Connectivity and Inference Problems for Temporal Networks

Abstract

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.

Temporal node centrality in complex networks

“…Kempe et al [6] proposed a model of temporal networks as static graphs where every edge is labeled with the time the interaction took place. Ferreira [7] also views a dynamic network as a sequence of static graphs and seeks to tackle the fundamental network problems such as routing metrics, connectivity, and spanning trees for dynamic networks.…”

Predicting and controlling infectious disease epidemics using temporal networks

“…The temporal network is an emerging modelling framework for understanding epidemic spreading on social networks [1,2]. The “atoms” making up a temporal network are the contacts (sometimes called events).…”

“…Lately, contact pattern data with precise temporal information of the contacts (i.e., timing and duration of contacts) between individuals have been collected in various situations. The aim in this report is to discuss the relevance of such data, collectively called the temporal networks [1,2], to theoretical epidemiology.…”

Error and attack vulnerability of temporal networks

“…[2,3], vastly overestimate the network robustness in temporal networks. The first attempt to express temporal network properties was made by Kempe et al [27], considering time labels of links, but neglecting temporary disconnected nodes. A thorough survey on temporal networks and existing analysis methods has been proposed by Holme and Saramäki [28].…”