A Finite Memory Interacting P\'{o}lya Contagion Network and its Approximating Dynamical Systems

TL;DR

This paper introduces a novel network model for contagion spread using finite memory Pólya urns, analyzes its stochastic properties, and develops dynamical system approximations to understand infection dynamics.

Contribution

It proposes a new finite memory interacting Pólya urn network model and derives both exact and approximate dynamical systems for contagion analysis.

Findings

Characterized the limiting infection state for homogeneous parameters.

Derived linear and nonlinear dynamical systems for different memory modes.

Validated approximations through simulation studies.

Abstract

We introduce a new model for contagion spread using a network of interacting finite memory two-color P\'{o}lya urns, which we refer to as the finite memory interacting P\'{o}lya contagion network. The urns interact in the sense that the probability of drawing a red ball (which represents an infection state) for a given urn, not only depends on the ratio of red balls in that urn but also on the ratio of red balls in the other urns in the network, hence accounting for the effect of spatial contagion. The resulting network-wide contagion process is a discrete-time finite-memory ($M$th order) Markov process, whose transition probability matrix is determined. The stochastic properties of the network contagion Markov process are analytically examined, and for homogeneous system parameters, we characterize the limiting state of infection in each urn. For the non-homogeneous case, given the complexity of the stochastic process, and in the same spirit as the well-studied SIS models, we use a mean-field type approximation to obtain a discrete-time dynamical system for the finite memory interacting P\'{o}lya contagion network. Interestingly, for $M=1$, we obtain a linear dynamical system which exactly represents the corresponding Markov process. For $M>1$, we use mean-field approximation to obtain a nonlinear dynamical system. Furthermore, noting that the latter dynamical system admits a linear variant (realized by retaining its leading linear terms), we study the asymptotic behavior of the linear systems for both memory modes and characterize their equilibrium. Finally, we present simulation studies to assess the quality of the approximation purveyed by the linear and non-linear dynamical systems.