Contagion processes are strongly linked to the network structures on which they propagate, and learning these structures is essential for understanding and intervention on complex network processes such as epidemics and (mis)information propagation. However, using contagion data to infer network structure is a challenging inverse problem. In particular, it is imperative to have appropriate measures of uncertainty in network structure estimates, however these are largely ignored in most machine-learning approaches. We present a probabilistic framework that uses samples from the distribution of networks that are compatible with the dynamics observed to produce network and uncertainty estimates. We demonstrate the method using the well known independent cascade model to sample from the distribution of networks P(G) conditioned on the observation of a set of infections C. We evaluate the accuracy of the method by using the marginal probabilities of each edge in the distribution, and show the bene ts of quantifying uncertainty to improve estimates and understanding, particularly with small amounts of data.
@cite_19: ABSTRACTGeneration of deviates from random graph models with nontrivial edge dependence is an increasingly important problem. Here, we introduce a method which allows perfect sampling from random g...
@cite_26: Abstract Exponential random graph models are extremely difficult models to handle from a statistical viewpoint, since their normalising constant, which depends on model parameters, is available only in very trivial cases. We show how inference can be carried out in a Bayesian framework using a MCMC algorithm, which circumvents the need to calculate the normalising constants. We use a population MCMC approach which accelerates convergence and improves mixing of the Markov chain. This approach improves performance with respect to the Monte Carlo maximum likelihood method of Geyer and Thompson (1992) .
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It is worth noting that Bayesian methods are extensively used in sampling exponential random graph models (ERGMs) and fitting coefficients of motifs to observed networks @cite_26 @cite_3 @cite_19 . Fitting ERGMs can be extended to the network inference problem by observing information cascades instead of observing an existing network to infer ERGM coefficients.