Large Deviations of Mean-Field Jump-Markov Processes on Structured Sparse Disordered Graphs

TL;DR

This paper establishes a Large Deviation Principle for jump-Markov processes on large, sparse, disordered networks embedded in space, showing that the likelihood of state transitions matches that of fully connected networks, with applications to epidemiological models.

Contribution

It proves a Large Deviation Principle for jump-Markov processes on disordered sparse networks and shows the rate function matches that of all-to-all networks, with applications to epidemiology.

Findings

Rate function identical to all-to-all networks

Derived Euler-Lagrange equations for transition paths

Applied results to stochastic SIS epidemic model

Abstract

We prove a Large Deviation Principle for {\color{blue} jump-Markov } Processes on sparse large disordered network with disordered connectivity. The network is embedded in a geometric space, with the probability of a connection a (scaled) function of the spatial positions of the nodes. This type of model has numerous applications, including neuroscience, epidemiology and social networks. We prove that the rate function (that indicates the asymptotic likelihood of state transitions) is the same as for a network with all-to-all connectivity. We apply our results to a stochastic $SIS$ epidemiological model on a disordered networks, and determine Euler-Lagrange equations that dictate the most likely transition path between different states of the network.