Expressivity of time-varying graphs

Time-varying graphs model in a natural way infrastructure-less highly dynamic systems, such as wireless ad-hoc mobile networks, robotic swarms, vehicular networks, etc. In these systems, a path from a node to another might still exist over time, rendering computing possible, even though at no time the path exists in its entirety. Some of these systems allow waiting (i.e., provide the nodes with store-carry-forward-like mechanisms such as local buffering) while others do not. In this paper, we focus on the structure of the time-varying graphs modelling these highly dynamical environments. We examine the complexity of these graphs, with respect to waiting, in terms of their expressivity; that is in terms of the language generated by the feasible journeys (i.e., the "paths over time"). We prove that the set of languages L_nowait when no waiting is allowed contains all computable languages. On the other end, using algebraic properties of quasi-orders, we prove that L_wait is just the family of regular languages, even if the presence of edges is controlled by some arbitrary function of the time. In other words, we prove that, when waiting is allowed, the power of the accepting automaton drops drastically from being as powerful as a Turing machine, to becoming that of a Finite-State machine. This large gap provides a measure of the impact of waiting. We also study bounded waiting; that is when waiting is allowed at a node for at most d time units. We prove that L_wait[d] = L_nowait; that is, the complexity of the accepting automaton decreases only if waiting is unbounded.