While a natural fit for modeling and understanding mobile networks, time-varying graphs remain poorly understood. Indeed, many of the usual concepts of static graphs have no obvious counterpart in time-varying ones. In this paper, we introduce the notion of temporal reachability graphs. A (tau,delta)-reachability graph is a time-varying directed graph derived from an existing connectivity graph. An edge exists from one node to another in the reachability graph at time t if there exists a journey (i.e., a spatiotemporal path) in the connectivity graph from the first node to the second, leaving after t, with a positive edge traversal time tau, and arriving within a maximum delay delta. We make three contributions. First, we develop the theoretical framework around temporal reachability graphs. Second, we harness our theoretical findings to propose an algorithm for their efficient computation. Finally, we demonstrate the analytic power of the temporal reachability graph concept by applying it to synthetic and real-life datasets. On top of defining clear upper bounds on communication capabilities, reachability graphs highlight asymmetric communication opportunities and offloading potential.
Several approaches to reachability in time-varying graphs exists. For strictly positive edge traversal times, a simple heuristic consists in dividing time into successive slots of length @math and keeping only edges that are persistently present during each slot @cite_7 . This provides a good lower-bound approximation for small values of @math (i.e., less than @math ) whereas our approach can handle arbitrary edge traversal times. From a given starting time @math , reachability among all pairs of nodes can be calculated by iterating over all edge UP DOWN events @cite_24 . This calculation can then be repeated for a sample of starting times @cite_28 . This approach yields static reachability graphs for a discrete sequence of starting times, whereas the temporal reachability graphs defined in this paper calculate reachability in continuous time.
@cite_24 @cite_28 @cite_7
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While a natural fit for modeling and understanding mobile networks, time-varying graphs remain poorly understood. Indeed, many of the usual concepts of static graphs have no obvious counterpart in time-varying ones. In this paper, we introduce the notion of temporal reachability graphs. A (tau,delta)-reachability graph is a time-varying directed graph derived from an existing connectivity graph. An edge exists from one node to another in the reachability graph at time t if there exists a journey (i.e., a spatiotemporal path) in the connectivity graph from the first node to the second, leaving after t, with a positive edge traversal time tau, and arriving within a maximum delay delta. We make three contributions. First, we develop the theoretical framework around temporal reachability graphs. Second, we harness our theoretical findings to propose an algorithm for their efficient computation. Finally, we demonstrate the analytic power of the temporal reachability graph concept by applying it to synthetic and real-life datasets. On top of defining clear upper bounds on communication capabilities, reachability graphs highlight asymmetric communication opportunities and offloading potential. An edge-Markovian process with birth-rate p and death-rate q generates sequences of graphs (G0,G1,G2,…) with the same node set [n] such that Gt is obtained from Gt−1 as follows: if e ∉ E(Gt−1) then e ∈ E(Gt) with probability p, and if e ∈ E(Gt−1) then e ∉ E(Gt) with probability q. (PODC 2008) analyzed thoroughly information dissemination in such dynamic graphs, by establishing bounds on their flooding time--flooding is the basic mechanism in which every node becoming aware of an information at step t forwards this information to all its neighbors at all forthcoming steps t∦ > t. In this paper, we establish tight bounds on the complexity of flooding for all possible birth rates and death rates, completing the previous results by Moreover, we note that despite its many advantages in term of simplicity and robustness, flooding suffers from its high bandwidth consumption. Hence we also show that flooding in dynamic graphs can be implemented in a more parsimonious manner, so that to save bandwidth, yet preserving efficiency in term of simplicity and completion time. For a positive integer k, we say that the flooding protocol is k-active if each node forwards an information only during the k time steps immediately following the step at which the node receives that information for the first time. We define the reachability threshold for the flooding protocol as the smallest integer k such that, for any source s ∈ [n], the k-active flooding protocol from s completes (i.e., reaches all nodes), and we establish tight bounds for this parameter. We show that, for a large spectrum of parameters p and q, the reachability threshold is by several orders of magnitude smaller than the flooding time. In particular, we show that it is even constant whenever the ratio p (p + q) exceeds log n n. Moreover, we also show that being active for a number of steps equal to the reachability threshold (up to a multiplicative constant) allows the flooding protocol to complete in optimal time, i.e., in asymptotically the same number of steps as when being perpetually active. These results demonstrate that flooding can be implemented in a practical and efficient manner in dynamic graphs. The main ingredient in the proofs of our results is a reduction lemma enabling to overcome the time dependencies in edge-Markovian dynamic graphs. We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of n vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex u to vertex v is a path from u to v where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a (u, v)-journey for any pair of vertices u, v, u ź v. We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can freely choose availability instances for all edges and aims for temporal connectivity with very small cost; the cost is the total number of availability instances used. We achieve this via a simple polynomial-time procedure which derives designs of cost linear in n. We also show that the above procedure is (almost) optimal when the underlying graph is a tree, by proving a lower bound on the cost for any tree. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead given a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless P = NP. On the positive side, we show that in dense graphs with random edge availabilities, there is asymptotically almost surely a very large number of redundant labels. A temporal design may, however, be minimal, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least nlogn labels. An edge-scheduled network N is a multigraph G = (V, E), where each edge e ϵ E has been assigned two real weights: a start time α(e) and a finish time β(e). Such a multigraph models a communication or transportation network. A multiedge joining vertices u and v represents a direct communication (transportation) link between u and v, and the edges of the multiedge represent potential communications (transportations) between u and v over a fixed period of time. For a, b ϵ V, and k a nonnegative integer, we say that N is k-failure ab-invulnerable for the time period [0, t] if information can be relayed from a to b within that time period, even if up to k edges are deleted, i.e., “fail.” The k-failure ab-vulnerability threshold νab(k) is the earliest time t such that N is k-failure ab-invulnerable for the time period [0, t] [where νab(k) = ∞ if no such t exists]. Let κ denote the smallest k such that νab(k) = ∞. In this paper, we present an O(κ|E|) algorithm for computing νab(i), i = 0, …, κ −1. The latter algorithm constructs a set of κ pairwise edge-disjoint schedule-conforming paths P0, …, Pκ −1 such that the finish time of Pi is νab(i), i = 0, 1, …, κ −1. (A path P = ae1u1e2 ··· Upp−1epb is schedule-conforming if the finish time of edge ei is no greater than the start time of the next edge ei + 1.) The existence of such paths when α(e) = β(e) = 0, for all e ϵ E, implies Menger's Theorem. In this paper, we also show that the obvious analogs of these results for either multiedge deletions or vertex deletions do not hold. In fact, we show that the problem of finding k schedule-conforming paths such that no two paths pass through the same vertex (multiedge) is NP-complete, even for k = 2. © 1996 John Wiley & Sons, Inc.