Network failure detection and graph connectivity

TLDR: In this article, the authors consider a model for monitoring the connectivity of a network subject to node or edge failures, and they show that for any graph G, there is an (e, k)-detection set of size bounded by a polynomial in k and e, independent of the size of G. They also show that detection set bounds can be made considerably stronger when parameterized by these connectivity values.

Abstract: We consider a model for monitoring the connectivity of a network subject to node or edge failures. In particular, we are concerned with detecting (e, k)-failures: events in which an adversary deletes up to network elements (nodes or edges), after which there are two sets of nodes A and B, each at least an e fraction of the network, that are disconnected from one another. We say that a set D of nodes is an (e k)-detection set if, for any (e k)-failure of the network, some two nodes in D are no longer able to communicate; in this way, D "witnesses" any such failure. Recent results show that for any graph G, there is an is (e k)-detection set of size bounded by a polynomial in k and e, independent of the size of G.In this paper, we expose some relationships between bounds on detection sets and the edge-connectivity λ and node-connectivity κ of the underlying graph. Specifically, we show that detection set bounds can be made considerably stronger when parameterized by these connectivity values. We show that for an adversary that can delete κλ edges, there is always a detection set of size O((κ/e) log (1/e)) which can be found by random sampling. Moreover, an (e, l for node failures, we develop a novel approach for working with the much more complex set of all minimum node-cuts of a graph.