Finding the Graph of Epidemic Cascades

We consider the problem of finding the graph on which an epidemic cascade spreads, given only the times when each node gets infected. While this is a problem of importance in several contexts -- offline and online social networks, e-commerce, epidemiology, vulnerabilities in infrastructure networks -- there has been very little work, analytical or empirical, on finding the graph. Clearly, it is impossible to do so from just one cascade; our interest is in learning the graph from a small number of cascades. For the classic and popular "independent cascade" SIR epidemics, we analytically establish the number of cascades required by both the global maximum-likelihood (ML) estimator, and a natural greedy algorithm. Both results are based on a key observation: the global graph learning problem decouples into $n$ local problems -- one for each node. For a node of degree $d$, we show that its neighborhood can be reliably found once it has been infected $O(d^2 \log n)$ times (for ML on general graphs) or $O(d\log n)$ times (for greedy on trees). We also provide a corresponding information-theoretic lower bound of $\Omega(d\log n)$; thus our bounds are essentially tight. Furthermore, if we are given side-information in the form of a super-graph of the actual graph (as is often the case), then the number of cascade samples required -- in all cases -- becomes independent of the network size $n$. Finally, we show that for a very general SIR epidemic cascade model, the Markov graph of infection times is obtained via the moralization of the network graph.Comment: To appear in Proc. ACM SIGMETRICS/Performance 201

Low-rank Matrix Completion using Alternating Minimization

Alternating minimization represents a widely applicable and empirically successful approach for finding low-rank matrices that best fit the given data. For example, for the problem of low-rank matrix completion, this method is believed to be one of the most accurate and efficient, and formed a major component of the winning entry in the Netflix Challenge. In the alternating minimization approach, the low-rank target matrix is written in a bi-linear form, i.e. $X = UV^\dag$; the algorithm then alternates between finding the best $U$ and the best $V$. Typically, each alternating step in isolation is convex and tractable. However the overall problem becomes non-convex and there has been almost no theoretical understanding of when this approach yields a good result. In this paper we present first theoretical analysis of the performance of alternating minimization for matrix completion, and the related problem of matrix sensing. For both these problems, celebrated recent results have shown that they become well-posed and tractable once certain (now standard) conditions are imposed on the problem. We show that alternating minimization also succeeds under similar conditions. Moreover, compared to existing results, our paper shows that alternating minimization guarantees faster (in particular, geometric) convergence to the true matrix, while allowing a simpler analysis