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37
Timevarying graphs and dynamic networks
 International Journal of Parallel, Emergent and Distributed Systems
"... The past few years have seen intensive research efforts carried out in some apparently unrelated areas of dynamic systems – delaytolerant networks, opportunisticmobility networks, social networks – obtaining closely related insights. Indeed, the concepts discovered in these investigations can be v ..."
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The past few years have seen intensive research efforts carried out in some apparently unrelated areas of dynamic systems – delaytolerant networks, opportunisticmobility networks, social networks – obtaining 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 timevarying 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 atemporal (as in the majority of existing studies) or temporal. Finally, we briefly discuss the introduction of randomness in TVGs.
Analyzing Network Coding Gossip Made Easy
, 2011
"... We introduce projection analysis – a new technique to analyze the stopping time of gossip protocols that are based on random linear network coding (RLNC). Projection analysis drastically simplifies, extends and strengthens previous results. We analyze RLNC gossip in a general framework for network a ..."
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Cited by 17 (9 self)
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We introduce projection analysis – a new technique to analyze the stopping time of gossip protocols that are based on random linear network coding (RLNC). Projection analysis drastically simplifies, extends and strengthens previous results. We analyze RLNC gossip in a general framework for network and communication models that encompasses and unifies the models used previously in this context. We show, in most settings for the first time, that the RLNC gossip converges with high probability in optimal time. Most stopping times are of the form O(k + T), where k is the number of messages to be distributed and T is the time it takes to disseminate one message. This means RLNC gossip achieves “perfect pipelining”. Our analysis directly extends to highly dynamic networks in which the topology can change completely at any time. This remains true, even if the network dynamics are controlled by a fully adaptive adversary that knows the complete network state. Virtually nothing besides simple O(kT) sequential flooding protocols was previously known for such a setting. While RLNC gossip works in this wide variety of networks our analysis remains the same and extremely simple. This contrasts with more complex proofs that were put forward to give less strong results for various special cases.
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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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 7 (6 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 6 (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.
The Complexity of Data Aggregation in Directed Networks
"... Abstract. We study problems of data aggregation, such as approximate counting and computing the minimum input value, in synchronous directed networks with bounded message bandwidthB = Ω(logn). In undirected networks of diameter D, many such problems can easily be solved in O(D) rounds, using O(logn) ..."
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Abstract. We study problems of data aggregation, such as approximate counting and computing the minimum input value, in synchronous directed networks with bounded message bandwidthB = Ω(logn). In undirected networks of diameter D, many such problems can easily be solved in O(D) rounds, using O(logn)size messages. We show that for directed networks this is not the case: when the bandwidth B is small, several classical data aggregation problems have a time complexity that depends polynomially on the size of the network, even when the diameter of the network is constant. We show that computing anǫapproximation to the size n of the network requires Ω(min { n,1/ǫ 2} /B) rounds, even in networks of diameter 2. We also show that computing a sensitive function (e.g., minimum and maximum) requires Ω ( √ n/B) rounds in networks of diameter 2, provided that the diameter is not known in advance to be o ( √ n/B). Our lower bounds are established by reduction from several wellknown problems in communication complexity. On the positive side, we give a nearly optimal Õ(D+ √ n/B)round algorithm for computing simple sensitive functions using messages of size B = Ω(logN), where N is a loose upper bound on the size of the network and D is the diameter. 1
One Packet Suffices  Highly Efficient Packetized Network Coding With Finite Memory
 in Proc. of the International Symposium on Information Theory (ISIT
, 2011
"... Abstract—Random Linear Network Coding (RLNC) has emerged as a powerful tool for robust highthroughput multicast. Projection analysis, a recently introduced technique, shows that the distributed packetized RLNC protocol achieves (order) optimal and perfectly pipelined information dissemination in ma ..."
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Cited by 4 (4 self)
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Abstract—Random Linear Network Coding (RLNC) has emerged as a powerful tool for robust highthroughput multicast. Projection analysis, a recently introduced technique, shows that the distributed packetized RLNC protocol achieves (order) optimal and perfectly pipelined information dissemination in many settings. In the original approach to RNLC intermediate nodes code together all available information. This requires intermediate nodes to keep considerable data available for coding. Moreover, it results in a coding complexity that grows linearly with the size of this data. While this has been identified as a problem, approaches that combine queuing theory and network coding have heretofore not provided a succinct representation of the memory needs of network coding at intermediates nodes. This paper shows the surprising result that, in all settings with a continuous stream of data, network coding continues to perform optimally even if only one packet per node is kept in active memory and used for computations. This leads to an extremely simple RLNC protocol variant with drastically reduced requirements on computational and memory resources. By extending the projection analysis, we show that in all settings in which the RLNC protocol was proven to be optimal its finite memory variant performs equally well. In the same way as the original projection analysis, our technique applies in a wide variety of network models, including highly dynamic topologies that can change completely at any time in an adversarial fashion. I.
Prioritized Gossip in Vehicular Networks
, 2010
"... We present a method for using real world mobility traces to identify tractable theoretical models for the study of distributed algorithms in mobile networks. We validate the method by deriving a vehicular ad hoc network model from a large corpus of position data generated by Bostonarea taxicabs. Un ..."
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We present a method for using real world mobility traces to identify tractable theoretical models for the study of distributed algorithms in mobile networks. We validate the method by deriving a vehicular ad hoc network model from a large corpus of position data generated by Bostonarea taxicabs. Unlike previous work, our model does not assume global connectivity or eventual stability; it instead assumes only that some subset of processes are connected through transient paths (e.g., paths that exist over time). We use this model to study the problem of prioritized gossip, in which processes attempt to disseminate messages of different priority. Specifically, we present CabChat, a distributed prioritized gossip algorithm that leverages an interesting connection to the classic Tower of Hanoi problem to schedule the broadcast of packets of different priorities. Whereas previous studies of gossip leverage strong connectivity or stabilization assumptions to prove the time complexity of global termination, in our model, with its weak assumptions, we instead analyze CabChat with respect to its ability to deliver a high proportion of high priority messages over the transient paths that happen to exist in a given execution.
Opportunistic Information Dissemination in Mobile Adhoc Networks: The Profit of Global Synchrony
"... The topic of this paper is the study of Information Dissemination in Mobile Adhoc Networks by means of deterministic protocols. We characterize the connectivity resulting from the movement, from failures and from the fact that nodes may join the computation at different times with two values, α an ..."
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The topic of this paper is the study of Information Dissemination in Mobile Adhoc Networks by means of deterministic protocols. We characterize the connectivity resulting from the movement, from failures and from the fact that nodes may join the computation at different times with two values, α and β, so that, within α time slots, some node that has the information must be connected to some node without it for at least β time slots. The protocols studied are classified into three classes: oblivious (the transmission schedule of a node is only a function of its ID), quasioblivious (the transmission schedule may also depend on a global time), and adaptive. The main contribution of this work concerns negative results. Contrasting the lower and upper bounds derived, interesting complexity gaps among protocolclasses are observed. More precisely, in order to guarantee any progress towards solving the problem, it is shown that β must be at least n − 1 in general, but that β ∈ Ω(n 2 / log n) if an oblivious protocol is used. Since quasioblivious protocols can guarantee progress with β ∈ O(n), this represents a significant gap,
Counting in Anonymous Dynamic Networks: An Experimental Perspective
"... Abstract. Counting is a fundamental problem of every distributed system as it represents a basic building block to implement high level abstractions. In anonymous dynamic networks, counting is far from being trivial as nodes have no identity and the knowledge about the network is limited to the loca ..."
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Abstract. Counting is a fundamental problem of every distributed system as it represents a basic building block to implement high level abstractions. In anonymous dynamic networks, counting is far from being trivial as nodes have no identity and the knowledge about the network is limited to the local perception of the process itself. Moreover, nodes have to cope with continuous changes of the topology imposed by an external adversary. A relevant example of such kind of networks is represented by wireless sensor networks characterized by the dynamicity of the communication links due to possible collisions or to the presence of dutycycles aimed at battery preservation. In a companion paper [14], two leaderbased algorithms, namely ANoK and ALCO, to count the number of processes in an anonymous dynamic network have been proposed. Such algorithms employ the notion of energy transfer to count the exact number of nodes by (i) having no knowledge on the network or (ii) having access to a local counting oracle reporting the exact number of neighbors at the beginning of a communication round. Let us notice that, while ALCO has a well defined terminating condition, ANoK only ensures that eventually the leader is able to count the exact number of processes but it is not able to identify when this happens. In this paper, we define a new algorithm A ∗ NoK by augmenting ANoK with a termination heuristics that allows the leader to guess when it should output the current count and we provide an experimental evaluation on different types of dynamic graphs for both ANoK and A ∗ NoK. In addition, we also extended ALCO by defining a new algorithm, namely A ∗ LCO, that is the basic ALCO augmented with a symmetry breaking condition that helps to speed up the convergence time. 1