  Information diffusion in networks can be used to model many real-world
phenomena, including rumor spreading on online social networks, epidemics in
human beings, and malware on the Internet. Informally speaking, the source
localization problem is to identify a node in the network that provides the
best explanation of the observed diffusion. Despite significant efforts and
successes over last few years, theoretical guarantees of source localization
algorithms were established only for tree networks due to the complexity of the
problem. This paper presents a new source localization algorithm, called the
Short-Fat Tree (SFT) algorithm. Loosely speaking, the algorithm selects the
node such that the breadth-first search (BFS) tree from the node has the
minimum depth but the maximum number of leaf nodes. Performance guarantees of
SFT under the independent cascade (IC) model are established for both tree
networks and the Erdos-Renyi (ER) random graph. On tree networks, SFT is the
maximum a posterior (MAP) estimator. On the ER random graph, the following
fundamental limits have been obtained: $(i)$ when the infection duration
$<\frac{2}{3}t_u,$ SFT identifies the source with probability one
asymptotically, where $t_u=\left\lceil\frac{\log n}{\log \mu}\right\rceil+2$
and $\mu$ is the average node degree, $(ii)$ when the infection duration
$>t_u,$ the probability of identifying the source approaches zero
asymptotically under any algorithm; and $(iii)$ when infection duration $<t_u,$
the BFS tree starting from the source is a fat tree. Numerical experiments on
tree networks, the ER random graphs and real world networks with different
evaluation metrics show that the SFT algorithm outperforms existing algorithms.
