Introduction Problem Approach Properties Measures Summary Measuring Segregation in Social Networks Micha� l Bojanowski Rense Corten ICS/Sociology, Utrecht University July 2, 2010 Sunbelt XXX, Riva del Garda
Introduction Problem Approach Properties Measures Summary Outline Properties 4 Introduction 1 Ties Homophily and segregation Nodes Problem 2 Network Approach 3 Measures Approach 5 Notation Summary 6
Introduction Problem Approach Properties Measures Summary 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).
Introduction Problem Approach Properties Measures Summary 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).
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 0 1 2 3 4 5 6 7 8 9 10 11 Neighborhood structure can be conceptualized as a network in 12 13 14 15 16 17 which links correspond to neigh- borhood proximities. 18 19 20 21 22 23 24 25 26 27 28 29
Introduction Problem Approach Properties Measures Summary Homophily and segregation Assumption In static terms homophily and segregation correspond to the same 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 different networks (different school classes, different cities etc.) we need a measure .
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
Introduction Problem Approach Properties Measures Summary 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.
Introduction Problem Approach Properties Measures Summary 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 (Erd¨ os-Renyi random graph, preferential attachment, small-world etc.). Generate a collection of networks. Proceed as in the item above.
Introduction Problem Approach Properties Measures Summary 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 (Erd¨ os-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.
Introduction Problem Approach Properties Measures Summary 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 (Erd¨ os-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.
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 = { 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 .
Introduction Problem Approach Properties Measures Summary 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 � � � � (1 − x ij ) m gh 1 = x ij m gh 0 = i ∈ G g j ∈ G h i ∈ G g j ∈ G h
Introduction Problem Approach Properties Measures Summary 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.
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 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.
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 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.
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 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 )
Introduction Problem Approach Properties Measures Summary 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?
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 Y together as a single network, namely: Z = [ z pq ] 2 N × 2 N such that ∀ p , q ∈ { 1 , . . . , N } z pq = x pq ∀ p , q ∈ { N + 1 , . . . , 2 N } z pq = y pq otherwise z pq = 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)
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