How will a virus propagate in a real network? How long does it take to disinfect a network given particular values of infection rate and virus death rate? What is the single best node to immunize? Answering these questions is essential for devising network-wide strategies to counter viruses. In addition, viral propagation is very similar in principle to the spread of rumors, information, and “fads,” implying that the solutions for viral propagation would also offer insights into these other problem settings. We answer these questions by developing a nonlinear dynamical system ( NLDS ) that accurately models viral propagation in any arbitrary network, including real and synthesized network graphs. We propose a general epidemic threshold condition for the NLDS system: we prove that the epidemic threshold for a network is exactly the inverse of the largest eigenvalue of its adjacency matrix. Finally, we show that below the epidemic threshold, infections die out at an exponential rate. Our epidemic threshold model subsumes many known thresholds for special-case graphs (e.g., Erdös--Rényi, BA powerlaw, homogeneous). We demonstrate the predictive power of our model with extensive experiments on real and synthesized graphs, and show that our threshold condition holds for arbitrary graphs. Finally, we show how to utilize our threshold condition for practical uses: It can dictate which nodes to immunize; it can assess the effects of a throttling policy; it can help us design network topologies so that they are more resistant to viruses.

Epidemic thresholds in real networks