Parameter identifiability in a class of random graph mixture models

Abstract

We prove identifiability of parameters for a broad class of random graph mixture models. These models are characterized by a partition of the set of graph nodes into latent (unobservable) groups. The connectivities between nodes are independent random variables when conditioned on the groups of the nodes being connected. In the binary random graph case, in which edges are either present or absent, these models are known as stochastic blockmodels and have been widely used in the social sciences and, more recently, in biology. Their generalizations to weighted random graphs, either in parametric or non-parametric form, are also of interest in many areas. Despite a broad range of applications, the parameter identifiability issue for such models is involved, and previously has only been touched upon in the literature. We give here a thorough investigation of this problem. Our work also has consequences for parameter estimation. In particular, the estimation procedure proposed by Frank and Harary for binary affiliation models is revisited in this article.

New Consistent and Asymptotically Normal Parameter Estimates for Random-Graph Mixture Models

“…In Allman et al. (2011), the possible solutions (with respect to α and β ) of this set of moment equations are examined.…”

“…We first mention that identifiability of all the parameters ( π , α , β ) in the model that is defined by expressions (1) and (4), i.e. relying on the full distribution over (comprising the marginal distributions of the random graphs over a set of n nodes, for any value of n ), is a difficult issue, for which only partial results have been obtained in Allman et al. (2011).…”

Nonparametric Identification in the Dynamic Stochastic Block Model

“…The proof is along the lines of Allman et al (2011, theorem 14). Since the proof of that result is only sketched in Allman et al (2011), for completeness we provide a detailed proof of the following theorem.…”

“…Assumption a. in Theorem 1 cannot be satisfied in these cases. We develop an alternative set of conditions for point identification, complementing the generic identification results in for the dynamic and in Allman et al (2011) for the static case. Let us introduce a relevant object for identification.…”

“…In Theorem 1 we consider nonparametric classes of distributions for weighted edges under the assumption that the true underlying distributions are linearly independent. Next, in Theorem 2 we provide a result in the static setting for binary and finitely weighted edges with few edge states along the lines of the arguments in Allman et al (2011), but formulated and proved without referring to generic conditions. This is extended to the dynamic SBM in Theorem 3.…”