroups based on complete mutuality(maximal complete subgraph)(clique)

Subgroups based on complete mutualityThree cliques:{1,2,3}{3,4,5,6}{1,3,5}clique?

cliques?

cliques

cliques?

Knokes information exchange

Knokes information exchange

Clique(1) Too stingy: clique(2) (sparse network)clique(3) cliquekk+1(4)

(1)(2)

(intermediaries)

n-n-cliquesn-n(maximal subgraph)

n=1n-clique

2-cliques:{2,3,4,5,6} {1,2,3,4,5}6n-clique2

Florentine marriage newtork

Florentine marriage newtork

Knokes information exchange

{1,2,3,4,5}2-cliques>2 n-cliquenn-cliquen-clann-clann-

n-clann-n-cland(i, j)nn-cliquen-clique

Strozzi Castellani Barbadori

(direct contact)(adjacency)

EricksonSeidman and Fostern-clique(robust)(robust)(vulnerable)

(robust) vs. (vulnerable)2-clique3

1234

k-plexk-plex gs(gs-k)(maximal subgraph)

ds(i)Gsi

k-plexk=11-plex = clique, gs-1kk-plexk-plexclique

K-plexK-plexkSediman and Fosterk

Ridolfi

k-corek-k-kk-plexk-coreGsk-core

k-corek-

K-core2-core

Salviati Salviati2-core

Strength, Frequency, Density, ClosenessWeakness, infrequency, sparseness, distance

NactorRGGsNNsNtNNsNtNsNNtNsNt = N Ns

(1) Ns(2) Nt(3)NsNt

NgNsgsNtgtg(g-1)/2Nsgs(gs-1)/2NsNt(gsgt)/2

Complete component(strong alliance)cliquecomplete maximalclique

Lambda sets(relatively robust in terms of its connectivity)

line-connectivity (G)(G)ll-

line-connectivityij(i, j)(line-disjoint or line-independent paths)(lines in common)(i, j)ijBorgattilambda set

Connectivity & CohesionFormalize the argument:If there is a path between every node in a graph, the graph is connected, and called a component.

In every component, the paths linking actors i and j must pass through a set of nodes, S, that if removed would disconnect the graph.

The number of nodes in the smallest S is equal to the number of independent paths connecting i and j.

Connectivity & CohesionComponents and cut-sets:Every path from 1 to 8 must go through 4. S(1,8) = 4, and N(1,8)=1. That is, the graph is a component.

Connectivity & CohesionComponents and cut-sets:In this graph, there are multiple paths connecting nodes 1 and 8.125836784678458But only 2 of them are independent.158123678N(1,8) = 2.

Connectivity & CohesionThe relation between cut-set size and number of paths leads to the two versions of our final definition: Def 3aA groups structural cohesion is equal to the minimum number of actors who, if removed from the group, would disconnect the group.Def 3bA groups structural cohesion is equal to the minimum number of independent paths linking each pair of actors in the group.These two definitions are equivalent.

Lambda setsNslambda setlambda setlambda set lambda set

Lambda sets(i, j)ijlambda set

disconnected

disconnecteddisconnected

Other measureBock & Husain (1950) Alba (1973)Iteratively to construct subgroups so that the ratio of the strength of ties within the subgroup to ties between the subgroups does not decrease appreciably with the addition of new members.

Other measureCohesive subgroup of actors can be constructed by successively adding members to an existing subgroups, so long as the additional members do not greatly decrease the value of this ratioAverage strength of ties within the subgroup. Centripetal: cohesiveness of a subgroupAverage strength of ties that are from subgroup members to outsiders. Centrifual: sparsity of ties to actors outside the subgroup

Subgroup Subgroup based on complete mutuality cliqueSubgroup based on reachability and diameter n-cliques, n-clanSubgroup based on nodal degree k-plexes, k-coresSubgroup based on within/outside subgroup comparison lambda sets

(1) (symmetrize)

6 cliques found.

1: 4 12 17 2: 12 17 21 3: 5 11 17 4: 11 15 19 5: 1 4 12 6: 5 11 19

connectivity in directional relationsPeay (1980): I and j isweakly n-connected: ijn(semipath)Unilaterally n-connected: ijnij (path) jiStrongly n-connected: ijnij (path) jiRecursively n-connected : strongly n-connected, ijji

connectivity in directional relationsijA weakly connected n-cliqueA unilaterally connected n-cliqueA strongly connected n-cliqueRecursively connected n-clique