  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 k>1 infection sources, generalize the Jordan center definition
to a k-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.
