We introduce a statistical inference framework for maximum likelihood estimation of the contagion source from a partially observed contagion spreading process on an arbitrary network structure. The framework is based on simulations of a contagion spreading process from a set of potential sources which were infected in the observed realization. We present a number of different likelihood estimators for determining the conditional probabilities of potential initial sources producing the observed epidemic realization, which are computed in scalable and parallel way. This statistical inference framework is applicable to arbitrary networks with different dynamical spreading processes.
Zhu et. al. adopted the SIR model and proposed a sample path counting approach for source detection @cite_15 . They prove that the source node on infinite trees minimizes the maximum distance to the infected nodes. They assume that the infected and susceptible nodes are indistinguishable. Lokhov et. al. use a dynamic message-passing algorithm to estimate the probability that a given node produces an observed snapshot. They use a mean-field-like approximation to compute the marginal probabilities and an assumption of sparse contact network @cite_7 .
@cite_15 @cite_7
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In this paper, we consider the problem of locating the information source with sparse observations. We assume that a piece of information spreads in a network following a heterogeneous susceptible-infected-recovered (SIR) model and that a small subset of infected nodes are reported, from which we need to find the source of the information. We adopt the sample path based estimator developed in [1], and prove that on infinite trees, the sample path based estimator is a Jordan infection center with respect to the set of observed infected nodes. In other words, the sample path based estimator minimizes the maximum distance to observed infected nodes. We further prove that the distance between the estimator and the actual source is upper bounded by a constant independent of the number of infected nodes with a high probability on infinite trees. Our simulations on tree networks and real world networks show that the sample path based estimator is closer to the actual source than several other algorithms. This paper studies the problem of detecting the information source in a network in which the spread of information follows the popular Susceptible-Infected-Recovered (SIR) model. We assume all nodes in the network are in the susceptible state initially except the information source which is in the infected state. Susceptible nodes may then be infected by infected nodes, and infected nodes may recover and will not be infected again after recovery. Given a snapshot of the network, from which we know all infected nodes but cannot distinguish susceptible nodes and recovered nodes, the problem is to find the information source based on the snapshot and the network topology. We develop a sample path based approach where the estimator of the information source is chosen to be the root node associated with the sample path that most likely leads to the observed snapshot. We prove for infinite-trees, the estimator is a node that minimizes the maximum distance to the infected nodes. A reverse-infection algorithm is proposed to find such an estimator in general graphs. We prove that for g-regular trees such that gq > 1, where g is the node degree and q is the infection probability, the estimator is within a constant distance from the actual source with high probability, independent of the number of infected nodes and the time the snapshot is taken. Our simulation results show that for tree networks, the estimator produced by the reverse-infection algorithm is closer to the actual source than the one identified by the closeness centrality heuristic. In this paper, we study the problem of detecting multiple information sources in networks under the Susceptible-Infected-Recovered (SIR) model. First, assuming the number of information sources is known, we develop a sample-path-based algorithm, named clustering and localization, for trees. For g-regular trees, the estimators produced by the proposed algorithm are within a constant distance from the real sources with a high probability. We further present a heuristic algorithm for general networks and an algorithm for estimating the number of sources when the number of real sources is unknown. Finding the infection sources in a network when we only know the network topology and infected nodes, but not the rates of infection, is a challenging combinatorial problem, and it is even more difficult in practice where the underlying infection spreading model is usually unknown a priori . In this paper, we are interested in finding a source estimator that is applicable to various spreading models, including the susceptible–infected (SI), susceptible–infected–recovered (SIR), susceptible–infected–recovered–infected (SIRI), and susceptible–infected–susceptible (SIS) models. We show that under the SI, SIR, and SIRI spreading models and with mild technical assumptions, the Jordan center is the infection source associated with the most likely infection path in a tree network with a single infection source. This conclusion applies for a wide range of spreading parameters, while it holds for regular trees under the SIS model with homogeneous infection and recovery rates. Since the Jordan center does not depend on the infection, recovery, and reinfection rates, it can be regarded as a universal source estimator. We also consider the case where there are @math infection sources, generalize the Jordan center definition to a @math -Jordan center set, and show that this is an optimal infection source set estimator in a tree network for the SI model. Simulation results on various general synthetic networks and real-world networks suggest that Jordan center-based estimators consistently outperform the betweenness, closeness, distance, degree, eigenvector, and pagerank centrality-based heuristics, even if the network is not a tree.