Restless reachability problems in temporal graphs

Abstract

We study a family of temporal reachability problems under waiting-time restrictions. In particular, given a temporal graph and a set of source vertices, we find the set of vertices that are reachable from a source via a time-respecting path, and such that the difference in timestamps between consecutive edges is at most a resting time. This kind of problems have several interesting applications in understanding the spread of a disease in a network, tracing contacts in epidemic outbreaks, and finding signaling pathways in the brain network.We present an algebraic algorithm based on constrained multilinear sieving for solving the restless reachability problems we propose. With an open-source implementation we demonstrate that the algorithm can scale to large temporal graphs with tens of millions of edges, despite the problem being NP-hard. The implementation is efficiently engineered and highly optimized. For instance, we can solve the restless reachability problem by restricting the path length to 9 in a real-world graph dataset with over 36 million directed edges in less than one hour on a 4-core Haswell desktop.

Directed Percolation in Temporal Networks

“…Connec-tivity is a central property of temporal networks too, with several recent techniques to characterise it, e.g. by using limited waiting time reachability [44][45][46][47][48]. However, previous works on the theory of temporal connectivity [39,49,50] fundamentally rely on different notions of connectivity of structure, such as static connectivity, deterministic walks, or networks consisting of discrete temporal layers.…”