1. Beyond Triangles: The Importance Of Diamonds In Networks Katherine Stovel Christine Fountain Yen-Sheng Chiang University of Washington

2. Roadmap  The Problem  Measuring 4-cycles  Generating Models  Empirical Examples

3. The Problem  Many observed social networks are distinctive because of their high degree of local clustering  Local clustering is often explained as a by- product of tendencies toward balance  However, local clustering may not always be the result of transitivity The Problem

4. i Triadic measures of clustering 3( ) C3 = The Problem Watts 1999, Dorogovtsev 2004, etc. C v = density of subgraph X containing i’s neighbors C = ∑C v /n { Require that k>=2

5. Triadic measures fail to capture clustering in the presence of local prohibitions The Problem

6. Heterosexual Nets Minimal Structure The Problem

7. Chains of Affection Bearman, Moody, Stovel 2004 12 9 63 Male Female 2 Empirical Analyses

8. Producer 2 Competitive or Stratified Worlds Producer 1 supplier consumer The Problem

9. Solution: Consider the relative frequency of diamonds Diamonds capture simultaneous preference for nearness and local prohibitions Classic Bi-partite graphs… The Problem

10. Measuring Diamonds  Bernoulli expectation  Census of observed diamonds  Variants for  directed graphs  time-ordered data Measuring 4-cycles

11. Bernoulli expectation 0.00.20.40.60.81.0 0 e+00 1 e+06 2 e+06 3 e+06 4 e+06 Network Density (p) Diamonds Expected Observed in simulated data N = 200; 5 nets per simulated point Measuring 4-cycles Undirected graphs

12. Bernoulli Expectation: Directed graphs Measuring 4-cycles

13. i l jk i l jk i l jk i l jk Symmetric Hierarchy (HI)Unique Hierarchy (HII) CycleIncoherence Measuring 4-cycles

15. Diamond Census  Count number of complete, partial, and empty diamonds in network  Variants for more complex graphs  Directed graphs  Time-ordered graphs  Coded in both R and Matlab Measuring 4-cycles

17. Two Generating Models  Attribute Sort Model (θ)  Variable strength prohibition against in-group ties  Basic assortative-disassortative mixing model  Burt Model (Ω)  Actors build networks that are rich in structural holes  Modification of Watts’ α model Generating Models

18. Attribute Sort Model Generating Models θ controls strength of mixing θ = 0 in-group ties prohibited θ =.5 no preference for in- or out-group ties θ = 1 out-group ties prohibited N nodes Mean degree = k Create matrix R that indexes similarity of nodes i and j If R ij = 1, If R ij = 0,

20. Generating Models

21. The Burt Model Generating Models Ω controls strength of preference for structural holes 0 ≤ Ω ≥ ∞ No preference ↔ Strong preference X is current tie matrix M indexes shared alters p = small tie probability k = mean degree if k > Mij, if Mij ≥ k

22. Generating Models

23. Diamonds and Triangles in Omega Graphs Generating Models

24. The Upshot:  Both generating models create far more diamonds than in comparable random graphs  In the absence of any preference for social closeness, effects are somewhat density dependent  Though density is an artificial means of imposing a closeness constraint Generating Models

25. Data Analysis  Strategic alliances  Academic Citation Patterns Empirical Analyses

26. Strategic Alliances Biotechnology, 1990-1992 Justin Baer 2002 Empirical Analyses

30. Academic Citation Patterns Empirical Analyses Lowell Hargens 2000

31. Prevalence of  s NpC3Exp (each type) Symmetric Hierarchy Unique Hierarchy Incoherent Celestial Masers 3840.044.24310410651652019103 Toni Morrison Criticism 3090.018.1837253523202 Empirical Analyses

32. Take Away Message  Transitivity is obviously not the only systematic form of local structuring  Local out-group preferences or strategic behavior may preclude triadic closure in real social networks  Combined with propinquity or a preference for nearness, these prohibitions may create diamond-like clustered local structures  Observing diamonds may be an indication of a normative prohibition against specific relations The End

34. Expected number of m-link cycles (total) in Bernoulli random graph Expected number of m-link chains in Bernoulli random graph

35. Diamond Census Measuring 4-cycles i jk l i jk l i jk l i jk l Empty Partial Complete