Learning Network Graph of SIR Epidemic Cascades Using Minimal Hitting Set Based Approach

We consider learning the underlying graph structure of a network in which infection spreads based on the observations of node infection times. We give an algorithm based on minimal hitting set to learn the exact underlying graph structure and provide sufficient condition on number of cascades required (i.e. sample complexity) for reliable recovery, which is shown to be O(logn), where n is the number of nodes in the graph. We then analytically evaluate performance of minimal hitting set approach in learning the degree distribution and detecting leaf nodes of a graph and provide a sufficient condition for its sample complexity which is shown to be lower than that of learning the whole graph. We also generalize the exact graph estimation problem to the problem of estimating the graph within a certain distortion, measured by edit distance. We show that this edit distance based graph estimator has a lower sample complexity. Our experimental results based on both synthetic network topologies and a real-world network trace show that our algorithm achieves superior performance than a previously proposed algorithm based on maximum likelihood.