Connectivity and inference problems for temporal networks

Many network problems are based on fundamental relationships involving time. Consider, for example, the problems of modeling the flow of information through a distributed network, studying the spread of a disease through a population, or analyzing the reachability properties of an airline timetable. In such settings, a natural model is that of a graph in which each edge is annotated with a time label specifying the time at which its endpoints "communicated." We will call such a graph a temporal network. To model the notion that information in such a network "flows" only on paths whose labels respect the ordering of time, we call a path time-respecting if the time labels on its edges are non-decreasing. The central motivation for our work is the following question: how do the basic combinatorial and algorithmic properties of graphs change when we impose this additional temporal condition? The notion of a path is intrinsic to many of the most fundamental algorithmic problems on graphs; spanning trees, connectivity, flows, and cuts are some examples. When we focus on time-respecting paths in place of arbitrary paths, many of these problems acquire a character that is different from the traditional setting, but very rich in its own right. We provide results on two types of problems for temporal networks. First, we consider connectivity problems, in which we seek disjoint time-respecting paths between pairs of nodes. The natural analogue of Menger's Theorem for node-disjoint paths fails in general for time-respecting paths; we give a non-trivial characterization of those graphs for which the theorem does hold in terms of an excluded subdivision theorem, and provide a polynomial-time algorithm for connectivity on this class of graphs. (The problem on general graphs is NP-complete.) We then define and study the class of inference problems, in which we seek to reconstruct a partially specified time labeling of a network in a manner consistent with an observed history of information flow.

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