How will a virus propagate in a real network? Does an epidemic threshold exist for a finite power-law graph, or any finite graph? How long does it take to disinfect a network given particular values of infection rate and virus death rate?

We answer the first question by providing equations that accurately model virus propagation in any network including real and synthesized network graphs. We propose a general epidemic threshold condition that applies to arbitrary graphs: we prove that, under reasonable approximations, the epidemic threshold for a network is closely related to the largest eigenvalue of its adjacency matrix. Finally, for the last question, we show that infections tend to zero exponentially below the epidemic threshold.

We show that our epidemic threshold model subsumes many known thresholds for special-case graphs (e.g., Erdös-Rényi, BA power-law, homogeneous); we show that the threshold tends to zero for infinite power-law graphs. Finally, we illustrate the predictive power of our model with extensive experiments on real and synthesized graphs. We show that our threshold condition holds for arbitrary graphs.