The study of SIS epidemics on networks has stressed the role of the network topology on the spreading process. However, accurate models of SIS epidemics rely on the complete knowledge of the network topology, which is often not available. This paper tackles the problem of inferring the network topology from observed infection time traces, especially where the network topology is partially known or known with some uncertainty. We propose a Bayesian method to infer the posterior probability of uncertain links in the network, and we derive closed form equations for these probabilities. We also propose a numerical approach based on a Gibbs sampling when the number of uncertain links is large such that using the closed form equations becomes impractical. Numerical results show the capability of the proposed approach to assign high probability to existing links and low probability to non-existing links of the network when the SIS traces are sufficiently long.
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