Hierarchical Models for Independence Structures of Networks

TL;DR

This paper introduces hierarchical network models that explicitly represent dependencies among network ties, generalizing classic models like Erd"os-Rényi and beta-models to include complex dependency structures.

Contribution

It presents a new family of models linking graphical models with network dependence, extending existing models to incorporate explicit dyadic dependencies.

Findings

Generalized Erd"os-Rényi and beta-models with hierarchical structures

Methods for parameter estimation in models with sparse dependency graphs

Simulation studies demonstrating model properties

Abstract

We introduce a new family of network models, called hierarchical network models, that allow us to represent in an explicit manner the stochastic dependence among the dyads (random ties) of the network. In particular, each member of this family can be associated with a graphical model defining conditional independence clauses among the dyads of the network, called the dependency graph. Every network model with dyadic independence assumption can be generalized to construct members of this new family. Using this new framework, we generalize the Erd\"os-R\'enyi and beta-models to create hierarchical Erd\"os-R\'enyi and beta-models. We describe various methods for parameter estimation as well as simulation studies for models with sparse dependency graphs.