index.md (1646B)
1 +++ 2 title = 'Communities' 3 template = 'page-math.html' 4 +++ 5 # Communities 6 Sociogram: graph-like representation of social structure 7 calculate stats like eccentricity, closeness, betweenness centrality 8 9 proximity prestige 10 11 - D is digraph with n vertices 12 - influence domain R-(v) of v is set of vertices from which v can be reached 13 - proximity prestige: (fraction of vertices that can reach v) / (average distance of those vertices to v) 14 15 ranked prestige 16 17 - A is adjacency matrix for digraph 18 - A[v,u] means how much v is appreciated by u 19 20 21 $\sum_{v \neq u} A[v, u] = 1$ for each vertex u 22 23 $p_{rank} (v) = \sum_{u \neq v} A[v, u] \times p_{rank} (u)$ 24 25 $\sum_{v} p_{rank} (v)^2 = 1$ 26 27 example: 28 29  30 31 structural balance 32 33 - a signed graph (edges labelled +/-) is balanced if all its cycles are positive (product of edge labels is positive) 34 - if the graph has no cycles, it is balanced 35 - signed graph is balanced iff its vertices can be partitioned into two disjoint subsets such that: 36 - each negative edge joins the subsets, and 37 - each positive edge joins vertices in the same subset 38 39 affiliation networks 40 41 - people are tied together through membership relations 42 - social structures consist of actors and events 43 - naturally bipartite, with two sets (Va actors, Ve events) 44 - represented with an actor-event matrix: 45 46  47 48 - number of events in which a and b participated 49 50 $NE[a, b] = \sum_{e \in V_e} AE[a, e] \times AE[b, e]$ 51 52 - number of actors participating in events e and f 53 54 $NA [e, f] = \sum_{a \in V_a} AE[a, e] \times AE[a, f]$