We consider the problem of learning the weighted edges of a graph by observing the noisy times of infection for multiple epidemic cascades on this graph. Past work has considered this problem when the cascade information, i.e., infection times, are known exactly. Though the noisy setting is well motivated by many epidemic processes (e.g., most human epidemics), to the best of our knowledge, very little is known about when it is solvable. Previous work on the no-noise setting critically uses the ordering information. If noise can reverse this-a node's reported (noisy) infection time comes after the reported infection time of some node it infected-then we are unable to see how previous results can be extended. We therefore tackle two versions of the noisy setting: the limited-noise setting, where we know noisy times of infections, and the extreme-noise setting, in which we only know whether or not a node was infected. We provide a polynomial time algorithm for recovering the structure of bidirectional trees in the extreme-noise setting, and show our algorithm matches lower bounds established in the no-noise setting, and hence is optimal. We extend our results for general degree-bounded graphs, where again we show that our (poly-time) algorithm can recover the structure of the graph with optimal sample complexity. We also provide the first efficient algorithm to learn the weights of the bidirectional tree in the limited-noise setting. Finally, we give a polynomial time algorithm for learning the weights of general bounded-degree graphs in the limited-noise setting. This algorithm extends to general graphs (at the price of exponential running time), proving the problem is solvable in the general case. All our algorithms work for any noise distribution, without any restriction on the variance. CCS Concepts: • Mathematics of computing → Graph theory; Graph algorithms; Probability and statistics; Computing most probable explanation; Combinatorics; • Networks → Network algorithms; • Theory of computation → Graph algorithms analysis; Sample complexity and generalization bounds; Random walks and Markov chains; Social networks.
Motivated by the study of controlling (curing) epidemics, we consider the spread of an SI process on a known graph, where we have a limited budget to use to transition infected nodes back to the susceptible state (i.e., to cure nodes). Recent work has demonstrated that under perfect and instantaneous information (which nodes are/are not infected), the budget required for curing a graph precisely depends on a combinatorial property called the CutWidth. We show that this assumption is in fact necessary: even a minor degradation of perfect information, e.g., a diagnostic test that is 99% accurate, drastically alters the landscape. Infections that could previously be cured in sublinear time now may require exponential time, or orderwise larger budget to cure. The crux of the issue comes down to a tension not present in the full information case: if a node is suspected (but not certain) to be infected, do we risk wasting our budget to try to cure an uninfected node, or increase our certainty by longer observation, at the risk that the infection spreads further? Our results present fundamental, algorithm-independent bounds that tradeoff budget required vs. uncertainty.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.