Beyond Triangles: The Beyond Triangles: The Importance Of Diamonds In Importance Of Diamonds In

Networks Networks Katherine StovelKatherine Stovel

Christine FountainChristine FountainYen-Sheng ChiangYen-Sheng Chiang

University of WashingtonUniversity of Washington

RoadmapRoadmap

The ProblemThe ProblemMeasuring 4-cyclesMeasuring 4-cyclesGenerating ModelsGenerating ModelsEmpirical ExamplesEmpirical Examples

The ProblemThe Problem

Many observed social networks are Many observed social networks are distinctive because of their high degree of distinctive because of their high degree of local clusteringlocal clustering

Local clustering is often explained as a by-Local clustering is often explained as a by-product of tendencies toward balanceproduct of tendencies toward balance

However, local clustering may not always However, local clustering may not always be the result of transitivitybe the result of transitivity

The Problem

i

Triadic measures of clusteringTriadic measures of clustering

3( )C3 =

The Problem

Watts 1999, Dorogovtsev 2004, etc.

Cv = density of subgraph X containing i’s neighbors

C = ∑Cv/n{Require that

k>=2

Triadic measures fail to capture Triadic measures fail to capture clustering in the presence of local clustering in the presence of local

prohibitionsprohibitions

The Problem

Heterosexual NetsHeterosexual Nets

Minimal Structure

The Problem

Chains of AffectionChains of AffectionBearman, Moody, Stovel 2004Bearman, Moody, Stovel 2004

12 9

63

Male

Female

2

Empirical Analyses

Producer2

Competitive or Stratified WorldsCompetitive or Stratified Worlds

Producer1

supplier

consumer

The Problem

Solution: Consider the relative Solution: Consider the relative frequency of diamondsfrequency of diamonds

Diamonds capture simultaneous Diamonds capture simultaneous preference for nearness and local preference for nearness and local prohibitionsprohibitions

Classic Bi-partite graphs…Classic Bi-partite graphs…

The Problem

Measuring DiamondsMeasuring Diamonds

Bernoulli expectationBernoulli expectationCensus of observed diamondsCensus of observed diamondsVariants for Variants for

directed graphs directed graphs time-ordered datatime-ordered data

Measuring 4-cycles

Bernoulli expectationBernoulli expectation

24 )1(34

ppn

Du

0.0 0.2 0.4 0.6 0.8 1.0

0

e+

00

1

e+

06

2

e+

06

3

e+

06

4

e+

06

Network Density (p)

Dia

mo

nd

s

ExpectedObserved in simulated data

N = 200; 5 nets per simulated point

Measuring 4-cycles

4 2

24

24

! (1 )

8 4 !

4! 1

4 8

3 14

n p p

n

n p p

np p

Undirected graphs

Bernoulli Expectation: Bernoulli Expectation: Directed graphsDirected graphs

44 )1(484

ppn

Dd

Measuring 4-cycles

i

l

j k

i

l

j k

i

l

j k

i

l

j k

4444

)1(6422

)1(!4

4pp

nppnDI

44

44

)1(644

)1(!4

4pp

nppnDC

4444

)1(1242

)1(!4

4pp

nppnDHI

Symmetric Hierarchy (HI) Unique Hierarchy (HII)

Cycle Incoherence

4444 )1(244

)1(!44

ppn

ppn

DSHI

Measuring 4-cycles

0.0 0.2 0.4 0.6 0.8 1.0

Expected Empty Diamonds by Density on Bernoulli Random Graphs

Network Density

Dia

mo

nd

sDirected GraphsUndirected Graphs

EDU3

N

4

p

41 p2

EDD48

N

4

p

41 p4

Maximum at p = 2/3

Maximum at p = 1/2

Measuring 4-cycles

Diamond CensusDiamond Census

Count number of complete, partial, and Count number of complete, partial, and empty diamonds in networkempty diamonds in network

Variants for more complex graphsVariants for more complex graphsDirected graphsDirected graphsTime-ordered graphsTime-ordered graphs

Coded in both R and MatlabCoded in both R and Matlab

Measuring 4-cycles

0.05 0.10 0.15 0.20

05

00

00

10

00

00

15

00

00

Relative Frequency of Diamonds by Density

Network Density

Nu

mb

er

of D

iam

on

ds

Expected RandomPreferential AttachmentWatts (Random)Watts (Clustered)Watts (Small World)

N = 200; 5 nets per point

Measuring 4-cycles

Two Generating ModelsTwo Generating Models

Attribute Sort Model (Attribute Sort Model (θθ))Variable strength prohibition against in-group Variable strength prohibition against in-group

tiestiesBasic assortative-disassortative mixing modelBasic assortative-disassortative mixing model

Burt Model (Burt Model (ΩΩ))Actors build networks that are rich in Actors build networks that are rich in

structural holesstructural holesModification of Watts’ Modification of Watts’ αα model model

Generating Models

Attribute Sort ModelAttribute Sort Model

jiR

P N

jij

ij

1

Generating Models

ji

RP N

jij

ij

1

11

1

θ controls strength of mixing

θ = 0 in-group ties prohibitedθ = .5 no preference for in- or out-group tiesθ = 1 out-group ties prohibited

N nodesMean degree = k

Create matrix R that indexes similarity of nodes i and j

If Rij = 1,

If Rij = 0,

0.0 0.2 0.4 0.6 0.8 1.0

10

00

15

00

20

00

25

00

Attribute Network: Number of Empty Diamonds by In-Group Preference

Dia

mo

nd

s

30 nets per point; N = 200, k=10

Random Expectation at N = 100Random Expectation at N = 200

0.05 0.10 0.15 0.20

0

e+

00

2

e+

05

4

e+

05

6

e+

05

Attribute Model: Relative Frequency of Diamonds by Density

Network Density

Nu

mb

er

of D

iam

on

ds

ExpectedAttribute (0)Attribute (0.25)Attribute (0.5)

N = 200; 5 nets per point

Generating Models

The Burt ModelThe Burt Model

Generating Models

Ω controls strength of preference for structural holes

0 ≤ Ω ≥ ∞No preference ↔ Strong preference

X is current tie matrixM indexes shared altersp = small tie probability

k = mean degree

if k > Mij,

pPij if Mij ≥ k

k

MP ijij 1

0.05 0.10 0.15 0.20

05

000

01

500

00

250

00

0

Burt Networks: Frequency of Empty Diamonds by Density

Network Density

Nu

mb

er

of D

iam

on

ds

40 15 5Bernoulli Expectation

N = 200; 5 nets per point

Generating Models

0 20 40 60 80 100

0.5

1.0

1.5

Small World Properties of Burt Graphs

Sca

led

Sta

tistic

5 graphs per point; N = 200, p = .15; statistics scaled to random expectation

ClusteringMedian GeodesicDiamonds

0 20 40 60 80 100

0.5

1.0

1.5

2.0

Small World Properties of Burt Graphs

Sca

led

Sta

tistic

5 graphs per point; N = 200, p = .25; statistics scaled to random expectation

ClusteringMedian GeodesicDiamonds

Diamonds and Triangles inDiamonds and Triangles inOmega GraphsOmega Graphs

Generating Models

The Upshot:The Upshot:

Both generating models create far more Both generating models create far more diamonds than in comparable random diamonds than in comparable random graphsgraphs

In the absence of any preference for social In the absence of any preference for social closeness, effects are somewhat density closeness, effects are somewhat density dependentdependentThough density is an artificial means of Though density is an artificial means of

imposing a closeness constraintimposing a closeness constraint

Generating Models

Data AnalysisData Analysis

Strategic alliances Strategic alliances Academic Citation PatternsAcademic Citation Patterns

Empirical Analyses

Strategic AlliancesStrategic AlliancesBiotechnology, 1990-1992Biotechnology, 1990-1992

Justin Baer 2002Empirical Analyses

0.0 0.5 1.0 1.5 2.0 2.5

01

23

4

Power-Law Degree Distribution

Logged Alliances

Lo

gg

ed

Fre

qu

en

cy

119 Focal Firms + 97 Orgs

1.76

R20.905

Empirical Analyses

Simulated Empty Diamonds on Three Random Graph Models

Diamonds

Fre

qu

en

cy

0 20 40 60 80 100

02

00

40

06

00

80

01

00

0

Baer Alliances: 119 Focal Firms + 97 Orgs

Obs = 13

Degree DistributionPreferential AttachmentBernoulli

Empirical Analyses

Simulated Clustering Coefficients on Three Random Graph Models

Diamonds

Fre

qu

en

cy

0.00 0.05 0.10 0.15 0.20 0.25

01

002

003

004

005

00

Baer Alliances: 119 Focal Firms + 97 Orgs

Obs = 0.02

Degree DistributionPreferential AttachmentBernoulli

Empirical Analyses

Academic Citation PatternsAcademic Citation Patterns

Empirical Analyses

Lowell Hargens 2000

Prevalence of Prevalence of ss

NN pp C3C3

ExpExp

(each (each type)type)

Symmetric Symmetric HierarchyHierarchy

Unique Unique HierarchyHierarchy

IncoherentIncoherent

Celestial Celestial MasersMasers 384384 0.0440.044 .24.24 31043104 1065110651 65206520 1910319103

Toni Toni Morrison Morrison CriticismCriticism 309309 0.0180.018 .18.18 3737 253253 5252 32023202

Empirical Analyses

Take Away MessageTake Away Message

Transitivity is obviously not the only systematic Transitivity is obviously not the only systematic form of local structuringform of local structuring

Local out-group preferences or strategic Local out-group preferences or strategic behavior may preclude triadic closure in real behavior may preclude triadic closure in real social networkssocial networks

Combined with propinquity or a preference for Combined with propinquity or a preference for nearness, these prohibitions may create nearness, these prohibitions may create diamond-like clustered local structuresdiamond-like clustered local structures

Observing diamonds may be an indication of a Observing diamonds may be an indication of a normative prohibition normative prohibition againstagainst specific relations specific relations

The End

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

!

2 1 !

mn p

n m

Expected number of m-link chains in Bernoulli random graph

Diamond CensusDiamond Census

Measuring 4-cycles

i

j k

li

j k

l

i

j k

li

j k

l

Empty

Partial

Complete