Random Walk Models of Network Formation and Sequential Monte Carlo Methods for Graphs

TL;DR

This paper introduces a new class of generative network models based on random walks, explores their theoretical properties, and develops sequential Monte Carlo methods for parameter estimation and inference.

Contribution

It proposes a novel random walk-based network model that is both flexible and tractable, extending preferential attachment, and develops SMC algorithms for inference.

Findings

Model converges to an extension of preferential attachment

Parameters can be estimated from a single graph

Random walk length influences interaction scale

Abstract

We introduce a class of generative network models that insert edges by connecting the starting and terminal vertices of a random walk on the network graph. Within the taxonomy of statistical network models, this class is distinguished by permitting the location of a new edge to explicitly depend on the structure of the graph, but being nonetheless statistically and computationally tractable. In the limit of infinite walk length, the model converges to an extension of the preferential attachment model---in this sense, it can be motivated alternatively by asking what preferential attachment is an approximation to. Theoretical properties, including the limiting degree sequence, are studied analytically. If the entire history of the graph is observed, parameters can be estimated by maximum likelihood. If only the final graph is available, its history can be imputed using MCMC. We develop a class of sequential Monte Carlo algorithms that are more generally applicable to sequential network models, and may be of interest in their own right. The model parameters can be recovered from a single graph generated by the model. Applications to data clarify the role of the random walk length as a length scale of interactions within the graph.