A Generic Framework for Impossibility Results in Time-Varying Graphs Nicolas Braud-Santoni IAIK, Graz University of Technology, Austria E-mail: nicolas.braud-santoni@iaik.tugraz.at Swan Dubois Mohamed-Hamza Kaaouachi Franck Petit Sorbonne Universit´ es, UPMC Universit´ e Paris 6, F-75005, Paris, France CNRS, UMR 7606, LIP6, F-75005, Paris, France Inria, ´ Equipe-projet REGAL, F-75005, Paris, France E-mail: firstname.lastname@lip6.fr Abstract—We address highly dynamic distributed systems modelled by time-varying graphs (TVGs). We are interested in proof of impossibility results that often use informal arguments about convergence. First, we provide a topological distance metric over sets of TVGs to correctly define the convergence of TVG sequences in such sets. Next, we provide a general framework that formally proves the convergence of the sequence of executions of any deterministic algorithm over TVGs of any convergent sequence of TVGs. Finally, we illustrate the relevance of the above result by proving that no deterministic algorithm exists to compute the underlying graph of any connected-over-time TVG, i.e., any TVG of the weakest class of long-lived TVGs. Keywords—Time-Varying Graph, Impossibility results, underly- ing graph I. I NTRODUCTION The availability of wireless communication has drastically increased in recent years and established new applications that make various communicating agents and terminals (e.g., robots, sensors, Unmanned Aerial Vehicles, ...) interact to- gether. A common feature of a vast majority of these net- works is their high dynamic, meaning that their topology keeps continuously changing over time. Classically, distributed systems are modelled by a static undirected connected graph where vertices are processes and edges represent bidirectional communication links. Clearly, such modelling is not suitable for highly dynamic networks. Numerous models taking in account topological changes over time have been proposed since several decades, to quote only a few, [1], [2], [3], [4], [5], [6], [7]. Some works aim at unifying most of the above approaches. For instance, in [8], the authors introduced the evolving graphs. They proposed modelling the time as a sequence of discrete time instants and the system dynamic by a sequence of static This work was performed within the Labex SMART, supported by French state funds managed by the ANR within the “Investissements d’Avenir” programme under reference ANR-11-LABX-65. graphs, one for each time instant. More recently, another graph formalism, called Time-Varying Graphs (TVG), has been provided in [9]. In contrast with evolving graphs, TVGs allow systems evolving within continuous time. Also in [9] and in companion papers [10], [11], TVGs are gathered and ordered into classes depending mainly on two main features: the quality of connectivity among the participating nodes and the possibility/impossibility to perform tasks. The core result of our paper consists of providing a basic block intended for formally prove impossibility results in TVGs. As many other proofs of impossibility results in distributed computing (e.g., [12], [13], [14], [15]), it is based on convergence of sequences of abstract objects (e.g., graphs, executions, causal DAGs,. . . ) built over the model describing the considered distributed system. More precisely, we first define a metric to compute a distance between any pair of TVGs based on the length of their longest common temporal prefix. Such distance allows to study the convergence of TVG sequences. Our main result consists of showing that, given an algorithm A designed for any TVG and a sequence of TVGs that converges toward a TVG g, then the sequence of executions of A on each TVG of the sequence also converges. Furthermore, the latter converges toward the execution of A over g. Next, we provide an example of use of this general result. In this paper, we consider long-lived TVGs, i.e., TVGS that ensure that every process is able to communicate infinitely often with any other process (possibly indirectly and not always along the same path). In this context, the weakest assumption on connectivity is captured by the connected-over-time TVGs class, i.e., the class of TVGs where any node can contact any other node infinitely often without any supplementary assumption (e.g., recurrence, periodicity,...). It can also be described as the family of TVGs such that the eventual underlying graph (i.e., the subgraph encompassing all edges that are infinitely often present) is connected. Then, we show that no deterministic 2015 IEEE International Parallel and Distributed Processing Symposium Workshops /15 $31.00 © 2015 IEEE DOI 10.1109/IPDPSW.2015.59 483 2015 IEEE International Parallel and Distributed Processing Symposium Workshop 978-1-4673-7684-6/15 $31.00 © 2015 IEEE DOI 10.1109/IPDPSW.2015.59 483