This paper studies the problem of locating multiple diffusion sources in networks with partial observations. We propose a new source localization algorithm, named Optimal-Jordan-Cover (OJC). The algorithm first extracts a subgraph using a candidate selection algorithm that selects source candidates based on the number of observed infected nodes in their neighborhoods. Then, in the extracted subgraph, OJC finds a set of nodes that "cover" all observed infected nodes with the minimum radius. The set of nodes is called the Jordan cover, and is regarded as the set of diffusion sources. Considering the heterogeneous susceptible-infected-recovered (SIR) diffusion in the Erdos-Renyi (ER) random graph, we prove that OJC can locate all sources with probability one asymptotically with partial observations. OJC is a polynomial-time algorithm in terms of network size. However, the computational complexity increases exponentially in @math the number of sources. We further propose a low-complexity heuristic based on the K-Means for approximating the Jordan cover, named Approximate-Jordan-Cover (AJC). Simulations on random graphs and real networks demonstrate that both AJC and OJC significantly outperform other heuristic algorithms.
@cite_14: We consider the problem of detecting the source of a rumor (information diffusion) in a network based on observations about which set of nodes possess the rumor. In a recent work [10], this question was introduced and studied. The authors proposed rumor centrality as an estimator for detecting the source. They establish it to be the maximum likelihood estimator with respect to the popular Susceptible Infected (SI) model with exponential spreading time for regular trees. They showed that as the size of infected graph increases, for a line (2-regular tree) graph, the probability of source detection goes to 0 while for d-regular trees with d ≥ 3 the probability of detection, say αd, remains bounded away from 0 and is less than 1 2. Their results, however stop short of providing insights for the heterogeneous setting such as irregular trees or the SI model with non-exponential spreading times. This paper overcomes this limitation and establishes the effectiveness of rumor centrality for source detection for generic random trees and the SI model with a generic spreading time distribution. The key result is an interesting connection between a multi-type continuous time branching process (an equivalent representation of a generalized Polya's urn, cf. [1]) and the effectiveness of rumor centrality. Through this, it is possible to quantify the detection probability precisely. As a consequence, we recover all the results of [10] as a special case and more importantly, we obtain a variety of results establishing the universality of rumor centrality in the context of tree-like graphs and the SI model with a generic spreading time distribution.
@cite_22: This paper addresses the problem of a single rumor source detection with multiple observations, from a statistical point of view of a spreading over a network, based on the susceptible-infectious model. For tree networks, multiple sequential observations for one single instance of rumor spreading cannot improve over the initial snapshot observation. The situation dramatically improves for multiple independent observations. We propose a unified inference framework based on the union rumor centrality, and provide explicit detection performance for degree-regular tree networks. Surprisingly, even with merely two observations, the detection probability at least doubles that of a single observation, and further approaches one, i.e., reliable detection, with increasing degree. This indicates that a richer diversity enhances detectability. For general graphs, a detection algorithm using a breadth-first search strategy is also proposed and evaluated. Besides rumor source detection, our results can be used in network forensics to combat recurring epidemic-like information spreading such as online anomaly and fraudulent email spams.
@cite_24: We provide a systematic study of the problem of finding the source of a rumor in a network. We model rumor spreading in a network with the popular susceptible-infected (SI) model and then construct an estimator for the rumor source. This estimator is based upon a novel topological quantity which we term rumor centrality. We establish that this is a maximum likelihood (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 nontrivial 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 of the true source across 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 rumor sources in networks which are not tree-like.
@cite_15: Consider a network where an unidentified source starts a rumor. The rumor spreads along the edges of the network to other nodes in the network. After a sufficiently long amount of time, we observe a subset of the nodes that have heard the rumor, and using this information wish to identify the source. Optimal estimators were recently proposed for regular (exponential growth) and irregular geometric (polynomial growth) trees when all nodes that heard the rumor reveal themselves. We provide the extension to the case in which nodes reveal whether they have heard the rumor with probability p, independent of each other. For geometric trees and p > 0, we achieve the same performance as the optimal estimator with p = 1. For regular trees, the estimator can achieve performance within e of the optimal, provided that p is larger than a threshold.
@cite_13: 
@cite_11: Identifying the infection sources in a network, including the index cases that introduce a contagious disease into a population network, the servers that inject a computer virus into a computer network, or the individuals who started a rumor in a social network, plays a critical role in limiting the damage caused by the infection through timely quarantine of the sources. We consider the problem of estimating the infection sources and the infection regions (subsets of nodes infected by each source) in a network, based only on knowledge of which nodes are infected and their connections, and when the number of sources is unknown a priori. We derive estimators for the infection sources and their infection regions based on approximations of the infection sequences count. We prove that if there are at most two infection sources in a geometric tree, our estimator identifies the true source or sources with probability going to one as the number of infected nodes increases. When there are more than two infection sources, and when the maximum possible number of infection sources is known, we propose an algorithm with quadratic complexity to estimate the actual number and identities of the infection sources. Simulations on various kinds of networks, including tree networks, small-world networks and real world power grid networks, and tests on two real data sets are provided to verify the performance of our estimators.
Shah and Zaman @cite_13 @cite_24 studied the rumor source detection problem, and proposed a new graph centrality called . They proved that the node with the maximum rumor centrality is the maximum likelihood estimator (MLE) of the diffusion source on regular trees under the continuous-time SI model. In addition, it has been proved that the detection probability is greater than zero on regular trees and approaches one for geometric trees. @cite_14 analyzed the detection probability of rumor centrality for general random trees. Later, the performance of rumor centrality has been studied under different models, including multiple sources @cite_11 , incomplete observations @cite_15 , multiple independent diffusion processes from the same source @cite_22 .