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Chapter 8 Embedding

Chapter 8 Embedding. Introduction. Density Reciprocity Transitivity Clustering Group-external and group-internal ties Krackhardt's graph theoretical dimensions of hierarchy . U nit of analysis. Micro. Actor: People Dyad Triad Subgroups Organizations Collectives/aggregates

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Chapter 8 Embedding

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  1. Chapter 8 Embedding

  2. Introduction Density Reciprocity Transitivity Clustering Group-external and group-internal ties Krackhardt's graph theoretical dimensions of hierarchy

  3. Unit of analysis Micro • Actor: • People • Dyad • Triad • Subgroups • Organizations • Collectives/aggregates • Communities • Nation-states 环境 组织 团体 个体 个体是镶嵌在网络中，而个体所镶嵌的网络是镶嵌在更高层及的网络中。 Macro/aggregates

  4. Dyad 两人组 • Binary ties-present, or absent • Directed relations • 有无关系？是否是双向关系？(reciprocal) • 还是单向(asymmetrical)？

  5. Density The more actors are connected to one another, the more dense the network will be. Binary data: the number of present ties/the number of all possible ties Undirected network: n(n-1)/2 = 2n-1 possible pairs of actors. Δ = Directed network: n(n-1)*2/2 = 2n-2possible lines. ΔD= Valued data: the average strength of ties across all possible ties

  6. Density Density=4/(3*2)

  7. Reciprocity • Directed data, four possible dyadic relations • Actor: reciprocity pair (AB)/ all possible pairs (AB, BC, AC) • Dyad method: • the number of pairs with a reciprocity tie/the number of pairs with any tie. (AB)/(AB, BC) • Relations: reciprocal tie(AB, BA)/all possible pairs (AB, BA, BC, CB, AC, CA) • Arc method: • The number of reciprocal ties/ total number of actual ties (AB,BA)/(AB, BA, BC)

  8. Reciprocity Dyad method

  9. Triad 三人组 Tu = the number of triads that belong to isomorphism class u Cg3 T=(T003, T012, ...,T300)‟ =(003, 012, 102, 021D, 021U, 021C, 111D, 111U, 030T, 030C, 201, 120D, 120U, 120C, 210, 300) Un-directed data, four possible types of triadic relations Directed-data, 16 possible types of triadic relations

  10. Transtivity

  11. Clustering “6-degrees” phenomenon, How close actors are together “Clique-like” local neighborhood, the tendency towards dense local neighborhoods. Clustering: the density in neighborhood

  12. Weight, possible number of pairs 

  13. Group-external and group-internal ties Measure of group based on comparing the numbers of ties within groups and between groups. E-I Index This value can range from 1 to -1, but for a given network density and group sizes its range may be restricted and so it can be rescaled. The index is also calculated for each group and for each individual actor.

  14. =36/(14+50) =42/(66+24)

  15. Krackhardt's graph theoretical dimensions of hierarchy Krackhardt argues that an ‘Outree” is the archetype of hierarchy. • Krackhardt focuses on 4 dimensions: • 1) Connectedness • 2) Digraph hierarchic • 3) digraph efficiency • 4) least upper bound (what are the allowed triad types for an out-tree?)

  16. Connectedness: The digraph is connected if the underlying graph is a component. We can measure the extent of connectedness through reachability. Where V is the number of pairs that are not reachable, and N is the number of people in the network.

  17. Reach: 1 2 3 4 5 1 0 1 2 1 0 2 1 0 1 2 0 3 2 1 0 3 0 4 1 2 3 0 0 5 0 0 0 0 0 Graph: 1 2 3 4 5 1 0 1 0 1 0 2 1 0 1 0 0 3 0 1 0 0 0 4 1 0 0 0 0 5 0 0 0 0 0 Digraph: 1 2 3 4 5 1 0 1 0 1 0 2 0 0 1 0 0 3 0 0 0 0 0 4 0 0 0 0 0 5 0 0 0 0 0 1 4 2 5 3 How to calculate Connectedness: V = # of zeros in the upper diagonal of Reach: V = 4. C = 1 - [4/((5*4)/2)] = 1 - 4/1 = .6

  18. Reachable: 1 2 3 4 5 1 0 1 1 1 0 2 1 0 1 1 0 3 1 1 0 1 0 4 1 1 1 0 0 5 0 0 0 0 0 Reach: 1 2 3 4 5 1 0 1 2 1 0 2 1 0 1 2 0 3 2 1 0 3 0 4 1 2 3 0 0 5 0 0 0 0 0 1 4 2 5 3 How to calculate Connectedness: This is equivalent to the density of the reachability matrix. D = SR/(N(N-1)) = 12 /(5*4) = .6

  19. Graph Hierarchy: The extent to which people are asymmetrically reachable. Where V is the number of symmetrically reachable pairs in the network. Max(V) is the number of pairs where i can reach j or j can reach i.

  20. 1 4 2 5 3 Graph Hierarchy: An example Dreachable 1 2 3 4 5 1 0 1 2 1 0 2 0 0 1 0 0 3 0 1 0 0 0 4 0 0 0 0 0 5 0 0 0 0 0 Digraph: 1 2 3 4 5 1 0 1 0 1 0 2 0 0 1 0 0 3 0 1 0 0 0 4 0 0 0 0 0 5 0 0 0 0 0 Dreach 1 2 3 4 5 1 0 1 2 1 0 2 0 0 1 0 0 3 0 1 0 0 0 4 0 0 0 0 0 5 0 0 0 0 0 V = 1 Max(V) = 4 H = 1/4 = .25

  21. Graph Efficiency: The extent to which there are extra lines in the graph, given the number of components. Where v is the number of excess links and max(v) is the maximum possible number of excess links

  22. 1 4 2 6 5 3 7 The minimum number of lines in a connected component is N-1 (assuming symmetry, only use the upper half of the adjacency matrix). In this example, the first component contains 4 nodes and thus the minimum required lines is 3. There are 4 lines, and thus V1= 4-3 = 1. The second component contains 3 nodes and thus minimum connectivity is = 2, there are 3 so V2 = 1. Summed over all components V=2. The maximum number of lines would occur if every node was connected to every other, and equals N(N-1)/2. For the first component Max(V1) = (6-3)=3. For the second, Max(V2) = (3-2)=1, so Max(V) = 4. Efficiency = (1- 2/4 ) = .5 Graph Efficiency: 1 2

  23. Graph Efficiency: Steps to calculate efficiency: a) identify all components in the graph b) for each component (i) do: i) calculate Vi = S(Gi)/2 - Ni-1; ii) calculate Max(Vi) = Ni(Ni-1) - (Ni-1) c) V = S(Vi), Max(V)= S(Max(Vi) d) efficiency = 1 - V/Max(V) Substantively, this must be a function of the average density of the components in the graph.

  24. Least Upper Boundedness: This condition looks at how many ‘roots’ there are in the tree. The LUB for any pair of actors is the closest person who can reach both of them. In a formal hierarchy, every pair should have at least one LUB. E In this case, E is the LUB for (A,D), B is the LUB for (F,G), H is the LUB for (D,C), etc. H B G C F A D

  25. Graph Theoretic Dimensions of Informal Organizations Least Upper Boundedness: You get a violation of LUB if two people in the organization do not have an (eventual) common boss. Here, persons 4 and 7 do not have an LUB.

  26. Distance matrix 1 2 3 4 5 6 7 8 9 1 1 1 1 2 2 2 2 1 1 1 3 1 1 4 1 5 1 6 1 1 1 2 7 1 1 8 1 9 1 Reachable matrix 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 2 1 1 1 3 1 1 4 1 5 1 6 1 1 1 1 7 1 1 8 1 9 1 Least Upper Boundedness: Calculate LUB by looking at reachability. (Note that I set the diagonal = 1) A violation occurs whenever a pair is not reachable by at least one common node. We can get common reachability through matrix multiplication

  27. Reachable matrix 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 2 1 1 1 3 1 1 4 1 5 1 6 1 1 1 1 7 1 1 8 1 9 1 Reachable Trans 1 2 3 4 5 6 7 8 9 1 1 2 1 1 3 1 1 4 1 1 1 5 1 1 1 6 1 7 1 1 8 1 1 9 1 1 1 1 1 Graph Theoretic Dimensions of Informal Organizations Least Upper Boundedness: Calculate LUB by looking at reachability. Common Reach 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 0 0 0 1 2 1 2 1 2 2 0 0 0 1 3 1 1 2 1 1 0 0 0 2 4 1 2 1 3 2 0 0 0 1 5 1 2 1 2 3 0 0 0 1 6 0 0 0 0 0 1 1 1 1 7 0 0 0 0 0 1 2 1 2 8 0 0 0 0 0 1 1 2 1 9 1 1 2 1 1 1 2 1 5 X = (R by S) (S by R) (R by R) Any place with a zero indicates a pair that does not have a LUB. R`*R = CR

  28. Least Upper Boundedness: Calculate LUB by looking at reachability. Where V = number of pairs that have no LUB, summed over all components, and:

  29. Bearman, Peter S., James Moody, and Katherine Stovel. "Chains of Affection: The Structure of Adolescent Romantic and Sexual Networks1." American Journal of Sociology 110.1 (2004): 44-91.

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