Time-varying graphs are a useful model for networks with dynamic connectivity such as vehicular networks, yet, despite their great modeling power, many important features of time-varying graphs are still poorly understood. In this paper, we study the survivability properties of time-varying networks against unpredictable interruptions. We first show that the traditional definition of survivability is not effective in time-varying networks, and propose a new survivability framework. To evaluate the survivability of time-varying networks under the new framework, we propose two metrics that are analogous to MaxFlow and MinCut in static networks. We show that some fundamental survivability-related results such as Menger's Theorem only conditionally hold in time-varying networks. Then we analyze the complexity of computing the proposed metrics and develop several approximation algorithms. Finally, we conduct trace-driven simulations to demonstrate the application of our survivability framework to the robust design of a real-world bus communication network.
Despite the extensive research on time-varying graphs, there is very little literature on survivability of time-varying networks. The closest work to ours was done by Berman @cite_18 and Kleinberg @cite_9 . They discussed vulnerability in so-called edge-scheduled networks" or temporal networks" where each link is active for exactly one slot and only permanent failures happen. Our work considers a more general graph model while leveraging the temporal features of failures, thus generalizing their results. Scellato @cite_13 investigated a similar problem in random time-varying graphs and proposed a metric called temporal robustness". By comparison, our framework is deterministic, thus guaranteeing the worst-case survivability. Li @cite_32 studied a related but different problem in time-varying networks; specifically, they proposed heuristic algorithms to find the the min-cost subgraph of a probabilistic time-varying graph such that the probability that the subgraph is temporally connected exceeds a certain threshold.
@cite_9 @cite_18 @cite_13 @cite_32
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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. Real complex systems are inherently time-varying. Thanks to new communication systems and novel technologies, today it is possible to produce and analyze social and biological networks with detailed information on the time of occurrence and duration of each link. However, standard graph metrics introduced so far in complex network theory are mainly suited for static graphs, i.e., graphs in which the links do not change over time, or graphs built from time-varying systems by aggregating all the links as if they were concurrent in time. In this paper, we extend the notion of connectedness, and the definitions of node and graph components, to the case of time-varying graphs, which are represented as time-ordered sequences of graphs defined over a fixed set of nodes. We show that the problem of finding strongly connected components in a time-varying graph can be mapped into the problem of discovering the maximal-cliques in an opportunely constructed static graph, which we name the affine graph. It is, therefore, an NP-complete problem. As a practical example, we have performed a temporal component analysis of time-varying graphs constructed from three data sets of human interactions. The results show that taking time into account in the definition of graph components allows to capture important features of real systems. In particular, we observe a large variability in the size of node temporal in- and out-components. This is due to intrinsic fluctuations in the activity patterns of individuals, which cannot be detected by static graph analysis. We introduce stochastic time-dependency in evolving graphs: starting from an arbitrary initial edge probability distribution, at every time step, every edge changes its state (existing or not) according to a two-state Markovian process with probabilities p (edge birth-rate) and q (edge death-rate). If an edge exists at time t then, at time t+1, it dies with probability q. If instead the edge does not exist at time t, then it will come into existence at time t+1 with probability p. Such evolving graph model is a wide generalization of time-independent dynamic random graphs [6] and will be called edge-Markovian dynamic graphs. We investigate the speed of information dissemination in such dynamic graphs. We provide nearly tight bounds (which in fact turn out to be tight for a wide range of probabilities p and q) on the completion time of the flooding mechanism aiming to broadcast a piece of information from a source node to all nodes. In particular, we provide: i) A tight characterization of the class of edge-Markovian dynamic graphs where flooding time is constant and, thus, it does not asymptotically depend on the initial probability distribution. ii) A tight characterization of the class of edge-Markovian dynamic graphs where flooding time does not asymptotically depend on the edge death-rate q.