Catch'Em All: Locating Multiple Diffusion Sources in Networks with   Partial Observations
  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 $m,$ 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. 
Kai Zhu, Zhen Chen, Lei Ying
cs.SI