Learning Mixtures of Graphs from Epidemic Cascades

TL;DR

This paper establishes the first necessary and sufficient conditions for efficiently learning the weighted edges of a mixture of two graphs from epidemic cascades, providing algorithms with optimal sample complexity under certain conditions.

Contribution

It introduces the first polynomial-time algorithms with rigorous guarantees for learning graph mixtures from epidemic data, covering both undirected and directed graphs with various priors.

Findings

Necessary and sufficient conditions for solvability on edge-separated graphs.

Efficient algorithms with optimal sample complexity when conditions are met.

Sample-optimal algorithms for directed graphs with out-degree at least three.

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