Random Walk Models of Network Formation and Sequential Monte Carlo
Methods for Graphs
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by
Benjamin Bloem-Reddy, Peter Orbanz
2018
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.
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