When Is a Network Epidemic Hard to Eliminate?

We consider the propagation of a contagion process (“epidemic”) on a network and study the problem of dynamically allocating a fixed curing budget to the nodes of the graph, at each time instant. For bounded degree graphs, we provide a lower bound on the expected time to extinction under any such dynamic allocation policy, in terms of a combinatorial quantity that we call the resistance of the set of initially infected nodes, the available budget, and the number of nodes n .Specifically, we consider the case of bounded degree graphs, with the resistance growing linearly in n . We show that if the curing budget is less than a certain multiple of the resistance, then the expected time to extinction grows exponentially with n . As a corollary, if all nodes are initially infected and the CutWidth of the graph grows linearly, while the curing budget is less than a certain multiple of the CutWidth, then the expected time to extinction grows exponentially in n . The combination of the latter with our prior work establishes a fairly sharp phase transition on the expected time to extinction (sublinear versus exponential) based on the relation between the CutWidth and the curing budget.

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