Random Walk Models of Network Formation and Sequential Monte Carlo Methods for Graphs
This work addresses the problem of modeling network formation with structural dependencies for researchers in network science, though it is incremental as it builds on existing preferential attachment models.
The authors introduced a generative network model where edges are added based on random walks on the graph, offering a tractable alternative to preferential attachment models. They developed sequential Monte Carlo algorithms for parameter estimation and applied the model to data, revealing the random walk length as a key interaction scale.
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.