The random subgraph model for the analysis of an ecclesiastical network in Merovingian Gaul

TL;DR

This paper introduces a novel stochastic model for analyzing typed, partitioned networks with latent clusters, demonstrated on a historical ecclesiastical network from Merovingian Gaul, with an inference algorithm and model selection method.

Contribution

The work develops a new random subgraph model incorporating known partitions and edge types, with a variational Bayes inference algorithm and model selection for cluster estimation.

Findings

Effective modeling of typed, partitioned networks.

Successful application to a historical ecclesiastical network.

Availability of R code for reproducibility.

Abstract

In the last two decades many random graph models have been proposed to extract knowledge from networks. Most of them look for communities or, more generally, clusters of vertices with homogeneous connection profiles. While the first models focused on networks with binary edges only, extensions now allow to deal with valued networks. Recently, new models were also introduced in order to characterize connection patterns in networks through mixed memberships. This work was motivated by the need of analyzing a historical network where a partition of the vertices is given and where edges are typed. A known partition is seen as a decomposition of a network into subgraphs that we propose to model using a stochastic model with unknown latent clusters. Each subgraph has its own mixing vector and sees its vertices associated to the clusters. The vertices then connect with a probability depending on the subgraphs only, while the types of edges are assumed to be sampled from the latent clusters. A variational Bayes expectation-maximization algorithm is proposed for inference as well as a model selection criterion for the estimation of the cluster number. Experiments are carried out on simulated data to assess the approach. The proposed methodology is then applied to an ecclesiastical network in Merovingian Gaul. An R code, called Rambo, implementing the inference algorithm is available from the authors upon request.