Exponential Random Graph models for Little Networks
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Statistical models for social networks have enabled researchers to study complex social phenomena that give rise to observed patterns of relationships among social actors and to gain a rich understanding of the interdependent nature of social ties and social actors. Much of this research has focused on social networks within medium to large social groups: from a couple of dozen students in a classroom, or colleagues in an organization; to larger social networks within schools, villages, and (online and offline) communities. To date, these advances in statistical models for social networks, and in particular, of Exponential-Family Random Graph Models (ERGMS), have rarely been applied to the study of small networks, despite small network data in teams, families, and personal (ego-centric) networks is common in many fields that study social phenomena. Furthermore, inferential degeneracy, which is a problem that arises in the case of Montecarlo quadrature needed to estimate this family of models, is one of the key issues that limit the usage of ERGMs in small networks. In this paper, we revisit the estimation of ERGMs for small networks and propose using exhaustive enumeration, and this, exact computation of likelihood functions to overcome the inference degeneracy problem that arises in the context of methods using approximations of it.
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