Learning Mixtures of Graphs from Epidemic Cascades

Jessica Hoffmann · Soumya Basu · Surbhi Goel · Constantine Caramanis

Keywords: [ Networks and Relational Learning ] [ Unsupervised and Semi-supervised Learning ] [ Sequential, Network, and Time-Series Modeling ]

Abstract:

We consider the problem of learning the weighted edges of a balanced mixture of two undirected graphs from epidemic cascades. While mixture models are popular modeling tools, algorithmic development with rigorous guarantees has lagged. Graph mixtures are apparently no exception: until now, very little is known about whether this problem is solvable.

 To the best of our knowledge, we establish the first necessary and sufficient conditions for this problem to be solvable in polynomial time on edge-separated graphs. When the conditions are met, i.e., when the graphs are connected with at least three edges, we give an efficient algorithm for learning the weights of both graphs with optimal sample complexity (up to log factors).

We give complementary results and provide sample-optimal (up to log factors) algorithms for mixtures of directed graphs of out-degree at least three, and for mixture of undirected graphs of unbalanced and/or unknown priors.