Homophily and Community Structure in Networks

• Author: PRITHA DEV
• Date: 01 April 2016
• DOI: http://doi.org/10.1111/jpet.12142
• Published date: 01 April 2016

HOMOPHILY AND COMMUNITY STRUCTURE IN NETWORKS

PRITHA DEV

ITAM

Abstract

This paper proposes a strategy to estimate the community structure for

a network accounting for the empirically established fact that commu-

nities and links are formed based on homophily. It presents a maxi-

mum likelihood approach to rank community structures where the set

of possible community structures depends on the set of salient charac-

teristics and the probability of a link between two nodes varies accord-

ing to the characteristics of the two nodes. This approach has good

large sample properties, which lead to a practical algorithm for the es-

timation. Toexemplify the approach it is applied to data collected from

four village clusters in Ghana.1

1. Introduction

This paper examines homophily in networks with a particular focus on extracting the

salient characteristics for the fragmentation of networks. Homophily has long been

documented in the sociology literature as evident from the seminal paper by Lazars-

feld and Merton (1954). McPherson, Smith-Lovin, and Cook (2001) provide an excel-

lent overview of the vast literature studying homophily.2This literature points to net-

works organizing by homophilous groups with more links between members of the

same homophilous group than across groups. In this paper, an empirical methodol-

ogy is outlined to estimate the salient characteristics that lead to the formation of these

homophilous groups.

In particular, this paper seeks to uncover the community structure underlying a net-

work while directly accounting for the fact that communities within a network arise due

to homophily. A community is defined as a collection of nodes in which each member

of the community is more likely to have links with nodes from the community than with

1I would like to thank Chris Udry and Markus Goldstein for providing free access to data collected by

them from four villages in Eastern Region of Ghana.

2In the field of economics, a recent paper by Currarini, Jackson, and Pin (2009) shows the presence of

homophily in friendship networks.

I am thankful to Hans Haller for his suggestion regarding the name of the paper and to Alberto Bisin,

Prabal De, Kaushal Kishore, Guido Ruta, Julia Schwenkenberg, and Joerg Stoye for helpful comments

and suggestions.

I would also like to thank seminar participants at the LSU Networks Conference 2013. Support from

the Asociaci´

on Mexicana de Cultur´

a A.C. is gratefully acknowledged.

Received September 21, 2014; Accepted October 13, 2014.

C2014 Wiley Periodicals, Inc.

Journal of Public Economic Theory, 18 (2), 2016, pp. 268–290.

268

Community Structure in Networks 269

Figure 1: Raw network data.

(a) Sorted by Color (b) Sorted by Shape

Figure 2: Rearranged data.

nodes outside the community. A community structure is then the collection of all such

communities in a population. The novelty of this paper lies in focusing only on those

community structures that satisfy homophily along some characteristics. This greatly

reduces the costs of searching for the community structure for any given network.

In this paper, each node in a network is assigned an identity where identity is

defined along different dimensions and each dimension is composed of a fixed set

of discrete characteristics. Each node’s identity characteristic vector3consists of one

characteristic from each of these dimensions. This paper then seeks to find the dimen-

sions that lead to the fragmentation of networks. To fix ideas consider the network in

Figure 1 where nodes have identity along the dimensions of Color (white/black) and

Shape (square/triangle). Judging by this figure it is not clear which, if any, dimension

of identity is more important in partitioning the network. In Figures 2(a) and (b)

the network data are rearranged by Color and by Shape, respectively. These figures

make it visually clear that Shape is more important in generating the link data. The

estimation strategy is a generalization of the ideas presented in Figures 1 and 2(a)

and (b). It involves attaching likelihood numbers to the various possible partitions

and probabilities of link formation (varying by identity) and picking the partition and

corresponding probabilities of link formation, which maximize the likelihood.

The most important contribution of the paper comes from the fact that by account-

ing for homophily, the space of possible community structures can be reduced dramat-

ically. Suppressing network data, suppose the nodes are of the form given in Figure 3.

As in the previous figures, the nodes have characteristics along the dimension of Shape

and Color. Disregarding their characteristics, any partition defined over the set of nodes

is a possible community structure. Two such examples of the community structure are

given in Figure 4. The number of all possible community structures is a function of

the number of nodes and is increasing at an increasing rate with this number. An ex-

ample of the number of possible community structures is presented in Argyle (2013),

where it is shown that with 10 nodes the possible community structures are 115,975 while

with 25 nodes the possible community structures are 4,638,590,332,330,743,949. A phys-

ical search over all the possible community structures becomes impractical for a large

3Note that the identity characteristic vector attached to any node need not be unique as per the defi-

nition used in this paper.