Dynamics and Inference for Voter Model Processes

TL;DR

This paper develops a framework for estimating node interaction parameters in a voter model process, providing error bounds and consensus time analysis for a process that reaches absorbing consensus states.

Contribution

It introduces a novel approach to parameter estimation in voter models, including error bounds and high-probability consensus time bounds, extending analysis beyond stationary processes.

Findings

Derived error bounds for maximum likelihood estimation of interaction parameters.

Established new bounds for all moments of consensus time.

Provided high-probability bounds for consensus time and sampling complexity lower bounds.

Abstract

We consider a discrete-time voter model process on a set of nodes, each being in one of two states, either 0 or 1. In each time step, each node adopts the state of a randomly sampled neighbor according to sampling probabilities, referred to as node interaction parameters. We study the maximum likelihood estimation of the node interaction parameters from observed node states for a given number of realizations of the voter model process. In contrast to previous work on parameter estimation of network autoregressive processes, whose long-run behavior is according to a stationary stochastic process, the voter model is an absorbing stochastic process that eventually reaches a consensus state. This requires developing a framework for deriving parameter estimation error bounds from observations consisting of several realizations of a voter model process. We present parameter estimation error bounds by interpreting the observation data as being generated according to an extended voter process that consists of cycles, each corresponding to a realization of the voter model process until absorption to a consensus state. In order to obtain these results, consensus time of a voter model process plays an important role. We present new bounds for all moments and a bound that holds with any given probability for consensus time, which may be of independent interest. In contrast to most existing work, our results yield a consensus time bound that holds with high probability. We also present a sampling complexity lower bound for parameter estimation within a prescribed error tolerance for the class of locally stable estimators.