fedea

Fundación de Estudios de Economía Aplicada

Random–Walk–Based Segregation Measures by 六 Coralio Ballester ** Marc Vorsatz Documento de Trabajo 2010-30

Serie Inmigración CÁTEDRA Fedea-Banco Popular

November 2010

* Universidad de Alicante. ** FEDEA.

Jorge Juan, 46 28001 Madrid -España Tel.: +34 914 359 020 Fax: +34 915 779 575 infpub@fedea.es

Random–Walk–Based Segregation Measures^∗

Coralio Ballester^† and Marc Vorsatz^‡

October 29, 2010

First incomplete draft

Abstract

In this paper, we propose an intuitive way of how to measure residential segregation. Individuals are located in diferent nodes on a network that are interconnected through links. Each period, an individual either advances to an adjacent node or she stops moving. In this setting, the segregation index is then defined as the probability that a randomly chosen individual meets an individual of the same social group in the neighborhood where her random-walk terminates. It is shown in a dual theorem that the segregation index is as a natural generalization of the isolation index to networks and that it is proportional to the PageRank index applied by Google in order to determine the importance of webpages. Finally, the segregation index is applied to the Spanish 2009 census tract data and compared with other prominent measures.

Keywords: Isolation, Network, PageRank, random-walk, Segregation.

JEL–Numbers: C0, D85, Z13.

^∗We are very grateful to FEDEA for providing us with the necessary data for this analysis.

^†Corresponding author. Departamento de Fundamentos del Analísis Econónomico, Universidad de Alicante, Campus de San Vicente, 03080 Alicante, Spain. Email: coralio@merlin.ua.es. Financial support from the Spanish Ministry of Education and Innovation and FEDER, through the project SEJ–2007–62656, is gratefully acknowledged.

^‡Fundación de Estudios de Economía Aplicada (FEDEA), Calle Jorge Juan 46, 28001 Madrid, Spain. Email: mvorsatz@fedea.es. Financial support from the Spanish Ministry of Education and Science, through the Ramón y Cajal program and the project ECO2009–07530, is gratefully acknowledged.

1 Introduction

Motivation Residential segregation –the degree to which social groups live physically separate– is known to have adverse efects on a variety of socioeconomic variables. For example, Benabou (1993) studies theoretically a model with ex ante identical individuals who have to choose their level of education (they become either high-skilled, low-skilled, or drop out of the labor market) and where to live. Since education is assumed to be a club good, there is a benefit from being close to those who share the same characteristics and therefore, high-skilled workers are willing to pay more for their houses in a neighborhood with more high-skilled. It is then shown that the resulting stratification may not only turn neighborhoods with lesser educated into unproductive ghettos, the overall eficiency may even be inversely related to the level of segregation. Borjas (1995), on the other hand, investigates empirically the relation between residential segregation and human capital and finds that those with a higher ethnic capital (the average skills of the ethnic group in the parents generation) are more mobile. Finally, and as a last example, Cutler and Glaeser (1997) examine the efects of segregation for schooling, employment, and single parenthood in the US. Controlling for the choice of where to live, it is shown that blacks in more segregated areas have significantly worse outcomes than blacks in less segregated areas.

The most basic framework that addresses the question of how to actually measure segregation takes as a starting point some urban space (for example, a city) that is divided into smaller units called census tracts. Also one takes as given the distribution of the diferent social groups over the census tracts. Within this setting, the classical work of Massey and Denton (1988) introduces the following five dimensions of residential segregation: evenness or the extent to which a group is distributed homogeneously; exposure or the degree of potential contact to other social groups; concentration or the amount of physical space a social group occupies; centralization or the degree to which a social group resides close to the city center; and clustering or the extent to which individuals from the same social group tend to live in neighborhoods next to each other. The authors then review a total of 20 measures from which they isolate with the help of a factor analysis five, one for each dimension, that represent segregation best.

In a more recent study, Echenique and Fryer (2007) applied graph theory to derive a measure of segregation. The social interaction framework they study is based on the assumption that individuals divide their time equally among their neighbors and the more time they spend with individuals from their group, the more segregated the group as a whole is. More formally, individuals are located on the vertices of a graph with the edges representing social relations. Using this as the only information, the Spectral Segregation Index (SSI) for a given group is calculated as follows. One determines first the subgraph that only contains the interactions between members of the considered group. Generally such a subgraph consists of more than one connected components –a connected component is the largest set of individuals such that there is a path between any two individuals belonging to the set–, so the approach followed by the authors is to determine the segregation for each connected component separately and aggregate the values afterwards through a weighted average. In particular, the SSI of a con nected component is equal to its spectral radius, the largest absolute value of the eigenvalues of the matrix representing the social ties in that component. One particular advantage of this approach is that one can also obtain a level of segregation for every individual. This is done by rescaling the entries of the eigenvector corresponding to the spectral radius in such a way that their average is exactly equal to the spectral radius.

Model Our setting departs from the framework of Echenique and Fryer (2007) only in that we allow multiple individuals (of probably diferent social groups) to be located in the same node. This allows for more straightforward empirical applications in situations when one social group is in majority in almost all census tracts of a city, which happens to be the case in many empirical applications like ours. The measure we develop within this model is diferent in nature than the SSI and, in particular, it is related to the following random-walk: pick any two individuals from a given group s at random and suppose that the first of the two individuals starts to move over the network (city) from her position (area of residence) in such a way that each period, she either advances to some adjacent node with probability α (for instance, if node i has k adjacent nodes, then each of these nodes might be reached with probability or the process stops with probability 1 − α. The normalized segregation index for group s, denoted by , is then defined as the probability that the first individual meets the second individual when her random-walk terminates.

The use of a random-walk based model is justified in this setting because social interactions are often random. People meet friends, friends of friends or higher-order acquaintances in a random fashion. Also, socioeconomic relations are usually recursive, self-enforcing and present feedback efects. This implies that social interactions are not necessarily explained by shortest-path arguments, and a model based on random-walks is a convenient tool in order to take into account both the randomness and the recursive nature of social interactions.

Intuitively, the segregation index is controlling for the group size given that both individuals are chosen at random. It is also invariant to the city size and the total population and in this sense, it can be regarded as a normalized or relative coeficient of segregation. However, hidden in these invariance properties there is an important feature: it is independent on how the boundaries of census tracts are defined. In particular, if we assume that a census tract is a basic unit of interaction where every pair of citizens interact, then the definition of the boundary of a census tract will not afect . The social network can then be interpreted as a network of people, where each node is an individual and not merely as a network of arbitrarily defined nodes. This makes the choice of boundaries more natural –less arbitrary– given that, in fact the frontier of social interactions start at every social actor.

Nevertheless, it can be convenient to use a group-size dependent measure of segregation in some applications, so we also provide the non-normalized segregation index The interpretation of is then akin to that of the isolation index (an important measure of exposure) introduced by Blau (1977); i.e., the probability that a randomly chosen individual of group s will meet any individual from the same group after stopping.

Contribution Our main theoretical result is a dual theorem (Propositions 1 and 2) showing that the segregation index can be interpreted in two diferent ways. First, it is a natural spatial² generalization of the isolation index to networks. In fact, reduces to the isolation index of group s if the socio-geographical network is empty or, alternatively, if the transition probability α is zero, but generally it takes into account social ties to neighborhoods that are not too far away. Thus, is accounting for the intensity of interactions within group s. It should also be noted here that the measurement of isolation could be correlated to the level of interaction of group s with other groups. To say it diferently, more intra-group interactions prevents the group from socializing with groups, but, at the same time, more social links also allow for more inter-group interactions. In this respect, we stress the role of isolation as a stand-alone dimension of segregation whose measurement can be complementary to the study

¹In the literature, there is a variety of measures that are either relative or absolute regarding to group size. For instance, the isolation index, the exposure index and the mutual information index are sensitive to group sizes, even though the first two allow for straightforward normalizations. By contrast, the dissimilarity index, the Gini and the Atkinson index do not depend on the group size.

²We refer here to an abstract notion of space. Space could be regarded as the geographical or social setting where individuals interact.

of other dimensions of segregation.

Second, the segregation index is also related to the dimension of centralization because groups that are closer to the socio-geographical center of the network are more likely to interact with individuals from the same group everything else equal. The relevance of centralization as a dimension of segregation has been reported in some countries like the US, where socioeconomic factors can cause some groups to move to the city center. We show that the segregation index of a group s is a weighted sum over the vector of group proportions (the fraction of individuals of any node that belong to group s), where the weight assigned to each node is directly related to its centrality in the network. In particular, the centrality of a node is assessed through the PageRank index used by Google to determine the importance of webpages in the World Wide Web. Thus, more central districts receive more weight when measuring the segregation of a group.

A third theoretical contribution of the paper is related to a drawback of the SSI introduced by Echenique and Fryer (2007). In the Appendix, we present an example showing that the SSI is not continuous in the strength of the social ties because in situations when two connected components are linked, the SSI converges to the spectral radius of the bigger component instead of a weighted average over both connected components. We believe that continuity is not only desirable but a necessary property of any measure. Social networks (and also spatial networks) change over time and are subject to a continuous addition and deletion of links. In particular, abrupt changes in the SSI may occur if two individuals from completely disconnected components decide to interact for the first time. By considering the network structure explicitly our approach maintains the postulate of Echenique and Fryer (2007) that segregation is formed through social interactions, but we overcome the drawback of continuity by considering a measure that depends on the whole network.

Finally, we apply the segregation index to the Spanish 2009 census tract data. Groups are determined at the level of nationalities because social interactions naturally emerge between individuals from the same nationality since crucial aspects like culture, language and patriotism are often shared. In the last decade, Spain has experienced a considerable flow of immigrants, mainly attracted by the favorable conditions of the labor market, resulting in a total of 10% of non-national residents in the country. During these years, incoming residents have settled at diferent geographical locations not only based on economic factors like the location of the workplace and housing prices, but also on purely social reasons like proximity to close friends or family or feelings of empathy towards the location’s social environment. Ours is the first attempt to explore this geographic or social close-knitness of foreign residents in Spain since the start of the migration flow. As in Echenique and Fryer (2007) we use geographical data of residents’ households in order to compute our measure. This will allow us to explain patterns on the geographical distribution of groups (nationalities) in the country. But it will also shed some light on the social clustering of these groups since closer ties between two social agents usually have attached a smaller physical distance between them.

It is shown that the southern part of the Mediterranean coast is the most segregated area in the country, while northern autonomous communities of Galicia and the Basque Country together with the southern region of Andalusia are the least segregated ones. With respect to the diferent nationalities we find that the English and the Germans are among the most segregated groups, immigrants from Latin America, on the other, turn out to be rather integrated. The empirical study also reveals that network efects are strong for some groups like the Pakistanis and immigrants from African countries but weak for immigrants from European countries. Hence, adding the network structure on top of the isolation index is crucial for understanding the segregation of some but not all groups. Finally, the segregation index is positively correlated to the SSI but not strongly so, and the correlation between the segregation index and both the dissimilarity index and the Gini-coeficient, two important measures that recover the dimension of evenness, is between 0.5 and 0.7 depending on the value of the transition probability α.

Remainder In Section 2, we introduce the necessary notation and the definition of the segregation index. We also relate the segregation index to the dimensions of isolation and centralization. In Section 3, we apply the segregation index to the Spanish 2009 census tract data and compare the results to other prevalent measures in the literature. In Section 4, we conclude. Some additional results are relegated to the Appendices.

2 Theoretical Model

2.1 Notation and main definitions

Consider a finite set of n individuals. For simplicity we will call N a society. Individuals live in a city that is composed of a finite set of m nodes representing neighborhoods or census tracts. To be consistent with that interpretation, we assume that every individual is located at exactly one node but that each node possibly inhabits multiple individuals. Also, there is a set of k groups that forms a partition of the society. One can think of a group as a subset of members of the population that share a particular attribute such as gender, religion, or nationality. For any group , the number of individuals located at node i is given by . The number of individuals belonging to group is then equal to . Similarly, is the number of individuals located at node . Finally, the column vectors and with the generic entries and are referred to as the vectors of group densities and group proportions, respectively.

The diferent nodes in a city are interconnected through links. Formally, A is an matrix such that if there is a connection between i and j and otherwise. The general intuition is that two nodes are connected if they are geographically adjacent, or if agents at these nodes socially interact with each other. However from a theoretical point of view there is no need to assume that the graph is symmetric.³ Let be the number of nodes i is connected to by . Finally, it is assumed that so that each node can be accessed from itself.⁴ This assumption directly implies that

An matrix G is said to be a row stochastic matrix associated with A whenever the following conditions hold: whenever whenever , and . One particular way of constructing a row stochastic matrix G associated with A is to set for all (i, j)–entries; that is, the entries in row i are normalized by the number of connections of node i. We will apply this particular construction technique in our later application, however from a theoretical point of view one can allow for arbitrary values in the associated matrix as long as the restrictions above are satisfied. With this notation at hand we can now formally define a city as a tuple

The advantage of working with a stochastic G is that it provides a natural interpretation in terms of node–to–node transition probabilities. Formally, a walk ω is a sequence of nodes of the city. It is assumed that a walk continues every period . to an adjacent node with probability and that it ends with probability 1 − α. In case of continuation, the walk currently at node i passes to node j with probability . Thus, we define an –random-walk as a random variable whose realization is a particular walk, where the probability of a walk of length is given by

³Asymmetric graphs occur for instance when A represents a social communication structure. Also, friendship networks do not necessarily satisfy the property of symmetry. Finally, the mobility between two diferent social strata could be more sticky depending on the direction of movement. Consequently, our framework can also be applied to more abstract settings like social relationships and is not restricted to the geographica dimension of segregation.

⁴Thus, individuals from the same neighborhood are assumed to interact with each other. This assumption is purely technical and simplifies the analysis without losing generality. Even more importantly, it reduces the computational time for calculating the spectral segregation index of Echenique and Fryer (2007) as done in Section 4.

\[\mathsf {P r o b} _ {\omega} (\alpha , \mathbf {G}) = \alpha^ {h} (1 - \alpha) g _ {\omega_ {0}, \omega_ {1}} g _ {\omega_ {1}, \omega_ {2}} \dots g _ {\omega_ {h - 1}, \omega_ {h}}.\]

Observe that an –random-walk is completely defined through the row stochastic matrix G and the continuation probability α. Since , a realization ω of an random-walk is also a finite–length walk with probability 1. Let be the set of all possible realizations of an –random-walk. The expected length of an random-walk

\[l (\alpha , \mathbf {G}) = \sum_ {k = 0} ^ {\infty} k \sum_ {\omega \in \Omega (\alpha , \mathbf {G}): | \omega | = k} \operatorname{Prob} _ {\omega} (\alpha , \mathbf {G}) = \frac {\alpha}{1 - \alpha}.\]

The choice of the α is therefore associated with the choice of the expected length of a walk. For instance, corresponds to an expected length of about 6 steps and to an expected length of about 99 steps. So, one can regard the choice of the parameter α regarded as a coeficient of social viscosity reflecting the expected flow of interactions that an agent is likely to face.⁶

Finally, given the transition matrix G and the continuation probability let (or simply, P) be the matrix where its generic entry (or simply, is the probability that an –walk starting at node i ends at node j. This matrix depends on the pattern of interactions G and the continuation probability α and one of our first objectives will be to provide a closed formula of P in terms of the parameters. However, it can already be seen at this point that if or if (all nodes are disconnected from each other), no interaction takes place across neighborhoods and P reduces to the identity matrix. When α tends to 1, the expected walk length grows arbitrarily large, and the notion of social proximity vanishes, because becomes independent of the origin i.

l(α, G)

(a_i > 0),

l(α, G) = α (1 + l(α, G))

⁵Note that can be rewritten recursively through the formula . This is so because every node has an out-link which implies that the expected length of a walk is independent of the origin. Consider now any starting node ω . The random-walk stops with probability having length 0 (1 − α) 0. It continues with probability α and the individual moves to some node ω₁. The random-walk starting at ω1 ω₁ has, by definition, an expected length of Consequently, . l(α, G). l(α, G) = (1 − α) · 0 + α(1 + l(α, G))

⁶Throughout, we are going to assume that α is identical for all groups. This is a simplifying assumption because the mobility of a group is likely to be related to various socioeconomic factors like income, education, and age. However, the model can be straightforwardly extended to take these kind of dependencies into account.

We are now ready to suggest a measure of residential segregation that is related to the level of within group interaction, taken into account through the matrix G. In particular, the segregation index for group in city C is defined as the probability that a randomly chosen individual of group s meets an individual of the same group in the node where she stops her random-walk.

Definition 1. Given a city and a continuation probability , the (non-normalized) segregation index of group is

\[\sigma_ {s} (C, \alpha) = \sum_ {i \in M} b _ {s, i} \sum_ {j \in M} p _ {i, j} c _ {s, j} = \mathbf {b} _ {s} ^ {\top} \mathbf {P} \mathbf {c} _ {s}.\]

It should be remarked that the measure depends on the number of individuals that belong to group s. To see this size dependence, take any city and consider the city that can be obtained from by doubling the fraction of individuals of group s in each node maintaining the total population in each node fixed. One easily sees that and that . Consequently, the segregation of group s has doubled simply because the group has grown but not because its relative structure in the society has changed. Intuitively, individuals from group s are now twice as likely to meet in city as in city given that the network structure did not change. This efect can be accounted for (as we do in the empirical application in Section 3) with a modified version of that normalizes the measure by the relative group size.

Definition 2. Given a city and a continuation probability , the normalized segregation index of group is

\[\bar {\sigma} _ {s} (C, \alpha) = \left(\frac {n _ {s}}{n}\right) ^ {- 1} \sum_ {i \in M} b _ {s, i} \sum_ {j \in M} p _ {i, j} c _ {s, j} = \left(\frac {n _ {s}}{n}\right) ^ {- 1} \mathbf {b} _ {s} ^ {\top} \mathbf {P} \mathbf {c} _ {s}.\]

This normalized segregation measure has an alternative interpretation that clearly reflects this size-independence: is related to the probability that a randomly chosen agent from group s will meet another randomly chosen agent from the same group. The fact that both agents are chosen at random prevents this probability from depending on the representativeness this group in the society. To see this, suppose that for all ; that is, the population is the same in all census tracts. We can then rewrite the normalized segregation index as

\[\bar {\sigma} _ {s} (C, \alpha) = m \sum_ {i \in M} b _ {s, i} \sum_ {j \in M} p _ {i j} (\alpha , \mathbf {G}) b _ {s, j} = m \mathbf {b} _ {s} ^ {\top} \mathbf {P} \mathbf {b} _ {s},\]

where denotes, as usual, the total number of nodes.

Finally, observe that the segregation index for city C can be straightforwardly defined as the weighted average over the segregation indices for each group; that is, . The normalized segregation index for city C is defined accordingly.

It will be shown next with the help of a dual theorem that the segregation index incorporates two of the five dimensions introduced by Massey and Denton (1988), namely those of exposure and centralization, directly. The segregation index of a group can on one hand be interpreted as the degree of within group interactions in the neighborhoods across the city and in particular we are going to show that the measure is a natural generalization of the well-known isolation index to networks (Proposition 1). On the other hand, the segregation index can also be expressed in terms of the centrality of a group. By this we mean that groups that are more central in the social network are more likely to interact not only with others but also with individuals from the same group. This implies that groups that are located more central are more segregated everything else equal. To formally relate the segregation index to the notion of centrality, we will present an equivalent definition on the basis of Google’s PageRank index that measures the importance of websites in the World Wide Web (Proposition 2). Observe that both results are essential as they derive specific formulae for the computation of the segregation index in real-life examples as in Section 3.

2.2 The dimension of isolation

Isolation refers to the tendency to interact with individuals from the same group. This notion is captured through the isolation index , which is defined as the average density of group s in the neighborhood of a typical individual from the same group:

\[I _ {s} (C) = \sum_ {i \in M} b _ {s, i} c _ {s, i} = \mathbf {b} _ {s} ^ {\top} \mathbf {c} _ {s}.\]

Even though the relation between the isolation index and the segregation index is clearly visible, the following proposition helps us to relate them more directly.

Proposition 1. Given a city and a continuation probability , the segregation index of group is

\[\sigma_ {s} (C, \alpha) = (1 - \alpha) \mathbf {b} _ {s} ^ {\top} (\mathbf {I} - \alpha \mathbf {G}) ^ {- 1} \mathbf {c} _ {s}.\]

Proof. By definition and therefore, it only remains to be shown that . Consider , which is the probability that a walk starting at node i ends at node . In the first period, the walk starts with probability α. Given that the walk starts, the individual reaches node at the end of period 1 with probability . Now, the probability that a walk starting at node ends up at node is, by definition, equal to . Therefore, we have that . Following a similar argument, one can show that . Rewriting these equations in matrix form yields , which is equivalent to □

It follows from Proposition 1 that the segregation index can be calculated as the average proportion of group s in the neighborhood that is reached by a typical s−member in a random-walk. Note that when all census tracts are disconnected —that is, when or when there is no node-to-node transition —that is, when the segregation index reduces to the isolation index . Hence, the segregation index generalizes the isolation index by taking the connectivity among census tracts in the city explicitly into account. Also, the measure can be interpreted as a weighted average over the entries of the vector , where the weight of each node is equal to . Since contains therefore the contributions of all census tracts to the segregation of group s, we will call it the vector of local isolations of group s. Since the proposition also provides us with a concrete formula of how to calculate the segregation index, we conclude with an example.

Example. Consider the city C depicted in Figure 1.

There are two ethnic groups, blacks and whites. It is assumed that 2 whites live in the nodes 1 and 3, 3 whites live in node 4 and 1 white lives in node 2. On the other hand, 1 black lives in the nodes 1, 2 and 4 while 3 blacks live in node 3. If we suppose that an individual moves to all adjacent nodes with the same probability, the transition matrix corresponding to the network structure in Figure 1 is

\[G = \left( \begin{array}{c c c c} 1 / 3 & 1 / 3 & 1 / 3 & 0 \\ 1 / 3 & 1 / 3 & 1 / 3 & 0 \\ 1 / 4 & 1 / 4 & 1 / 4 & 1 / 4 \\ 0 & 0 & 1 / 2 & 1 / 2 \end{array} \right).\]

Observe also that the distribution of individuals across nodes equals ， , and . One can then verify that and that

2.3 The dimension of centralization

Generally speaking the centrality of a node in a network captures its well–connectedness. Depending on the specific context, it can for example be assessed using the notions of degree (the number of connections a node has), betweeness (determine the shortest paths between any two nodes and calculate then for each node on how many of these shortest paths it belongs to), or closeness (the mean geodesic distance between a node and all nodes reachable from it). In our case, the segregation index belongs to the family of eigenvector-based measures and it is in particular related to the PageRank index of Brin and Page (1998) that underlies Google’s search engine.

The main idea of the PageRank index is that a webpage is more relevant given a query and therefore listed higher up by the search engine when a random surfer on the internet is more likely to arrive to that webpage. What diferentiates this approach from our problem is that the random surfer starts a walk without stopping probability. Instead, at each moment in time, the surfer continues her random-walk to a linked webpages or she is teleported away to another node. Afterwards, the same process continues.

To formally introduce the PageRank index, let S be a row stochastic matrix that governs the transition of the random surfer between any two webpages. The parameter denotes the probability of teleportation, the chance that the random surfer is taken away to a randomly chosen node. In case of teleportation, each node i is taken with probability . The vector q is called the personalization vector. The PageRank index r is then defined recursively as the solution to the systems of equations . Observe that the vector r is unique and that it corresponds to the stationary probabilities with which the random surfer arrives to each node under the described Markov chain. In this respect, a webpage that receive more visits in the stationary state attains a higher PageRank index.

One easily relates the components of our setting to the problem of how to rank webages. The graph G derived from the city map corresponds to the transition matrix S between webpages and our termination probability α corresponds to the probability that a random surfer is teleported to a diferent webpage. Since we can also use the vector of group densities for teleportation —that is, in this random-walk the individual is teleported to the neighborhood of a randomly chosen individual from the same group— the PageRank index for group s is to be defined as the solution to the system of linear equations

Definition 3. Given a city and a continuation probability , the PageRank index of group is

The following proposition is the dual of Proposition 1 stating that the segregation index of a group s is a weighted sum over the PageRank index for each node.

Proposition 2. Given a city and a continuation probability

, the segregation index of group s is

\[\sigma_ {s} (C, \alpha) = \mathbf {c} _ {s} ^ {\top} \mathbf {w} _ {s} (C, \alpha).\]

The PageRank index can also be calculated as the principal eigenvector of the matrix

Proof. By definition , which can be rewritten as . Following the very same line of argumentation as in Proposition 1, it can then be shown that . Hence, it only remains to be shown that the principal eigenvector x of the matrix is equal to . Since T is column stochastic, its principal eigenvalue is 1 and x satisfies the equation . Taking an x whose coordinates sum up to 1, we get that . By definition, □

Proposition 2 establishes formally that social groups that are located more central in the network have a higher segregation index. We conclude this part of the paper with an example showing that the spectral segregation index of Echenique and Fryer (2007) is not related to the dimension of centralization.

Example. Consider the city C depicted in Figure 2.

There are eleven individuals belonging to three ethnic groups (blacks, whites, and diamonds). At every node in the network, there is exactly one individual. The particularity of this social network structure is that the blacks and the whites are allocated in a very similar way. Every black is connected to two other blacks plus herself, one white, and one diamond. Also, every white is connected to two other whites plus herself, one black, and one diamond. The important diference between the two groups is that all blacks are connected to the same diamond (individual 11), while two whites are connected to diamond 9 and two whites to diamond 10. Hence, diamond 11 is the most central individual of her group. This implies that the blacks are located relative more central than the whites and are thus more segregated. Since the vectors of group proportions and are completely symmetric, the segregation index incorporates the diferences between the groups entirely through the PageRank indexes and . In particular, we obtain for that whenever i is a black and that whenever i is a white. As a consequence,

The SSI on the other hand is not related to the dimension of centrality because it only considers the network of within group interactions, and thereby misses that the blacks are connected to a more central diamond than the whites. Indeed, in this example the sub networks are identical for both groups so that

3 Application

3.1 Segregation in Spain

In this section, the segregation index is applied to the Spanish census tract data from January 2009. Among the developed countries, Spain is particularly interesting to look at because the country attracted a lot of immigrants from many diferent parts of the world over the last decade. During the boom years, it was easy for the young South Americans to find a job in the construction or the (private) service sector because they were relatively cheaper to hire and had the advantage of speaking the same native language. But also immigrants from the Eastern European countries that recently entered the European Union or Africa were also attracted by the vast job opportunities in the labor market. At the same time, Europeans from countries with a relatively higher GDP like England and Germany invested into the new residential areas at the Mediterranean coast. These combined efects led to the current situation that more than 10 % of the 45 million residents in Spain are foreigners.

Table 1: Residents in Spain as of January 2009 according to the country of origin. The data is made available by the National Statistical Institute (INE) of Spain.

Table 1 presents the population shares of the 29 nationalities with the highest number of residents. It can been seen that the Rumanians form the largest foreign group with 1.74 % of the total population followed by the Moroccans who account for 1.54 % of the residents. The high number Moroccans in Spain is of no surprise given their long tradition in the country. Next are Ecuador (0.92 %), Great Britain (0.82 %), Colombia (0.65 %) and Germany (0.41 %). Finally, the smallest international group are the Nigerians who amount to only 0.09 % of the total population.

The Spanish territory is oficially divided into 52 provinces, 47 of those are on the Iberian Peninsula. The remaining 5 ones are the Balearic Islands in the Mediterranean Sea, the Canary Islands in the Atlantic Ocean (2 provinces), and the autonomous cities Ceuta and Melilla in North Africa. We abstained from incorporating Ceuta and Melilla in our analysis because they are too small —each of the two cities has only about 75.000 residents and very few non-Africans. As of January 2009, the National Statistical Institute of Spain (INE) divides the 50 main provinces into a total of 35,757 census tracts. The mean number of residents per census tract is 1,284.95 with a typical standard deviation of 657.13. As in Echenique and Fryer (2007), in order to define the network, we use the geographical location of the centroids of each census tract. In particular, we define two census tracts to be connected if the distance between their centroids is less than 400 meters.⁷ A node has then on average 4.50 connections with a standard deviation of 5.48. Also, there are 18,114 isolated nodes. To see that this neighborhood radius is appropriately defined, note that the capital of Madrid is divided into a total of 2,397 census tracts and that its biggest connected component consists of 1,951 census tracts. Also, Echenique and Fryer (2007) apply a neighborhood radius of 1,000 meters in the United States, which is far bigger and less densely populated. In this case the neighborhood radius is set to 1,000 meters, the average number of links increases to 20.78 with a corresponding standard deviation of 27.10. Still there are 13,789 isolated nodes.

⁷Some clarifications are in order. First, we use of purely geographical positioning data in order to construct the network: the network is geographical. Nevertheless, we point out that it also proxies true social interactions because geographical proximity is naturally correlated to close relationships, even though it is true that new era of information has made weak social links less dependent on physical distance (see for instance Goldenberg and Levy, and Mok et al.). Thus, in our application we are analyzing not only geographical clustering, but also social clustering of residents of diferent nationalities. Second, the choice of 400 meters is arbitrary, based on the fact that actual neighboring census tracts should also be connected in the constructed network. Nevertheless, our analysis remains robust to changes in this choice. In particular, we also performed calculations for radii of 1000 and 1500 meters, and our results remain the same.

To compare the segregation index across provinces, we calculate for each province separately; that is C is defined by the borders of the provinces. We set α equal to 0.85 because it is a prominent choice in other application related to the PageRank index as well.⁸ The results are graphically represented in Figure 3.

We can see that the segregation index is lowest in the autonomous communities of Galicia (provinces 15, 27, 32, and 36), Asturias (province 33), Cantabria (province 39) and the Basque Country (provinces 01, 20, and 48) that are all in northern part of Spain touching the Cantabrian Sea and the Atlantic Ocean. Also, some parts of Andalusia such as Cordoba,

α ∈ {0.00; 0.25; 0.50; 0.70; 0.99}

⁸The numerical results for can be found in the Appendix.

Jaen, and Seville (provinces 14, 23, and 41) have a segregation index between 1.00 and 1.15. The largest part of the country has a moderate segregation index between 1.16 and 1.30. It includes, for example, Madrid (province 28), Barcelona (province 08), and Valencia (province 46), which are the countries biggest cities. The segregation index takes slightly higher values within the autonomous community of Catalonia (provinces 17, 25 and 43) in northeastern part of the country and the Balearic Islands (province 07). The segregation index is high —between 1.46 and 1.60— in the Canary Islands (provinces 35 and 38) and Murcia (province 30). It reaches its maximum in the coastal sides of Alicante (province 03) and Almeria (province 04).

Figure 3 uncovers how segregated diferent parts of the country are, but so far we have not analyzed which groups are causing the results we see. To say it in diferent words, we still have to investigate which groups are, on average, more segregated. To detail on this, we calculate the normalized segregation index for each of the 29 groups in the whole country; that is, C consists now of all 35,757 census tracts. Obviously, we take α to be equal to 0.85 again. The respective results are presented in Figure 4.

Figure 4: Normalized segregation in Spain by groups as of January 2009 for and a neighborhood radius of 400 meters. The ordering of the diferent groups corresponds to that in Table 1.

It can be seen that the Spanish are the least segregated group. This is because they are evenly distributed over all census tracts. The most segregated groups are by far the British followed by the Pakistanis and the Germans . The Rumanians and the immigrants from the African countries of Algeria, Nigeria, and Senegal have a segregation index of about 10 and are therefore less segregated than the formerly mentioned groups, however they still tend to cluster in the same neighborhoods. The immigrants of the South American countries, on the other hand, are the most integrated ones as their segregation index lies between 2.32 in the case of the Colombians and 5.94 in the case of the Uruguayans. Finally, one can use Figure 2 to explain why Alicante is the most segregated province. Overall, 0.82 % of the residents are British and 0.41 % are German. But, in Alicante 6.91 % are British and 1.97 % are German. Consequently, the segregation in Alicante is high because the more segregated groups are over-represented in this province.

3.2 Network efects

One important question at this point is whether the incorporation of the social network as an additional dimension adds to our understanding of segregation. To investigate this, one has to compare the segregation index for α = 0.85 presented above with the segregation index for when there is no network efect and the measure reduces to the normalized version of the isolation index.

Figure 5 presents the normalized isolation index for Spain as a whole. Comparing it with Figure 4, it can be seen that the network efect is substantial for some groups but negligible for others. The clearest efect can be identified for the Pakistanis: introducing the social network reduces the normalized segregation index of this group from more than 50 to about

Figure 5: Normalized segregation in Spain by groups as of January 2009 for and a neighborhood radius of 400 meters. The ordering of the diferent groups corresponds to that in Table 1.

25. Other countries that present a significant network efect are, for example, Nigeria and Senegal. However, it is not the absolute diference that matters (the segregation index is decreasing in α for all groups), more important is the fact the Pakistanis are by far the most segregated group if , while the British are the most segregated group if Actually, the network efect for the British and the German is rather small as the segregation index of these two groups hardly changes between and

We now proceed by calculating for all groups the correlation between the vectors of local isolations and to quantify the network efect. The correlation between the two vectors is highest in case of the British (0.9964) and Germans (0.9945). Consequently, and as we have argued before, the social network only plays a minor role in determining the segregation of these groups. The correlation is lowest (0.8172) in case of the Pakistanis. So, network efects are essential for understanding the segregation of this group. The average (median) correlation between the two vectors calculated over all groups is 0.9122 (0.9055). The correlations above are calculated at the country level, but they change to some extent if we study the provinces on their own. For example, the average correlation between the two vectors across groups tends to be on the lower end in provinces with bigger cities like Catalonia (0.8304), Madrid (0.8337), Valencia (0.8359), and Vizcaya (0.8056) but high in provinces with few inhabitants like Almeria (0.9549), Avíla (0.9711), Guadalajara (0.9666) and Segovia (0.9700).

3.3 Correlation with other measures

Before coming to a comparison with measures of evenness, we focus on the SSI proposed in Echenique and Fryer (2007). The studies are intimately related because they both calculate the eigenvector of a (sub-)stochastic matrix. Even though Echenique and Fryer (2007) considers a setting with a one–to–one mapping between nodes and individuals, it is possible to redefine their framework in such a way that a direct comparison to our segregation index becomes possible.

As laid out before, the basic idea behind the SSI is that individuals in a given node interact with their “neighbors”. If one node corresponds to one individual, as it is the case in Echenique and Fryer (2007), the set of neighbors of an individual naturally includes all nodes she is connected to. We allow for multiple individuals living in the same node, however the two frameworks become comparable if one defines the set of neighbors as all those individuals that can be reached within one step. Following this idea, let be the total population around node i. Remember that whenever the nodes i and j are connected and that otherwise. A social interaction network H_s for group s is then defined by setting

\[h _ {i, j} = \left\{ \begin{array}{c l} \frac {n _ {s , j}}{\hat {n} _ {i}} & \text {if a_{i,j} = 1} \\ 0 & \text {if a_{i,j} = 0} \end{array} \right..\]

This implies that an individual from group s living in node i interacts with all s−members in her neighborhood with equal probability. Note that is a sub-stochastic matrix because it reflects interactions restricted to members of s and therefore, its rows do not generally sum up to one. In order to capture the segregation of group s, the SSI measures the density of the restricted interactions within group s. In particular, one first computes for each connected component of the spectral radius of the sub-matrix corresponding to this component. ⁹ We denote the SSI of group s within γ by . The total segregation of group s is then defined as the weighted average over all connected components:

\[S S I (\mathbf {H} _ {s}) = \sum_ {\gamma} b _ {s} ^ {\gamma} \cdot S S I (\mathbf {H} _ {s} ^ {\gamma}),\]

where the weight corresponds to the fraction of individuals of group s living in component γ. Also, the spectral radius for each connected component turns out to be equal to the weighted average of the corresponding eigenvector; that is,

\[S S I (\mathbf {H} _ {s}) = \sum_ {\gamma} b _ {s} ^ {\gamma} \sum_ {i \in \gamma} b _ {s, i} ^ {\gamma} \cdot S S I _ {i} (\mathbf {H} _ {s} ^ {\gamma}),\]

where is the fraction of individuals of group s from component who are located at node i and is the i−th entry of the eigenvector corresponding to the spectral radius SSI(H^γ_s ). We can then rewrite the former equation as

\[S S I (\mathbf {H} _ {s}) = \sum_ {\gamma} \sum_ {i \in \gamma} b _ {s} ^ {\gamma} \cdot b _ {s, i} ^ {\gamma} \cdot S S I _ {i} (\mathbf {H} _ {s} ^ {\gamma}) = \sum_ {i \in M} b _ {s, i} \cdot S S I _ {i} (\mathbf {H} _ {s} ^ {\gamma (i)}).\]

Consequently, the spectral segregation index for a group s can be envisioned as a weighted average over the spectral segregation indexes of all nodes . Since the SSI is not invariant to the size of the group s, we have to compare it to the size dependent version of the segregation index, which according to Proposition 1 is equal to

\[\sigma_ {s} (C, \alpha) = \sum_ {i \in M} b _ {s, i} v _ {s, i}.\]

⁹The spectral radius of a matrix is the largest absolute value of the elements in its spectrum.

Consequently, we proceed by calculating the correlation between the two vectors and of size 32,757 (corresponding to the census tracts of Spain as a whole) for each of the 29 nationalities taking α = 0.85 and a direct neighborhood radius of 400 meters. The results are presented in Figure 6.

Figure 6: Correlation between the segregation index and the SSI in Spain by groups as of January 2009 for and a neighborhood radius of 400 meters. The ordering of the diferent groups corresponds to that in Table 1.

It can be seen that there is a positive correlation between the SSI and the segregation index for all 29 nationalities. For some groups like the Italians, the Polish, the Portuguese and the British, the correlation is considerable and lies between 0.38 and 0.48. For the immigrants from Morrocco and Bolivia, on the other hand, the correlation is rather small.

In the final part of the paper, we relate the segregation index to the dimension of evenness, which according to Echenique and Fryer (2007) is an important dimension of segregation to be considered in socio-economic applications because one ultimately cares about how segregation afects social interactions dimension. This dimension is related to how a social group is distributed across nodes and therefore, all measures belonging to that class should be a function of the vector of group densities . The most important measures reviewed by Massey and Denton (1988) satisfying this criterion are the dissimilarity index of Jahn et al.

(1947) and Gini-coeficient (see, Gini (1921)). The dissimilarity index corresponds to the fraction of individuals from a social group that have to be reallocated so that the fraction of individuals of that group in each node corresponds to the percentage the group accounts for in the whole city. Formally, the dissimilarity index for group s in city C is defined as

\[D _ {s} (C) = \frac {1}{2} \sum_ {i = 1} ^ {m} \left| \frac {n _ {s , i}}{n _ {s}} - \frac {n _ {i} - n _ {s , i}}{n - n _ {s}} \right| = \frac {1}{2} \sum_ {i = 1} ^ {m} \left| b _ {s, i} - \frac {n _ {i} - n _ {s , i}}{n - n _ {s}} \right|.\]

The Gini-coeficient, on the other hand, is traditionally applied to assess the inequality of an income distribution and it corresponds to the area below the Lorenz curve. Fortunately, the Gini-coeficient can be adapted to our setting if we reinterpret nodes to be individuals and the vector of group densities to be income shares. Under this interpretation, we can apply the formula derived by Deaton (1997) that calculates the Gini-coeficient for group s in city C as

\[G _ {s} (C) = \frac {m + 1}{m - 1} - \frac {2}{m - 1} \sum_ {i = 1} ^ {m} i \cdot \bar {b} _ {s, i},\]

where is the i–th highest value of the vector

In order to study how these to measure relate to the segregation index, observe that for each group s we have a total of fifty observations of and , one for each province. Consequently, we calculate the correlation between each of these two vectors and The average of these correlations over the 29 nationalities are presented in Table 2.

Table 2: Average correlation between the normalized segregation index and measures of evenness.

The numbers reveal that the average correlation between the segregation index and both the dissimilarity index and the Gini-coeficient is positive and greater than 0.50 for the whole range of α’s considered. Also, the average correlations tend to be (slightly) decreasing in α.

4 Conclusion

In this paper, we have developed a new measure of residential segregation based on social interactions. In our theoretical model, the nodes of a network represent neighborhoods or census tracts and links indicate which census tracts are adjacent in the urban space. It also assumed that every individual belonging to the society is located at only one node but that multiple individuals (from possibly diferent social groups) can be located at the very same node. Using this information as the only primitive of our analysis, we study the following Markov: every period, an individual advances to an adjacent node with a given probability or the individuals stops moving over the network. The segregation index is then defined as the probability that a randomly chosen individual from a given group meets an individual from the same social group in the node where her random-walk terminates.

It is shown that the segregation index has several favorable aspects. First, the measure reduces to the isolation index in case the network is empty or in case the exogenous probability that the random-walk stops is one. Consequently, the segregation index can be interpreted as a natural generalization of the isolation index to networks. Second, the segregation index is the first measure that easy to apply and incorporates the idea that social groups that are located closer to the city center are more segregated everything else equal. In particular, the segregation index turns out to be proportional to the PageRank index applied by Google to determine the importance of webpages in the World Wide Web. Finally, the segregation index is a continuous function in the social ties. The SSI suggested by Echenique and Fryer (2007), who have been the first to develop a measure residential segregation on the basis of social interactions, fails to satisfy this important criterion.

In our empirical application, we study the Spanish 2009 census tract data. Our main results show here that the provinces on the Mediterranean coast are the most segregated areas, mainly because the most segregated nationalities, the British and the Germans, are overrepresented in these regions. Also, network efects are crucial to understand the segregation of some of the some smaller nationalities like the Pakistanis and immigrants from African countries like Nigeria and Senegal but not for the bigger communities of the British and the Germans. On the other hand, network efects are substantial in provinces with bigger cities like Madrid, Barcelona, Valencia, and Bilbao. Finally, the segregation index is positively correlated to the SSI and to traditional measures of evenness such as the dissimilarity index and the Gini-coeficient.

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Appendix

The following example shows in an intuitive way that the SSI is not continuos in the network structure (contrary to what is stated in Proposition 5 of Echenique and Fryer (2007)). The panel on the left-hand side in Figure 7 corresponds to the motivating example in Echenique and Fryer (2007). The society living in City 1 is composed of two ethnic groups, blacks and whites. In this setting, each dot in the panel embodies a census tract, and its representative individual is one of the majority group in that particular census tract. Therefore, it only remains to define the social ties to be able to calculate the SSI. For the sake of simplicity it is assumed that individuals only interact with their horizontal and vertical neighbors. So, for example, individual (A,1) spends 50% of her time with each (A,2) and (B,1), while individual (C,2) interacts 25% of her time with each (C,1), (C,3), (B,2) and (D,2). As a consequence, the subgraphs restricted to within group interactions consists of two connected components for the blacks and one big component for the whites.

Figure 7: Examples for determining the SSI.

To finally determine the SSI for the blacks as a group one has to take a weighted average over the spectral radii of the two connected components. The upper part of Table 3 shows that the SSI in the two relevant components of City 1 equals 0.72 and 0.25, respectively. Using that 80% of the blacks reside in component 1 and 20% in component 2, the SSI for the blacks as a group in City 1 equals . The individual segregation levels are finally obtained by rescaling the eigenvectors corresponding to the spectral radii in such a way that their averages are equal to the spectral radii. In particular, one can see that individual (D,1) is the most segregated black in City 1.

Table 3: Calculation of the SSI.

To see that the SSI is not continuous in the strength of the social ties suppose that (D,4) starts communicating 1% of her time with (C,5). It does not matter which neighbor(s) receive now relative less attention from (D,4), for the sake of the example just suppose that this extra time is taken away from the white neighbors so that all interactions with the other blacks remain unafected. This small change in the network leads to the integrated City 2, which is depicted in the right-hand panel of Figure 7. As it can be seen in the lower part of Table 3, the efect of this change on the SSI of the blacks is substantial. In particular, the spectral radius of the integrated component converges to the spectral radius of the originally bigger component. As a consequence, individuals (B,5) and (C,5) receive a weight close to 0 while the relative weight between any two blacks who originally formed part of component 1 remains unchanged.

Table 4 : Normalized segregation index in Spain as of January 2009 for α = 0 . 00 (normalized isolation index) and a neighborhood radius of 400 meters .

Table 5 Normalized segregation index in Spain as of January 2009 for α = 0 . 25 and a neighborhood radius of 400 meters .

Table 6 Normalized segregation index in Spain as of January 2009 for α = 0 . 50 and a neighborhood radius of 400 meters .

Table 7: Normalized segregation index in Spain as of January 2009 for α = 0 . 70 and a neighborhood radius of 400 meters .

Table 8 Normalized segregation index in Spain as of January 2009 for α = 0 . 85 and a neighborhood radius of 400 meters .

Table 9 Normalized segregation index in Spain as of January 2009 for α = 0 . 99 and a neighborhood radius of 400 meters .

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