# A Polya Contagion Model for Networks

## TL;DR

This paper introduces a network contagion model based on a modified Polya urn scheme, analyzing its stochastic properties, asymptotic behavior, and comparing it with traditional epidemic models.

## Contribution

It develops a novel Polya-based network contagion model that accounts for spatial infection and provides analytical and empirical insights into its behavior.

## Key findings

- Model captures non-stationary contagion dynamics.
- Analytical approximations fit well across parameters.
- Model compares favorably with SIS epidemic models.

## Abstract

A network epidemics model based on the classical Polya urn scheme is investigated. Temporal contagion processes are generated on the network nodes using a modified Polya sampling scheme that accounts for spatial infection among neighbouring nodes. The stochastic properties and the asymptotic behaviour of the resulting network contagion process are analyzed. Unlike the classical Polya process, the network process is noted to be non-stationary in general, although it is shown to be time-invariant in its first and some of its second-order statistics and to satisfy martingale convergence properties under certain conditions. Three classical Polya processes, one computational and two analytical, are proposed to statistically approximate the contagion process of each node, showing a good fit for a range of system parameters. Finally, empirical results compare and contrast our model with the well-known discrete time SIS model.