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, social networksobtaining 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 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 TVG (for time-varying graphs). 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 a-temporal (as in the majority of existing studies) or temporal. Finally, we briefly discuss the introduction of randomness in TVGs.

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Computing a (short) path between two vertices is one of the most fundamental primitives in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. A path is time-respecting, or temporal, if it uses edges with non-decreasing time stamps. We investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration $$\varDelta $$ Δ , referred to as $$\varDelta $$ Δ -restless temporal paths. This constraint arises naturally in the modeling of real-world processes like packet routing in communication networks and infection transmission routes of diseases where recovery confers lasting resistance. While finding temporal paths without waiting time restrictions is known to be doable in polynomial time, we show that the “restless variant” of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by the distance to disjoint path of the underlying graph, which implies W[1]-hardness for many other parameters like feedback vertex number and pathwidth. A natural question is thus whether the problem becomes tractable in some natural settings. We explore several natural parameterizations, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.

Besides the complexity in time or in number of messages, a common approach for analyzing distributed algorithms is to look at the assumptions they make on the underlying network. We investigate this question from the perspective of network dynamics. In particular, we ask how a given property on the evolution of the network can be rigorously proven as necessary or sufficient for a given algorithm. The main contribution of this paper is to propose the combination of two existing tools in this direction: local computations by means of graph relabelings, and evolving graphs. Such a combination makes it possible to express fine-grained properties on the network dynamics, then examine what impact those properties have on the execution at a precise, intertwined, level. We illustrate the use of this framework through the analysis of three simple algorithms, then discuss general implications of this work, which include (i) the possibility to compare distributed algorithms on the basis of their topological requirements, (ii) a formal hierarchy of dynamic networks based on these requirements, and (iii) the potential for mechanization induced by our framework, which we believe opens a door towards automated analysis and decision support in dynamic networks.Proof. (using Lemma 1). Following from Lemma 1 and the initial states (I for the emitter, N for all other vertices), we have O A1 =⇒ C 1 , and thus ¬C 1 =⇒ ¬O A1 Condition 2 ∀v ∈ V, emitter st v Proposition 2 Under Progression Hypothesis 1 (P H 1 , defined in the previous section), Condition 2 (C 2 ) is sufficient on G to guarantee that A 1 will reach O A1 .Proof. (1):Now, because λ t0 (emitter) = I, we have C 2 (G) =⇒ ∀X ∈ X A/G , P 1 (G k )

Abstract. Most highly dynamic infrastructure-less networks have in common that the assumption of connectivity does not necessarily hold at a given instant. Still, communication routes can be available between any pair of nodes over time and space. These networks (variously called delay-tolerant, disruptive-tolerant, challenged) are naturally modeled as time-varying graphs (or evolving graphs), where the existence of an edge is a function of time. In this paper we study deterministic computations under unstructured mobility, that is when the edges of the graph appear infinitely often but without any (known) pattern. In particular, we focus on the problem of broadcasting with termination detection. We explore the problem with respect to three possible metrics: the date of message arrival (foremost), the time spent doing the broadcast (fastest), and the number of hops used by the broadcast (shortest). We prove that the solvability and complexity of this problem vary with the metric considered, as well as with the type of knowledge a priori available to the entities. These results draw a complete computability map for this problem when mobility is unstructured.

Highly dynamic networks rarely offer end-to-end connectivity at a given time. Yet, connectivity in these networks can be established over time and space, based on temporal analogues of multi-hop paths (also called journeys). Attempting to optimize the selection of the journeys in these networks naturally leads to the study of three cases: shortest (minimum hop), fastest (minimum duration), and foremost (earliest arrival) journeys. Efficient centralized algorithms exists to compute all cases, when the full knowledge of the network evolution is given.In this paper, we study the distributed counterparts of these problems, i.e. shortest, fastest, and foremost broadcast with termination detection (TDB), with minimal knowledge on the topology. We show that the feasibility of each of these problems requires distinct features on the evolution, through identifying three classes of dynamic graphs wherein the problems become gradually feasible: graphs in which the re-appearance of edges is recurrent (class R), bounded-recurrent (B), or periodic (P), together with specific knowledge that are respectively n (the number of nodes), ∆ (a bound on the recurrence time), and p (the period). In these classes it is not required that all pairs of nodes get in contact -only that the overall footprint of the graph is connected over time.Our results, together with the strict inclusion between P, B, and R, implies a feasibility order among the three variants of the problem, i.e. TDB[f oremost] requires weaker assumptions on the topology dynamics than TDB[shortest], which itself requires less than TDB[f astest]. Reversely, these differences in feasibility imply that the computational powers of Rn, B ∆ , and Pp also form a strict hierarchy.

Abstract-Delay-tolerant networks (DTNs) are characterized by a possible absence of end-to-end communication routes at any instant. In most cases, however, a form of connectivity can be established over time and space. This particularity leads to consider the relevance of a given route not only in terms of hops (topological length), but also in terms of time (temporal length). The problem of measuring temporal distances between individuals in a social network was recently addressed, based on a posteriori analysis of interaction traces. This paper focuses on the distributed version of this problem, asking whether every node in a network can know precisely and in real time how out-ofdate it is with respect to every other. Answering affirmatively is simple when contacts between the nodes are punctual, using the temporal adaptation of vector clocks provided in [23]. It becomes more difficult when contacts have a duration and can overlap in time with each other. We demonstrate that the problem remains solvable with arbitrarily long contacts and non-instantaneous (though invariant and known) propagation delays on edges. This is done constructively by extending the temporal adaptation of vector clocks to non-punctual causality. The second part of the paper discusses how the knowledge of temporal lags could be used as a building block to solve more concrete problems, such as the construction of foremost broadcast trees or network backbones in periodically-varying DTNs.

The consensus problem is a fundamental paradigm in distributed systems, because it captures the difficulty to solve other agreement problems. Many current systems evolve with time, e.g., due to node mobility, and consensus has been little studied in these systems so far. Specifically, it is not well established how to define an appropriate set of assumptions for consensus in dynamic distributed systems. This paper studies a hierarchy of three classes of time-varying graphs, and provides a solution for each class to the problem of Terminating Reliable Broadcast (TRB). The classes introduce increasingly stronger assumptions on timeliness, so that the trade-off between weakness versus implementability and efficiency can be analysed. Being TRB equivalent to consensus in synchronous systems, the paper extends this equivalence to dynamic systems.