Toward understanding spatial dependence on epidemic thresholds in networks

Social influence in online social networks bears resemblance to epidemic spread in networks and has been studied through epidemiological models. 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 exactly the inverse of the largest eigenvalue of its adjacency matrix. In this work, however, we indicate that such a threshold ignores spatial dependence among nodes and hence underestimates the actual epidemic threshold. Focusing on regular graphs, we analytically derive a more accurate epidemic threshold based on spatial Markov dependence. Our model shows that the epidemic threshold indeed depends on the spatial correlation coefficient between neighboring nodes and decreases with the death rate. Through both analysis and simulations, we show that our proposed epidemic threshold incorporates a certain spatial dependence and thus achieves a greater accuracy in characterizing the actual epidemic threshold in regular graphs. Moreover, we extend our study to irregular graphs by conjecturing a new epidemic threshold and show that such a threshold performs significantly better than previous work.