A Finite Memory Interacting Pólya Contagion Network and its Approximating Dynamical Systems
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We introduce a new model for contagion spread using a network of interacting finite memory two-color Pólya urns, which we refer to as the finite memory interacting Pólya 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 Markov process, whose transition probability matrix is determined. The stochastic properties of the network contagion Markov process are analytically examined, and in the particular case of memory one with homogeneous urns, we characterize the limiting state of infection in each urn. Given the complexity of this 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 Pólya contagion network. Interestingly, this dynamical system is linear in the memory one case, and is nonlinear for the general memory case. Furthermore, noting that the latter case 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 simulations to assess the quality of the approximation purveyed by the linear and non-linear dynamical systems, which illustrate the key role that memory plays in improving its accuracy.
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