Disentangling Mixtures of Epidemics on Graphs

We consider the problem of learning the weighted edges of a mixture of two graphs from epidemic cascades. This is a natural setting in the context of social networks, where a post created by one user will not spread on the same graph if it is about basketball or if it is about politics. However, very little is known about whether this problem is solvable. To the best of our knowledge, we establish the first conditions under which this problem can be solved, and provide conditions under which the problem is provably not solvable. When the conditions are met, i.e. when the graphs are connected, with distinct edges, and have at least three edges, we give an efficient algorithm for learning the weights of both graphs with almost optimal sample complexity (up to log factors). We extend the results to the setting in which the priors of the mixture are unknown and obtain similar guarantees.