Beyond Triangles: The Importance Of Diamonds In Networks Katherine Stovel Christine Fountain...
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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
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RoadmapRoadmap
The ProblemThe ProblemMeasuring 4-cyclesMeasuring 4-cyclesGenerating ModelsGenerating ModelsEmpirical ExamplesEmpirical Examples
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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
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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
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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
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Heterosexual NetsHeterosexual Nets
Minimal Structure
The Problem
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Chains of AffectionChains of AffectionBearman, Moody, Stovel 2004Bearman, Moody, Stovel 2004
12 9
63
Male
Female
2
Empirical Analyses
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Producer2
Competitive or Stratified WorldsCompetitive or Stratified Worlds
Producer1
supplier
consumer
The Problem
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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
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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
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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
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Bernoulli Expectation: Bernoulli Expectation: Directed graphsDirected graphs
44 )1(484
ppn
Dd
Measuring 4-cycles
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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
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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
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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
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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
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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
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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,
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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
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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
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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
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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
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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
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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
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Data AnalysisData Analysis
Strategic alliances Strategic alliances Academic Citation PatternsAcademic Citation Patterns
Empirical Analyses
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Strategic AlliancesStrategic AlliancesBiotechnology, 1990-1992Biotechnology, 1990-1992
Justin Baer 2002Empirical Analyses
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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
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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
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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
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Academic Citation PatternsAcademic Citation Patterns
Empirical Analyses
Lowell Hargens 2000
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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
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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
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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
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Diamond CensusDiamond Census
Measuring 4-cycles
i
j k
li
j k
l
i
j k
li
j k
l
Empty
Partial
Complete