We provide a dynamic policy for the rapid containment of a contagion process modeled as an SIS epidemic on a bounded degree undirected graph with n nodes. We show that if the budget r of curing resources available at each time is Ω(W ), where W is the CutWidth of the graph, and also of order Ω(log n), then the expected time until the extinction of the epidemic is of order O(n/r), which is within a constant factor from optimal, as well as sublinear in the number of nodes. Furthermore, if the CutWidth increases only sublinearly with n, a sublinear expected time to extinction is possible with a sublinearly increasing budget r.* Research partially supported by the Draper Laboratories and NSF grant CMMI-1234062. A preliminary version of some of the results in this paper are included in a paper submitted to the 53rd IEEE Conference on Decision and Control, December 2014.1 Our results actually are easily generalized to the case of directed graphs.
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Abstract-We consider an infinite collection of agents who make decisions, sequentially, about an unknown underlying binary state of the world. Each agent, prior to making a decision, receives an independent private signal whose distribution depends on the state of the world. Moreover, each agent also observes the decisions of its last K immediate predecessors. We study conditions under which the agent decisions converge to the correct value of the underlying state.We focus on the case where the private signals have bounded information content and investigate whether learning is possible, that is, whether there exist decision rules for the different agents that result in the convergence of their sequence of individual decisions to the correct state of the world. We first consider learning in the almost sure sense and show that it is impossible, for any value of K. We then explore the possibility of convergence in probability of the decisions to the correct state. Here, a distinction arises: if K = 1, learning in probability is impossible under any decision rule, while for K ≥ 2, we design a decision rule that achieves it.We finally consider a new model, involving forward looking strategic agents, each of which maximizes the discounted sum (over all agents) of the probabilities of a correct decision. (The case, studied in previous literature, of myopic agents who maximize the probability of their own decision being correct is an extreme special case.) We show that for any value of K, for any equilibrium of the associated Bayesian game, and under the assumption that each private signal has bounded information content, learning in probability fails to obtain.
We provide a dynamic policy for the rapid containment of a contagion process modeled as an SIS epidemic on a bounded degree undirected graph with n nodes. We show that if the budget r of curing resources available at each time is Ω(W ), where W is the CutWidth of the graph, and also of order Ω(log n), then the expected time until the extinction of the epidemic is of order O(n/r), which is within a constant factor from optimal, as well as sublinear in the number of nodes. Furthermore, if the CutWidth increases only sublinearly with n, a sublinear expected time to extinction is possible with a sublinearly increasing budget r.1. Introduction Many contagion processes over large networks can lead to costly cascades unless controlled by outside intervention. Examples include epidemics spreading over a population of individuals, viruses attacking a network of connected computers, or financial contagion in a network of banks. In this paper we study how this type of contagion can be prevented or contained by dynamically curing some of the infected nodes under a budget constraint on the amount of curing resources that can be deployed at each time.More specifically, we consider a canonical SIS epidemic model on an undirected graph 1 with n nodes, with a common infection rate along any edge that connects an infected and a healthy node, and node-specific curing rates ρ v (t) at each node v. The curing rates are to be chosen according to a curing policy which is based on the past history of the process and the network structure, subject to an upper bound on the total curing rate v ρ v (t).In a companion paper, [6] we characterize the cases for which a contagion process can be rapidly contained, i.e., the expected time to extinction can be made sublinear in the number n of nodes using a sublinear curing budget. Our characterization involves the CutWidth of the underlying graph. Intuitively, the CutWidth measures the required budget of curing resources in a simpler deterministic curing problem, in which infected nodes are cured one at a time, subject to the constraint that the number of edges between healthy and infected nodes is at all times less than or equal to the budget of curing resources. In [6], we establish that if the CutWidth increases at least linearly with n, then a sublinear (in n) expected time to extinction is impossible with a sublinear budget r. On the other hand, [6] provides a nonconstructive proof that for graphs with sublinear CutWidth and bounded degree, there exists a dynamic policy that achieves sublinear expected time to extinction using only a sublinear budget, for any set of initially infected nodes. The main contribution of the present paper is the construction of a specific policy with the latter desirable properties.Our policy is based on a combinatorial result which states the following. Given an initial set of infected nodes, nodes can be removed from that set, one at a time, in way that the maximum cut (number of edges) between healthy and infected nodes encountered during this process is upper...
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