# Random Networks, Graphical Models, and Exchangeability

## TL;DR

This paper explores the structure of exchangeable random network models, characterizing their Markov properties, and identifying classes of models based on subgraph densities, degree distributions, and consistency under subnetworks.

## Contribution

It provides a comprehensive characterization of Markov structures for finitely exchangeable networks and introduces new classes of models linked to bidirected Kneser graphs.

## Key findings

- Empirical subgraph densities are maximum likelihood estimates.
- Identified a new class of Markov network models for exchangeable graphs.
- Only certain graph structures are consistent under subnetwork formation.

## Abstract

We study conditional independence relationships for random networks and their interplay with exchangeability. We show that, for finitely exchangeable network models, the empirical subgraph densities are maximum likelihood estimates of their theoretical counterparts. We then characterize all possible Markov structures for finitely exchangeable random graphs, thereby identifying a new class of Markov network models corresponding to bidirected Kneser graphs. In particular, we demonstrate that the fundamental property of dissociatedness corresponds to a Markov property for exchangeable networks described by bidirected line graphs. Finally we study those exchangeable models that are also summarized in the sense that the probability of a network only depends onthe degree distribution, and identify a class of models that is dual to the Markov graphs of Frank and Strauss (1986). Particular emphasis is placed on studying consistency properties of network models under the process of forming subnetworks and we show that the only consistent systems of Markov properties correspond to the empty graph, the bidirected line graph of the complete graph, and the complete graph.