Measuring Segregation in Social Networks

Transcription

1 Measuring Segregation in Social Networks Micha l Bojanowski Rense Corten ICS/Sociology, Utrecht University July 2, 2010 Sunbelt XXX, Riva del Garda

2 Outline 1 Introduction Homophily and segregation 2 Problem 3 Approach Approach Notation 4 Properties Ties Nodes Network 5 Measures 6 Summary

3 Homophily and segregation Homophily and segregation Homophily Contact between similar people occurs at a higher rate than among dissimilar people (McPherson, Smith-Lovin, & Cook, 2001). Segregation Nonrandom allocation of people who belong to different groups into social positions and the associated social and physical distances between groups (Bruch & Mare, 2009).

4 Homophily and segregation Homophily and segregation Homophily Contact between similar people occurs at a higher rate than among dissimilar people (McPherson, Smith-Lovin, & Cook, 2001). Segregation Nonrandom allocation of people who belong to different groups into social positions and the associated social and physical distances between groups (Bruch & Mare, 2009).

5 Homophily and segregation Homophily: Friendship selection in school classes Moody (2001)

6 Homophily and segregation Residential segregation in Seattle Blacks Asians Whites Source: Seattle Civil Rights and Labor History Project

7 Homophily and segregation Segregation in network terms Neighborhood structure can be conceptualized as a network in which links correspond to neighborhood proximities

8 Homophily and segregation Assumption In static terms homophily and segregation correspond to the same network phenomenon. We will stick with the segregation label.

9 Measurement problem To be able to compare the levels of segregation of different networks (different school classes, different cities etc.) we need a measure.

10 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?

11 Approach Possible approaches

12 Approach Possible approaches Empirical Assemble a large set of empirical datasets. Calculate the measures for all of them. Look how they correlate. Perhaps through PCA or alike.

13 Approach Possible approaches Empirical Assemble a large set of empirical datasets. Calculate the measures for all of them. Look how they correlate. Perhaps through PCA or alike. Theo-pirical Take a set of probabilistic models of networks (Erdös-Renyi random graph, preferential attachment, small-world etc.). Generate a collection of networks. Proceed as in the item above.

14 Approach Possible approaches Empirical Assemble a large set of empirical datasets. Calculate the measures for all of them. Look how they correlate. Perhaps through PCA or alike. Theo-pirical Take a set of probabilistic models of networks (Erdös-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 measures might (or might not) posses. Evaluate the differences between the measures in terms of satisfying (or not) certain properties.

15 Approach Possible approaches Empirical Assemble a large set of empirical datasets. Calculate the measures for all of them. Look how they correlate. Perhaps through PCA or alike. Theo-pirical Take a set of probabilistic models of networks (Erdös-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 measures might (or might not) posses. Evaluate the differences between the measures in terms of satisfying (or not) certain properties.

16 Notation Actors Actors N = {1, 2,..., i,..., N} Groups of actors Actors are assigned into K exhaustive and mutually exclusive groups. G = {G 1,..., G k,..., G K }. Group membership is denoted with type vector : t = [t 1,..., t i,..., t N ] where t i {1,..., K} t i = group of actor i Let T be a set of all possible type vectors for N.

17 Notation Network Network Actors form an undirected network which is a square binary matrix X = [x ij ] N N. Let X be a set of all possible networks over actors in N. Mixing matrix A three-dimensional array M = [m ghy ] K K 2 defined as m gh1 = x ij m gh0 = (1 x ij ) j G h j G h i G g i G g

18 Notation Segregation index Segregation measure A generic segregation index S( ): S : X T R For a given network and type vector assign a real number.

19 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 of nodes, a type vector t, and two nodes i and j such that t i t j, x ij = 0, and y ij = 1. For all the other nodes p, q i, j x pq = y pq, i.e. the networks X and Y are identical. Network segregation index S is monotonic in between-group ties iff S(X, t) S(Y, t) In words: adding a between-group tie cannot increase segregation.

20 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 of nodes, a type vector t, and two nodes i and j such that t i = t j, x ij = 0 and y ij = 1. For all the other nodes p, q i, j x pg = y pg, i.e. the networks X and Y are identical. Network segregation index S is monotonic in within-group ties iff S(X, t) S(Y, t) In words: adding a within-group tie to the network cannot decrease segregation.

21 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 three nodes i, j and k such that 1 x ij = 1 and t i t j 2 y ij = 0, y ik = 1, and t i = t k That is, an between-group tie ij in X is rewired to a within-group tie ik in Y. Network segregation index S is monotonic in rewiring iff S(X, t) S(Y, t)

22 Nodes Adding isolates Property (Effect of adding isolates: ISO) Define two networks X = [x ij ] N N and Y = [y pq ] N+1 N+1 and associated type vectors u and w which are identical for the N actors and differ by an (N + 1)-th node which is an isolate: 1 p, q 1..N y pq = x pq 2 N+1 p=1 y p N+1 = N+1 q=1 y N+1 q = 0. 3 k 1..N w k = u k. S(X, u)? S(X, w) In words: how does the segregation level change if isolates are added to the network?

23 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 Y together as a single network, namely: Z = [z pq ] 2N 2N such that p, q {1,..., N} p, q {N + 1,..., 2N} otherwise z pq = 0 z pq = x pq z pq = y pq

24 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)

25 Measure Level Network type Scale Freeman network U [0; 1] SSI node U [0; ] g Assortativity network D/U [ 1 ; 1] g pg+p+g Gupta-Anderson-May network D/U [ 1 G 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 [ ; ]

26 Freeman (1978) Given two groups S Freeman = 1 p π where p is the observed proportion of between-group ties and π is the expected proportion given that ties are created randomly. It varies between 0 (random network) and 1 (full segregation of groups).

27 Assortativity Coefficient, Newman (2003) Based on a contact layer of the mixing matrix p gh = m gh1 /m ++1. S Newman = K g=1 p gg K g=1 p g+p +g 1 K g=1 p g+p +g Maximum of 1 for perfect segregation; 0 for random network. Negative values for dissasortative networks. Minimum depends on the density.

28 Gupta, Anderson & May 1989 Also based on contact layer of the mixing matrix S GAM = K g=1 λ g 1 K 1 Where λ g are eigenvalues of p gh. It varies between 1/(K 1) and 1

29 Coleman, 1958 Expected number of ties within group g mgg = n g 1 η i N 1 i G g S g Coleman = m gg m gg i G g η i m gg S g Coleman = m gg m gg m gg where m gg >= m gg (1) where m gg < m gg (2)

30 Segregation matrix index, Freshtman 1997 S SMI = d 11 d 12 d 11 + d 12 (3) where d 11 is the density of within-group ties and d 12 is the density of between-group ties.

31 Conditional Log-Linear Models (Koehly et al, 2004) log m gh1 = µ + λ A g + λ B h + λuhom gh log m gh1 = µ + λ A g + λ B h + λdhom gh { λ UHOM gh = λ UHOM g = h λ UHOM gh = 0 g h { λ DHOM gh = λ DHOM g g = h λ DHOM gh = 0 g h Parameters λ UHOM and λ DHOM g homophily/segregation. as measures of

32 ERGM Exponential Random Graph models log log ( mgh1 m gh0 ( mgh1 m gh0 ) ) = α + β A g + β B h + βuhom gh = µ + β A g + β B h + βdhom gh { β UHOM gh = β UHOM g = h β UHOM gh = 0 g h { βgh DHOM = βg DHOM g = h βgh DHOM = 0 g h Parameters β UHOM and βg DHOM homophily/segregation. as measures of

33 Spectral Segregation Index, Echenique & Fryer (2007) Segregation level of individual i in group g in component B: s g i (B) = 1 S g C i r ij s g j (B) (4) where r ij are entries in a row-normalized adjacency matrix. Segregation of individual i j S i SSI = l i l λ (5) where λ is the largest eigenvalue of B, and l is the corresponding eigenvector

34 SSI (2) Node segregation in White's kinship data Men Women Sister's Daughter Sister Sister's Husband Sister's Son Brother's Son Mother Father Brother Brother's Daughter Brother's Wife

35 Summary Measure MBG ( ) MWG ( ) MR ( ) ISO S ( ) Freeman SSI Assortativity Gupta-Anderson-May Coleman Segregation Matrix Index Uniform homophily (CLL) Differential homophily (CLL) Uniform homophily (ERGM) Differential homophily (ERGM)

36 Summary Measures on different levels: individuals, groups, global network 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 and Freeman decrease.

37 Summary: adding isolates Measures based on contact layer of mixing matrix are insensitive to isolates. SSI is the only one that always decreases The effect on others depend on relative group sizes.

38 Summary Measures based on contact layer of the mixing matrix summarize probability of node attribute combination given that the tie exists (CLL, assortativity, GAM): explaining attributes given the network. Measures that take also disconnected dyads into account. (ERGM, Freeman, SSI): explaining tie formation given the attributes.

39 Further questions Stricter formal analysis (axiomatizations). SSI is the only measure 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).

40 Thanks Thanks!