Characterising spatial dependence on epidemic thresholds in networks

Abstract

Epidemic processes are an important security research topic for both the internet and social networks. The epidemic threshold is a fundamental metric used to evaluate epidemic spread in networks. Previous work has shown that the epidemic threshold of a network is 1/λmax(A), i.e., the inverse of the largest eigenvalue of its adjacency matrix. In this work, however, we indicate that such a theoretical threshold ignores spatial dependence among nodes and hence underestimates the actual epidemic threshold. Moreover, inspired by the Markov random field, we analytically derive a more accurate epidemic threshold based on a spatial Markov dependence assumption. Our model shows that the epidemic threshold is indeed 1/λmax(A)(1 − ρ), where ρ is the average spatial correlation coefficient between neighbouring nodes. We then apply simulations to compare the performance of these two theoretical epidemic thresholds in different networks, including regular graphs, synthesised irregular graphs, and a real topology. We find that our proposed epidemic threshold incorporates a certain spatial dependence and thus achieves greater accuracy in characterising the actual epidemic threshold in networks.