Introduction Problem Approach Properties Measures Summary

Measuring Segregation in Social Networks

Micha l Bojanowski Rense Corten

ICS/Sociology, Utrecht University

July 2, 2010Sunbelt XXX, Riva del Garda

Introduction Problem Approach Properties Measures Summary

Outline

1 IntroductionHomophily and segregation

2 Problem3 Approach

ApproachNotation

4 PropertiesTiesNodesNetwork

5 Measures

6 Summary

Introduction Problem Approach Properties Measures Summary

Homophily and segregation

Homophily and segregation

Homophily Contact between similar people occurs at a higherrate than among dissimilar people (McPherson,Smith-Lovin, & Cook, 2001).

Segregation Nonrandom allocation of people who belong todifferent groups into social positions and theassociated social and physical distances betweengroups (Bruch & Mare, 2009).

Introduction Problem Approach Properties Measures Summary

Homophily and segregation

Homophily and segregation

Homophily Contact between similar people occurs at a higherrate than among dissimilar people (McPherson,Smith-Lovin, & Cook, 2001).

Segregation Nonrandom allocation of people who belong todifferent groups into social positions and theassociated social and physical distances betweengroups (Bruch & Mare, 2009).

Introduction Problem Approach Properties Measures Summary

Homophily and segregation

Homophily: Friendship selection in school classes

Moody (2001)

Introduction Problem Approach Properties Measures Summary

Homophily and segregation

Residential segregation in Seattle

Blacks Asians Whites

Source: Seattle Civil Rights and Labor History Project

Introduction Problem Approach Properties Measures Summary

Homophily and segregation

Segregation in network terms

Neighborhood structure can beconceptualized as a network inwhich links correspond to neigh-borhood proximities.

0 1 2 3 4 5

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Introduction Problem Approach Properties Measures Summary

Homophily and segregation

Assumption

In static terms homophily and segregation correspond to thesame network phenomenon.

We will stick with the segregation label.

Introduction Problem Approach Properties Measures Summary

Measurement problem

To be able to compare the levels of segregation of differentnetworks (different school classes, different cities etc.) we need ameasure.

Introduction Problem Approach Properties Measures Summary

Problems with measures

There exist an abundance of measures in the literature, but:

Stem from different research streams

Follow different logics

Hardly ever refer to each other

Lead to different conclusions given the same problems (data)

So, the problems are:

Which one to select in a given setting?

On what grounds such selection should be performed?

Introduction Problem Approach Properties Measures Summary

Approach

Possible approaches

Empirical Assemble a large set of empirical datasets. Calculatethe measures for all of them. Look how theycorrelate. Perhaps through PCA or alike.

Theo-pirical Take a set of probabilistic models of networks(Erdos-Renyi random graph, preferential attachment,small-world etc.). Generate a collection of networks.Proceed as in the item above.

Theoretical Come-up with a set of properties that the measuresmight (or might not) posses. Evaluate the differencesbetween the measures in terms of satisfying (or not)certain properties.

Introduction Problem Approach Properties Measures Summary

Approach

Possible approaches

Empirical Assemble a large set of empirical datasets. Calculatethe measures for all of them. Look how theycorrelate. Perhaps through PCA or alike.

Theo-pirical Take a set of probabilistic models of networks(Erdos-Renyi random graph, preferential attachment,small-world etc.). Generate a collection of networks.Proceed as in the item above.

Theoretical Come-up with a set of properties that the measuresmight (or might not) posses. Evaluate the differencesbetween the measures in terms of satisfying (or not)certain properties.

Introduction Problem Approach Properties Measures Summary

Approach

Possible approaches

Empirical Assemble a large set of empirical datasets. Calculatethe measures for all of them. Look how theycorrelate. Perhaps through PCA or alike.

Theo-pirical Take a set of probabilistic models of networks(Erdos-Renyi random graph, preferential attachment,small-world etc.). Generate a collection of networks.Proceed as in the item above.

Theoretical Come-up with a set of properties that the measuresmight (or might not) posses. Evaluate the differencesbetween the measures in terms of satisfying (or not)certain properties.

Introduction Problem Approach Properties Measures Summary

Approach

Possible approaches

Empirical Assemble a large set of empirical datasets. Calculatethe measures for all of them. Look how theycorrelate. Perhaps through PCA or alike.

Theo-pirical Take a set of probabilistic models of networks(Erdos-Renyi random graph, preferential attachment,small-world etc.). Generate a collection of networks.Proceed as in the item above.

Theoretical Come-up with a set of properties that the measuresmight (or might not) posses. Evaluate the differencesbetween the measures in terms of satisfying (or not)certain properties.

Introduction Problem Approach Properties Measures Summary

Approach

Possible approaches

Empirical Assemble a large set of empirical datasets. Calculatethe measures for all of them. Look how theycorrelate. Perhaps through PCA or alike.

Theo-pirical Take a set of probabilistic models of networks(Erdos-Renyi random graph, preferential attachment,small-world etc.). Generate a collection of networks.Proceed as in the item above.

Theoretical Come-up with a set of properties that the measuresmight (or might not) posses. Evaluate the differencesbetween the measures in terms of satisfying (or not)certain properties.

Introduction Problem Approach Properties Measures Summary

Notation

Actors

Actors N = {1, 2, . . . , i , . . . ,N}Groups of actors Actors are assigned into K exhaustive and

mutually exclusive groups.G = {G1, . . . ,Gk , . . . ,GK}.Group membership is denoted with “type vector”:

t = [t1, . . . , ti , . . . , tN ] where ti ∈ {1, . . . ,K}

ti = group of actor iLet T be a set of all possible type vectors for N .

Introduction Problem Approach Properties Measures Summary

Notation

Network

Network Actors form an undirected network which is a squarebinary matrix X = [xij ]N×N . Let X be a set of allpossible networks over actors in N .

Mixing matrix A three-dimensional array M = [mghy ]K×K×2

defined as

mgh1 =∑i∈Gg

∑j∈Gh

xij mgh0 =∑i∈Gg

∑j∈Gh

(1− xij)

Introduction Problem Approach Properties Measures Summary

Notation

Segregation index

Segregation measure A generic segregation index S(·):

S : X × T 7→ <

For a given network and type vector assign a realnumber.

Introduction Problem Approach Properties Measures Summary

Ties

Adding between-group ties

Property (Monotonicity in between-group ties: MBG)

Let there be two networks X and Y defined on the same set ofnodes, a type vector t, and two nodes i and j such that ti 6= tj ,xij = 0, and yij = 1. For all the other nodes p, q 6= i , j xpq = ypq,i.e. the networks X and Y are identical.Network segregation index S is monotonic in between-groupties iff

S(X , t) ≥ S(Y , t)

In words: adding a between-group tie cannot increase segregation.

Introduction Problem Approach Properties Measures Summary

Ties

Adding within-group ties

Property (Monotonicity in within-group ties: MWG)

Let there be two networks X and Y defined on the same set ofnodes, a type vector t, and two nodes i and j such that ti = tj ,xij = 0 and yij = 1. For all the other nodes p, q 6= i , j xpg = ypg ,i.e. the networks X and Y are identical.Network segregation index S is monotonic in within-group tiesiff

S(X , t) ≤ S(Y , t)

In words: adding a within-group tie to the network cannot decreasesegregation.

Introduction Problem Approach Properties Measures Summary

Ties

Rewiring between-group tie to within-group

Property (Monotonicity in rewiring: MR)

Let there be two networks X and Y , a type vector t and threenodes i , j and k such that

1 xij = 1 and ti 6= tj2 yij = 0, yik = 1, and ti = tk

That is, an between-group tie ij in X is rewired to a within-grouptie ik in Y .Network segregation index S is monotonic in rewiring iff

S(X , t) ≤ S(Y , t)

Introduction Problem Approach Properties Measures Summary

Nodes

Adding isolates

Property (Effect of adding isolates: ISO)

Define two networks X = [xij ]N×N and Y = [ypq]N+1×N+1 andassociated type vectors u and w which are identical for the Nactors and differ by an (N + 1)-th node which is an isolate:

1 ∀p, q ∈ 1..N ypq = xpq

2∑N+1

p=1 yp N+1 =∑N+1

q=1 yN+1 q = 0.

3 ∀k ∈ 1..N wk = uk .

S(X ,u) ? S(X ,w)

In words: how does the segregation level change if isolates areadded to the network?

Introduction Problem Approach Properties Measures Summary

Network

Duplicating the network

Property (Symmetry: S)

Define two identical networks X and Y and some type vector t.Network segregation index S satisfies symmetry iff

S(X , t) = S(Y , t) = S(Z , z)

where the network Z is constructed by considering X and Ytogether as a single network, namely: Z = [zpq]2N×2N such that

∀p, q ∈ {1, . . . ,N} zpq = xpq

∀p, q ∈ {N + 1, . . . , 2N} zpq = ypq

otherwise zpq = 0

Introduction Problem Approach Properties Measures Summary

Measures

Freeman’s segregation index (Freeman, 1978)

Spectral Segregation Index (Echenique & Fryer, 2007)

Assortativity coefficient (Newman, 2003)

Gupta-Anderson-May’s Q (Gupta et al, 1989)

Coleman’s Homophily Index (Coleman, 1958)

Segregation Matrix index (Freshtman, 1997)

Exponential Random Graph Models (Snijders et al, 2006)

Conditional Log-linear models for mixing matrix (Koehly,Goodreau & Morris, 2004)

Introduction Problem Approach Properties Measures Summary

Measure LevelNetwork

typeScale

Freeman network U [0; 1]SSI node U [0;∞]

Assortativity network D/U [−∑

g pg+p+g

1−∑

g pg+p+g; 1]

Gupta-Anderson-May network D/U [− 1G−1

; 1]

Coleman group D [−1; 1]Segregation Matrix Index group D/U [−1; 1]Uniform homophily (CLL) network D/U [−∞;∞]Differential homophily (CLL) group D/U [−∞;∞]Uniform homophily (ERGM) network D/U [−∞;∞]Differential homophily (ERGM) group D/U [−∞;∞]

Introduction Problem Approach Properties Measures Summary

Freeman (1978)

Given two groups

SFreeman = 1− p

π

where p is the observed proportion of between-group ties and π isthe expected proportion given that ties are created randomly. Itvaries between 0 (random network) and 1 (full segregation ofgroups).

Introduction Problem Approach Properties Measures Summary

Assortativity Coefficient, Newman (2003)

Based on a contact layer of the mixing matrix pgh = mgh1/m++1.

SNewman =

∑Kg=1 pgg −

∑Kg=1 pg+p+g

1−∑K

g=1 pg+p+g

Maximum of 1 for perfect segregation; 0 for random network.Negative values for “dissasortative” networks. Minimum dependson the density.

Introduction Problem Approach Properties Measures Summary

Gupta, Anderson & May 1989

Also based on contact layer of the mixing matrix

SGAM =

∑Kg=1 λg − 1

K − 1

Where λg are eigenvalues of pgh. It varies between −1/(K − 1)and 1

Introduction Problem Approach Properties Measures Summary

Coleman, 1958

Expected number of ties within group g

m∗gg =∑i∈Gg

ηing − 1

N − 1

SgColeman =

mgg −m∗gg∑i∈Gg

ηi −m∗ggwhere mgg >= m∗gg (1)

SgColeman =

mgg −m∗ggm∗gg

where mgg < m∗gg (2)

Introduction Problem Approach Properties Measures Summary

Segregation matrix index, Freshtman 1997

SSMI =d11 − d12

d11 + d12(3)

where d11 is the density of within-group ties and d12 is the densityof between-group ties.

Introduction Problem Approach Properties Measures Summary

Conditional Log-Linear Models (Koehly et al, 2004)

log mgh1 = µ+ λAg + λBh + λUHOMgh

{λUHOMgh = λUHOM g = h

λUHOMgh = 0 g 6= h

log mgh1 = µ+ λAg + λBh + λDHOMgh

{λDHOMgh = λDHOM

g g = h

λDHOMgh = 0 g 6= h

Parameters λUHOM and λDHOMg as measures of

homophily/segregation.

Introduction Problem Approach Properties Measures Summary

ERGM

Exponential Random Graph models

log

(mgh1

mgh0

)= α + βAg + βBh + βUHOM

gh

{βUHOMgh = βUHOM g = h

βUHOMgh = 0 g 6= h

log

(mgh1

mgh0

)= µ+ βAg + βBh + βDHOM

gh

{βDHOMgh = βDHOM

g g = h

βDHOMgh = 0 g 6= h

Parameters βUHOM and βDHOMg as measures of

homophily/segregation.

Introduction Problem Approach Properties Measures Summary

Spectral Segregation Index, Echenique & Fryer (2007)

Segregation level of individual i in group g in component B:

sgi (B) =1

SgCi

∑j

rijsgj (B) (4)

where rij are entries in a row-normalized adjacency matrix.Segregation of individual i

S iSSI =

li

lλ (5)

where λ is the largest eigenvalue of B, and l is the correspondingeigenvector

Introduction Problem Approach Properties Measures Summary

SSI (2)

01

2

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5

67

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Node segregation in White's kinship data

Mother

Sister

Brother's Wife

Sister's Daughter

Brother's Daughter

Father Brother

Sister's Husband

Brother's SonSister's Son

●

MenWomen

Introduction Problem Approach Properties Measures Summary

Summary

Measure MBG (↘) MWG (↗) MR (↗) ISO S (→)

Freeman l l ↗ l ↘SSI ↘ ↗ ↗ ↘ →Assortativity ↘ ↗ ↗ → →Gupta-Anderson-May ↘ ↗ ↗ → →Coleman ↘ ↗ ↗ l ↘Segregation Matrix Index ↘ ↗ ↗ l →Uniform homophily (CLL) ↘ ↗ ↗ → →Differential homophily (CLL) ↘ ↗ ↗ → →Uniform homophily (ERGM) ↘ ↗ ↗ l →Differential homophily (ERGM) ↘ ↗ ↗ l →

Introduction Problem Approach Properties Measures Summary

Summary

Measures on different levels: individuals, groups, globalnetwork

Different zero points: random graph, proportionate mixing,full integration

MBW, MWG not very informative, all measures satisfy them.

Symmetry: All but two measures satisfy it, Coleman andFreeman decrease.

Introduction Problem Approach Properties Measures Summary

Summary: adding isolates

Measures based on contact layer of mixing matrix areinsensitive to isolates.

SSI is the only one that always decreases

The effect on others depend on relative group sizes.

Introduction Problem Approach Properties Measures Summary

Summary

Measures based on contact layer of the mixing matrixsummarize probability of node attribute combination giventhat the tie exists (CLL, assortativity, GAM): explainingattributes given the network.

Measures that take also disconnected dyads into account.(ERGM, Freeman, SSI): explaining tie formation given theattributes.

Introduction Problem Approach Properties Measures Summary

Further questions

Stricter formal analysis (axiomatizations). SSI is the onlymeasure derived axiomatically.

Link to behavioral models: how the segregation comes about.For example

Network formation game further justifying Bonacich centrality(Ballester et al., 2006)Coleman’s index in Currarini et al. (2010).

Introduction Problem Approach Properties Measures Summary

Thanks

Thanks!

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