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

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

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

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

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

Heterosexual Nets Minimal Structure The Problem

Chains of Affection Bearman, Moody, Stovel Male Female 2 Empirical Analyses

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

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

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

Bernoulli expectation 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

Bernoulli Expectation: Directed graphs Measuring 4-cycles

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

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

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

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,

Generating Models

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

Generating Models

Diamonds and Triangles in Omega Graphs Generating Models

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

Data Analysis  Strategic alliances  Academic Citation Patterns Empirical Analyses

Strategic Alliances Biotechnology, Justin Baer 2002 Empirical Analyses

Academic Citation Patterns Empirical Analyses Lowell Hargens 2000

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

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

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

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