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).
Both viewpoints have been extensively employed in the analysis of basic problems such as routing, broadcasting, gossiping and other forms of information spreading (e.g., @cite_14 @cite_29 @cite_15 @cite_17 @cite_24 ); to study problems of exploration in vehicular networks with periodic routes @cite_6 @cite_27 ; to examine failure detectors @cite_23 and consensus @cite_1 @cite_31 ; for the probabilistic analysis of informations spreading (e.g., @cite_8 @cite_19 ); and in the investigations of emerging properties in social networks (e.g., @cite_3 @cite_10 ). A characterization of classes of TVGs with respect to properties typically assumed in the research can be found in @cite_7 . The related investigations on dynamic networks include also the extensive work on population protocols (e.g., @cite_18 @cite_25 ); interestingly, the setting over which population protocols are defined is a particular class of time-varying graphs (recurrent interactions over a connected underlying graph). The impact of bounded waiting in dynamic networks has been investigated for exploration @cite_27 .
@cite_31 @cite_14 @cite_18 @cite_7 @cite_8 @cite_29 @cite_1 @cite_6 @cite_3 @cite_24 @cite_19 @cite_27 @cite_23 @cite_15 @cite_10 @cite_25 @cite_17
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We present a method for using real world mobility traces to identify tractable theoretical models for the study of distributed algorithms in mobile networks. We validate the method by deriving a vehicular ad hoc network model from a large corpus of position data generated by Boston-area taxicabs. Unlike previous work, our model does not assume global connectivity or eventual stability; it instead assumes only that some subset of processes are connected through transient paths (e.g., paths that exist over time). We use this model to study the problem of prioritized gossip, in which processes attempt to disseminate messages of different priority. Specifically, we present CabChat, a distributed prioritized gossip algorithm that leverages an interesting connection to the classic Tower of Hanoi problem to schedule the broadcast of packets of different priorities. Whereas previous studies of gossip leverage strong connectivity or stabilization assumptions to prove the time complexity of global termination, in our model, with its weak assumptions, we instead analyze CabChat with respect to its ability to deliver a high proportion of high priority messages over the transient paths that happen to exist in a given execution. The dynamic behavior of a network in which information is changing continuously over time requires robust and efficient mechanisms for keeping nodes updated about new information. Gossip protocols are mechanisms for this task in which nodes communicate with one another according to some underlying deterministic or randomized algorithm, exchanging information in each communication step. In a variety of contexts, the use of randomization to propagate information has been found to provide better reliability and scalability than more regimented deterministic approaches.In many settings, such as a cluster of distributed computing hosts, new information is generated at individual nodes, and is most "interesting" to nodes that are nearby. Thus, we propose distance-based propagation bounds as a performance measure for gossip mechanisms: a node at distance d from the origin of a new piece of information should be able to learn about this information with a delay that grows slowly with d, and is independent of the size of the network.For nodes arranged with uniform density in Euclidean space, we present natural gossip mechanisms, called spatial gossip, that satisfy such a guarantee: new information is spread to nodes at distance d, with high probability, in O(log1 + e d) time steps. Such a bound combines the desirable qualitative features of uniform gossip, in which information is spread with a delay that is logarithmic in the full network size, and deterministic flooding, in which information is spread with a delay that is linear in the distance and independent of the network size. Our mechanisms and their analysis resolve a conjecture of [1987].We further show an application of our gossip mechanisms to a basic resource location problem, in which nodes seek to rapidly learn the location of the nearest copy of a resource in a network. This problem, which is of considerable practical importance, can be solved by a very simple protocol using Spatial Gossip, whereas we can show that no protocol built on top of uniform gossip can inform nodes of their approximately nearest resource within poly-logarithmic time. The analysis relies on an additional useful property of spatial gossip, namely that information travels from its source to sinks along short paths not visiting points of the network far from the two nodes. 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. We study the computability and complexity of the exploration problem in a class of highly dynamic graphs: periodically varying (PV) graphs, where the edges exist only at some (unknown) times defined by the periodic movements of carriers. These graphs naturally model highly dynamic infrastructure-less networks such as public transports with fixed timetables, low earth orbiting (LEO) satellite systems, security guards' tours, etc. We establish necessary conditions for the problem to be solved. We also derive lower bounds on the amount of time required in general, as well as for the PV graphs defined by restricted classes of carriers movements: simple routes, and circular routes. We then prove that the limitations on computability and complexity we have established are indeed tight. We do so constructively presenting two worst case optimal solution algorithms, one for anonymous systems, and one for those with distinct nodes ids.