Corruption Detection on Networks

We consider the problem of corruption detection on networks. In this model each vertex of a directed graph can be either truthful or corrupt. Each vertex reports about the types (truthful or corrupt) of all his out-neighbors. If he is truthful, he reports the truth, whereas if he is corrupt he reports adversarially. This model, considered in [15] motivated by the desire to identify the faulty components of a digital system by having the other components checking them, became known as the PMC model. The main known results for this model characterize networks in which all corrupt (that is, faulty) vertices can be identified, when there is a known upper bound on their number. We are interested in the investigation of networks in which most of the corrupt vertices can be identified. We show that the main relevant parameter here is graph expansion. This implies that in contrast to the known results about the PMC model that imply that in order to identify all corrupt vertices when their number is t all indegrees have to be at least t, there are bounded degree graphs in which almost all corrupt and almost all truthful vertices can be identified, whenever there is a majority of truthful vertices. We also show that expansion is necessary for obtaining such a corruption detection and discuss algorithms and the computational hardness of the problem. Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel and School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540. [email protected]. Research supported in part by a USA-Israeli BSF grant 2012/107, by an ISF grant 620/13, by the Israeli I-Core program and by the Fund for Mathematics. Department of Statistics, University of Pennsylvania, 3730 Walnut Street, Philadelphia, PA 19104 and Departments of Statistics and Computer Science, U.C. Berkeley, 367 Evans Hall, Berkeley CA 94720. [email protected]. Research supported by NSF grant CCF 1320105, DOD ONR grant N0001414-1-0823, and grant 328025 from the Simons Foundation. Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA, [email protected]. Research supported in part by NSF grant # DMS-1209117