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 @math -regular trees such that @math where @math is the node degree and @math is the infection probability, the estimator is within a constant distance from the actual source with a 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. We then further evaluate the performance of the reverse infection algorithm on several real world networks.
@cite_5 @cite_17 @cite_7 proved that the node with the maximum closeness centrality is the an MLE on regular-trees. We define the infection closeness centrality to be the inverse of the sum of distances to infected nodes. Our simulations show that the sample path based estimator is closer to the actual source than the nodes with the maximum infection closeness.
@cite_5 @cite_7 @cite_17
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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. We provide a systematic study of the problem of finding the source of a computer virus in a network. We model virus spreading in a network with a variant of the popular SIR model and then construct an estimator for the virus source. This estimator is based upon a novel combinatorial quantity which we term rumor centrality. We establish that this is an ML estimator for a class of graphs. We find the following surprising threshold phenomenon: on trees which grow faster than a line, the estimator always has non-trivial detection probability, whereas on trees that grow like a line, the detection probability will go to 0 as the network grows. Simulations performed on synthetic networks such as the popular small-world and scale-free networks, and on real networks such as an internet AS network and the U.S. electric power grid network, show that the estimator either finds the source exactly or within a few hops in different network topologies. We compare rumor centrality to another common network centrality notion known as distance centrality. We prove that on trees, the rumor center and distance center are equivalent, but on general networks, they may differ. Indeed, simulations show that rumor centrality outperforms distance centrality in finding virus sources in networks which are not tree-like. Simulation studies using susceptible-infectious-recovered models were conducted to estimate individuals’ risk of infection and time to infection in small-world and randomly mixing networks. Infection transmitted more rapidly but ultimately resulted in fewer infected individuals in the small-world, compared with the random, network. The ability of measures of network centrality to identify high-risk individuals was also assessed. ‘‘Centrality’’ describes an individual’s position in a population; numerous parameters are available to assess this attribute. Here, the authors use the centrality measures degree (number of contacts), random-walk betweenness (a measure of the proportion of times an individual lies on the path between other individuals), shortest-path betweenness (the proportion of times an individual lies on the shortest path between other individuals), and farness (the sum of the number of steps between an individual and all other individuals). Each was associated with time to infection and risk of infection in the simulated outbreaks. In the networks examined, degree (which is the most readily measured) was at least as good as other network parameters in predicting risk of infection. Identification of more central individuals in populations may be used to inform surveillance and infection control strategies.