Modeling Network Contagion Via Interacting Finite Memory Pólya Urns

We construct a system of interacting finite memory Pólya urns to model contagion spread in a network. The urns, which are composed of red and black balls (representing degrees of infection and healthiness, respectively) interact in the sense that the probability at any time instant of drawing a red ball for a given urn not only depends on that urn’s ratio of red balls, but also on the ratio of red balls in the other urns of the network, hence accounting for the effect of spatial contagion. The urns have a finite memory, M, in the sense that reinforcing (black or red) balls added to each urn at time t are only kept in that urn for M future time instants (until time t + M). The resulting vector of all urn drawing variables forms an Mth order time-invariant irreducible and aperiodic Markov chain. We analytically examine the properties of the underlying Markov process and derive its asymptotic behaviour for the case of homogeneous system parameters. We further use mean-field approximation to obtain a class of approximating linear and nonlinear dynamical systems for the non-homogeneous case. Finally, we present simulations to assess the quality of these mean-field approximations.