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Coordinated Consensus in Dynamic Networks
"... We study several variants of coordinated consensus in dynamic networks. We assume a synchronous model, where the communication graph for each round is chosen by a worstcase adversary. The network topology is always connected, but can change completely from one round to the next. The model captures ..."
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Cited by 19 (1 self)
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We study several variants of coordinated consensus in dynamic networks. We assume a synchronous model, where the communication graph for each round is chosen by a worstcase adversary. The network topology is always connected, but can change completely from one round to the next. The model captures mobile and wireless networks, where communication can be unpredictable. In this setting we study the fundamental problems of eventual, simultaneous, and ∆coordinated consensus, as well as their relationship to other distributed problems, such as determining the size of the network. We show that in the absence of a good initial upper bound on the size of the network, eventual consensus is as hard as computing deterministic functions of the input, e.g., the minimum or maximum of inputs to the nodes. We also give an algorithm for computing such functions that is optimal in every execution. Next, we show that simultaneous consensus can never be achieved in less than n−1 rounds in any execution, where n is the size of the network; consequently, simultaneous consensus is as hard as computing an upper bound on the number of nodes in the network. For ∆coordinated consensus, we show that if the ratio between nodes with input 0 and input 1 is bounded away from 1, it is possible to decide in timen−Θ ( √ n∆), where∆bounds the time from the first decision until all nodes decide. If the dynamic graph has diameterD, the time to decide ismin{O(nD/∆),n−Ω(n∆/D)}, even if D is not known in advance. Finally, we show that (a) there is a dynamic graph such that for every input, no node can decide before timen−O( ∆ 0.28 n 0.72); and (b) for any diameterD=O(∆), there is an execution with diameter D where no node can decide before time Ω(nD/∆). To our knowledge, our work constitutes the first study of ∆coordinated consensus in general graphs.
Measuring Temporal Lags in DelayTolerant Networks
, 2011
"... Delaytolerant networks (DTNs) are characterized by a possible absence of endtoend 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 h ..."
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Cited by 13 (11 self)
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Delaytolerant networks (DTNs) are characterized by a possible absence of endtoend 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 outofdate 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 noninstantaneous (though invariant and known) propagation delays on edges. This is done constructively by extending the temporal adaptation of vector clocks to nonpunctual 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 periodicallyvarying DTNs.
On the power of waiting when exploring public transportation systems
 in Proc. of 15th Intl. Conf. On Principles Of Distributed Systems (OPODIS
, 2011
"... Abstract. We study the problem of exploration by a mobile entity (agent) of a class of dynamic networks, namely the periodicallyvarying graphs (the PVgraphs, modeling public transportation systems, among others). These are defined by a set of carriers following infinitely their prescribed route al ..."
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Cited by 11 (1 self)
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Abstract. We study the problem of exploration by a mobile entity (agent) of a class of dynamic networks, namely the periodicallyvarying graphs (the PVgraphs, modeling public transportation systems, among others). These are defined by a set of carriers following infinitely their prescribed route along the stations of the network. Flocchini, Mans, and Santoro [FMS09] (ISAAC 2009) studied this problem in the case when the agent must always travel on the carriers and thus cannot wait on a station. They described the necessary and sufficient conditions for the problem to be solvable and proved that the optimal number of steps (and thus of moves) to explore a nnode PVgraph of k carriers and maximal period p is in Θ(k · p 2) in the general case. In this paper, we study the impact of the ability to wait at the stations. We exhibit the necessary and sufficient conditions for the problem to be solvable in this context, and we prove that waiting at the stations allows the agent to reduce the worstcase optimal number of moves by a multiplicative factor of at least Θ(p), while the time complexity is reduced to Θ(n · p). (In any connected PVgraph, we have n ≤ k · p.) We also show some complementary optimal results in specific cases (same period for all carriers, highly connected PVgraphs). Finally this new ability allows the agent to completely map the PVgraph, in addition to just explore it.
Temporal reachability graphs
 ACM MobiCom
, 2012
"... While a natural fit for modeling and understanding mobile networks, timevarying graphs remain poorly understood. Indeed, many of the usual concepts of static graphs have no obvious counterpart in timevarying ones. In this paper, we introduce the notion of temporal reachability graphs. A (τ, δ)rea ..."
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Cited by 11 (1 self)
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While a natural fit for modeling and understanding mobile networks, timevarying graphs remain poorly understood. Indeed, many of the usual concepts of static graphs have no obvious counterpart in timevarying ones. In this paper, we introduce the notion of temporal reachability graphs. A (τ, δ)reachability graph is a timevarying 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 τ, and arriving within a maximum delay δ. 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 reallife datasets. On top of defining clear upper bounds on communication capabilities, reachability graphs highlight asymmetric communication opportunities and offloading potential.
Temporal Network Optimization Subject to Connectivity Constraints
, 2013
"... In this work we consider temporal networks, i.e. networks defined by a labeling λ assigning to each edge of an underlying graph G asetofdiscrete timelabels. The labels of an edge, which are natural numbers, indicate the discrete time moments at which the edge is available. We focus on path problem ..."
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Cited by 8 (6 self)
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In this work we consider temporal networks, i.e. networks defined by a labeling λ assigning to each edge of an underlying graph G asetofdiscrete timelabels. The labels of an edge, which are natural numbers, indicate the discrete time moments at which the edge is available. We focus on path problems of temporal networks. In particular, we consider timerespecting paths, i.e. paths whose edges are assigned by λ a strictly increasing sequence of labels. We begin by giving two efficient algorithms for computing shortest timerespecting paths on a temporal network. We then prove that there is a natural analogue of Menger’s theorem holding for arbitrary temporal networks. Finally, we propose two cost minimization parameters for temporal network design. One is the temporality of G, in which the goal is to minimize the maximum number of labels of an edge, and the other is the temporal cost of G, inwhich the goal is to minimize the total number of labels used. Optimization of these parameters is performed subject to some connectivity constraint. We prove several lower and upper bounds for the temporality and the temporal cost of some very basic graph families such as rings, directed acyclic graphs, and trees.
Causality, Influence, and Computation in Possibly Disconnected Synchronous Dynamic Networks
, 2013
"... In this work, we study the propagation of influence and computation in dynamic distributed computing systems that are possibly disconnected at every instant. We focus on a synchronous message passing communication model with broadcast and bidirectional links. Our network dynamicity assumption is a w ..."
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Cited by 6 (6 self)
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In this work, we study the propagation of influence and computation in dynamic distributed computing systems that are possibly disconnected at every instant. We focus on a synchronous message passing communication model with broadcast and bidirectional links. Our network dynamicity assumption is a worstcase dynamicity controlled by an adversary scheduler, which has received much attention recently. We replace the usual (in worstcase dynamic networks) assumption that the network is connected at every instant by minimal temporal connectivity conditions. Our conditions only require that another causal influence occurs within every timewindow of some given length. Based on this basic idea we define several novel metrics for capturing the speed of information spreading in a dynamic network. We present several results that correlate these metrics. Moreover, we investigate termination criteria in networks in which an upper bound on any of these metrics is known. We exploit our termination criteria to provide efficient (and optimal in some cases) protocols that solve the fundamental counting and alltoall token dissemination (or gossip) problems.
On the Exploration of TimeVarying Networks
, 2011
"... We study the computability and complexity of the exploration problem in a class of highly dynamic networks: carrier graphs, where the edges between sites exist only at some (unknown) times defined by the periodic movements of mobile carriers among the sites. These graphs naturally model highly dynam ..."
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Cited by 6 (2 self)
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We study the computability and complexity of the exploration problem in a class of highly dynamic networks: carrier graphs, where the edges between sites exist only at some (unknown) times defined by the periodic movements of mobile carriers among the sites. These graphs naturally model highly dynamic infrastructureless networks such as public transports with fixed timetables, low earth orbiting (LEO) satellite systems, security guards ’ tours, etc. We focus on the opportunistic exploration of these graphs, that is by an agent that exploits the movements of the carriers to move in the network. 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 carrier graphs defined by restricted classes of carriers movements. We then prove that the limitations on computability and complexity we have established are indeed tight. In fact we prove that all necessary conditions are also sufficient and all lower bounds on costs are tight. We do so
Agreement in Directed Dynamic Networks
 In Proceedings of the 19th International Colloquium on Structural Information and Communication Complexity (SIROCCO
, 2012
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Naming and counting in anonymous unknown dynamic networks
 In 15th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS
, 2013
"... Abstract. In this work, we study the fundamental naming and counting problems (and some variations) in networks that are anonymous, unknown, and possibly dynamic. In counting, nodes must determine the size of the network n and in naming they must end up with unique identities. By anonymous we mean ..."
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Cited by 4 (4 self)
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Abstract. In this work, we study the fundamental naming and counting problems (and some variations) in networks that are anonymous, unknown, and possibly dynamic. In counting, nodes must determine the size of the network n and in naming they must end up with unique identities. By anonymous we mean that all nodes begin from identical states apart possibly from a unique leader node and by unknown that nodes have no a priori knowledge of the network (apart from some minimal knowledge when necessary) including ignorance of n. Network dynamicity is modeled by the 1interval connectivity model [KLO10], in which communication is synchronous and a (worstcase) adversary chooses the edges of every round subject to the condition that each instance is connected. We first focus on static networks with broadcast where we prove that, without a leader, counting is impossible to solve and that naming is impossible to solve even with a leader and even if nodes know n. These impossibilities carry over to dynamic networks as well. We also show that a unique leader suffices in order to solve counting in linear time. Then we focus on dynamic networks with broadcast. We conjecture that dynamicity renders nontrivial computation impossible. In view of this, we let the nodes know an upper bound on the maximum degree that will ever appear and show that in this case the nodes can obtain an upper bound on n. Finally, we replace broadcast with onetoeach, in which a node may send a different message to each of its neighbors. Interestingly, this natural variation is proved to be computationally equivalent to a fullknowledge model, in which unique names exist and the size of the network is known. 1
Ephemeral Networks with Random Availability of Links: Diameter and Connectivity
, 2014
"... In this work we consider temporal networks, the links of which are available only at random times (randomly available temporal networks). Our networks are ephemeral: their links appear sporadically, only at certain times, within a given maximum time (lifetime of the net). More specifically, our tem ..."
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Cited by 4 (0 self)
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In this work we consider temporal networks, the links of which are available only at random times (randomly available temporal networks). Our networks are ephemeral: their links appear sporadically, only at certain times, within a given maximum time (lifetime of the net). More specifically, our temporal networks notion concerns networks, whose edges (arcs) are assigned one or more random discretetime labels drawn from a set of natural numbers. The labels of an edge indicate the discrete moments in time at which the edge is available. In such networks, information (e.g., messages) have to follow temporal paths, i.e., paths, the edges of which are assigned a strictly increasing sequence of labels. We first examine a very hostile network: a clique, each edge of which is known to be available only