Monitoring Properties of Large, Distributed, Dynamic Graphs

The following is a very common question in numerous theoretical and application-related domains: given a graph G, does it satisfy some given property? For example, is G connected? Is its diameter smaller than a given threshold? Is its average degree larger than a certain threshold? Traditionally, algorithms to quickly answer such questions were developed for static and centralized graphs (i.e. G is stored in a central server and the list of its vertices and edges is static and quickly accessible). Later, as dictated by practical considerations, a great deal of attention was given to on-line algorithms for dynamic graphs (where vertices and edges can be added and deleted); the focus of research was to quickly decide whether the new graph still satisfies the given property. Today, a more difficult version of this problem, referred to as the distributed monitoring problem, is becoming increasingly important: large graphs are not only dynamic, but also distributed, that is, G is partitioned between a few servers, none of which "sees" G in its entirety. The question is how to define local conditions, such that as long as they hold on the local graphs, it is guaranteed that the desired property holds for the global G. Such local conditions are crucial for avoiding a huge communication overhead. While defining local conditions for linear properties (e.g. average degree) is relatively easy, they are considerably more difficult to derive for non-linear functions over graphs. We propose a solution and a general definition of solution optimality, and demonstrate how to apply it to two important graph properties - the spectral gap and the number of triangles. We also define an absolute lower bound on the communication overhead for distributed monitoring, and compare our algorithm to it, with excellent results. Last but not least, performance improves as the graph becomes larger and denser - that is, when distributing it is more important.