In infrastructure-less highly dynamic networks, computing and performing even basic tasks (such as routing and broadcasting) is a very challenging activity du e to the fact that connectivity does not necessarily hold, and the network may actually be disconnected at every time instant. Clearly the task of designing protocols for these networks is less difficult if t he environment allows waiting (i.e., it provides the nodes with store-carry-forward-like mechanisms such as local buffering) than if waiting is not feasible. No quantitative corroborations of this fact exis t (e.g., no answer to the question: how much easier?). In this paper, we consider these qualitative questions ab out dynamic networks, modeled as time-varying (or evolving) graphs, where edges exist only at some times. We examine the difficulty of the environment in terms of the expressivity of the corresponding time-varying graph; that is in terms of the language generated by the feasible journeys in the graph. We prove that the set of languages Lnowait when no waiting is allowed contains all computable languages. On the other end, using algebraic properties of quasi-orders, we prove that Lwait is just the family of regular languages. In other words, we prove that, when waiting is no longer forbidden, the power of the accepting automaton (difficulty of the envir onment) drops drastically from being as powerful as a Turing machine, to becoming that of a Finite-State machine. This (perhaps surprisingly large) gap is a measure of the computational power of waiting. We also study bounded waiting; that is when waiting is allowed at a node only for at most d time units. We prove the negative result that Lwait[d] = Lnowait; that is, the expressivity decreases only if the waiting is finite but unpredictable (i.e., under the co ntrol of the protocol designer and not of the environment).
New technologies and the deployment of mobile and nomadic services are driving the emergence of complex communications networks, that have a highly dynamic behavior. This naturally engenders new route-discovery problems under changing conditions over these networks. Unfortunately, the temporal variations in the topology of dynamic networks are hard to be effectively captured in a classical graph model. In this paper, we use evolving graphs, which helps capture the dynamic characteristics of such networks, in order to compute multicast trees with minimum overall transmission time for a class of wireless mobile dynamic networks. We first show that computing different types of strongly connected components in evolving digraphs is NP-Complete, and then propose an algorithm to build all rooted directed minimum spanning trees in strongly connected dynamic networks. New technologies and the deployment of mobile and nomadic services are driving the emergence of complex communications networks, that have a highly dynamic behavior. This naturally engenders new route-discovery problems under changing conditions over these networks. Unfortunately, the temporal variations in the network topology are hard to be effectively captured in a classical graph model. In this paper, we use and extend a recently proposed graph theoretic model, which helps capture the evolving characteristic of such networks, in order to propose and formally analyze least cost journey (the analog of paths in usual graphs) in a class of dynamic networks, where the changes in the topology can be predicted in advance. Cost measures investigated here are hop count (shortest journeys), arrival date (foremost journeys), and time span (fastest journeys). The past few years have seen intensive research efforts carried out in some apparently unrelated areas of dynamic systems – delay-tolerant networks, opportunistic-mobility networks and social networks – obtaining closely related insights. Indeed, the concepts discovered in these investigations can be viewed as parts of the same conceptual universe, and the formal models proposed so far to express some specific concepts are the components of a larger formal description of this universe. The main contribution of this paper is to integrate the vast collection of concepts, formalisms and results found in the literature into a unified framework, which we call time-varying graphs TVGs. Using this framework, it is possible to express directly in the same formalism not only the concepts common to all those different areas, but also those specific to each. Based on this definitional work, employing both existing results and original observations, we present a hierarchical classification of TVGs; each class corresponds to a significant property examined in the distributed computing literature. We then examine how TVGs can be used to study the evolution of network properties, and propose different techniques, depending on whether the indicators for these properties are atemporal as in the majority of existing studies or temporal. Finally, we briefly discuss the introduction of randomness in TVGs. Wireless technologies and the deployment of mobile and nomadic services are driving the emergence of complex ad hoc networks that have a highly dynamic behavior. Modeling such dynamics and creating a reference model on which results could be compared and reproduced, was stated as a fundamental issue by a recent NSF workshop on networking. In this article we show how the modeling of time-changes unsettles old questions and allows for new insights into central problems in networking, such as routing metrics, connectivity, and spanning trees. Such modeling is made possible through evolving graphs, a simple combinatorial model that helps capture the behavior or dynamic networks over time.