Approximate Consensus in Highly Dynamic Networks: The Role of Averaging Algorithms

In this paper, we investigate the approximate consensus problem in highly dynamic networks in which topology may change continually and unpredictably. We prove that in both synchronous and partially synchronous systems, approximate consensus is solvable if and only if the communication graph in each round has a rooted spanning tree, i.e., there is a coordinator at each time. The striking point in this result is that the coordinator is not required to be unique and can change arbitrarily from round to round. Interestingly, the class of averaging algorithms, which are memoryless and require no process identifiers, entirely captures the solvability issue of approximate consensus in that the problem is solvable if and only if it can be solved using any averaging algorithm. Concerning the time complexity of averaging algorithms, we show that approximate consensus can be achieved with precision of $\varepsilon$ in a coordinated network model in $O(n^{n+1} \log\frac{1}{\varepsilon})$ synchronous rounds, and in $O(\Delta n^{n\Delta+1} \log\frac{1}{\varepsilon})$ rounds when the maximum round delay for a message to be delivered is $\Delta$. While in general, an upper bound on the time complexity of averaging algorithms has to be exponential, we investigate various network models in which this exponential bound in the number of nodes reduces to a polynomial bound. We apply our results to networked systems with a fixed topology and classical benign fault models, and deduce both known and new results for approximate consensus in these systems. In particular, we show that for solving approximate consensus, a complete network can tolerate up to 2n-3 arbitrarily located link faults at every round, in contrast with the impossibility result established by Santoro and Widmayer (STACS '89) showing that exact consensus is not solvable with n-1 link faults per round originating from the same node

Diffusive clock synchronization in highly dynamic networks

International audienceThis paper studies the clock synchronization problem in highly dynamic networks. We show that diffusive synchronization algorithms are well adapted to environments in which the network topology may change unpredictably. In a diffusive algorithm, each node repeatedly (i) estimates the clock difference to its neighbors via broadcast of zero-bit messages, and (ii) updates its local clock according to a weighted average of the estimated differences. The system model allows for drifting local clocks, running at possibly different frequencies. We show that having a rooted spanning tree in the network at every time instance suffices to solve clock synchronization. We do not require any stability of the spanning tree, nor do we impose that the links of the spanning tree be known to the nodes. Explicit bounds on the convergence speed are obtained. In particular, our results settle an open question posed by Simeone and Spagnolini to reach clock synchronization in dynamic networks in the presence of nonzero clock drift. We also identify certain reasonable assumptions that allow for a significant higher convergence speed, e.g., bidirectional networks or random graph models

Brief Announcement: Local Distributed Algorithms in Highly Dynamic Networks

We define a generalization of local distributed graph problems to (synchronous round-based) dynamic networks and present a framework for developing algorithms for these problems. We require two properties from our algorithms: (1) They should satisfy non-trivial guarantees in every round. The guarantees should be stronger the more stable the graph has been during the last few rounds and they coincide with the definition of the static graph problem if no topological change appeared recently. (2) If a constant neighborhood around some part of the graph is stable during an interval, the algorithms quickly converge to a solution for this part of the graph that remains unchanged throughout the interval. We demonstrate our generic framework with two classic distributed graph, namely (degree+1)-vertex coloring and maximal independent set (MIS)

Greedy Forwarding in Dynamic Scale-Free Networks Embedded in Hyperbolic Metric Spaces

We show that complex (scale-free) network topologies naturally emerge from hyperbolic metric spaces. Hyperbolic geometry facilitates maximally efficient greedy forwarding in these networks. Greedy forwarding is topology-oblivious. Nevertheless, greedy packets find their destinations with 100% probability following almost optimal shortest paths. This remarkable efficiency sustains even in highly dynamic networks. Our findings suggest that forwarding information through complex networks, such as the Internet, is possible without the overhead of existing routing protocols, and may also find practical applications in overlay networks for tasks such as application-level routing, information sharing, and data distribution

Clock Synchronization and Distributed Estimation in Highly Dynamic Networks: An Information Theoretic Approach

International audienceWe consider the External Clock Synchronization problem in dynamic sensor networks. Initially, sensors obtain inaccurate estimations of an external time reference and subsequently collaborate in order to synchronize their internal clocks with the external time. For simplicity, we adopt the drift-free assumption, where internal clocks are assumed to tick at the same pace. Hence, the problem is reduced to an estimation problem, in which the sensors need to estimate the initial external time. This work is further relevant to the problem of collective approximation of environmental values by biological groups. Unlike most works on clock synchronization that assume static networks, this paper focuses on an extreme case of highly dynamic networks. Specifically, we assume a non-adaptive scheduler adversary that dictates in advance an arbitrary, yet independent, meeting pattern. Such meeting patterns fit, for example, with short-time scenarios in highly dynamic settings, where each sensor interacts with only few other arbitrary sensors. We propose an extremely simple clock synchronization algorithm that is based on weighted averages, and prove that its performance on any given independent meeting pattern is highly competitive with that of the best possible algorithm, which operates without any resource or computational restrictions, and knows the meeting pattern in advance. In particular, when all distributions involved are Gaussian, the performances of our scheme coincide with the optimal performances. Our proofs rely on an extensive use of the concept of Fisher information. We use the Cramér-Rao bound and our definition of a Fisher Channel Capacity to quantify information flows and to obtain lower bounds on collective performance. This opens the door for further rigorous quantifications of information flows within collaborative sensors

Semi-probabilistic Routing for Highly Dynamic Networks

Abstract. In this paper we describe a semi-probabilistic routing approach designed to enable content-based publish-subscribe on highly dynamic networks, e.g., mobile, peer-to-peer, or wireless sensor networks. We present the rationale and high level strategy of our approach, and then show its application in a link-based graph overlay as well as in a broadcast-based sensor network. Simulation results confirm that, in both scenarios, our semi-probabilistic approach strikes a balance between entirely deterministic and entirely probabilistic solutions, achieving high reliability with low overhead

Self-Modeling Based Diagnosis of Software-Defined Networks

Networks built using SDN (Software-Defined Networks) and NFV (Network Functions Virtualization) approaches are expected to face several challenges such as scalability, robustness and resiliency. In this paper, we propose a self-modeling based diagnosis to enable resilient networks in the context of SDN and NFV. We focus on solving two major problems: On the one hand, we lack today of a model or template that describes the managed elements in the context of SDN and NFV. On the other hand, the highly dynamic networks enabled by the softwarisation require the generation at runtime of a diagnosis model from which the root causes can be identified. In this paper, we propose finer granular templates that do not only model network nodes but also their sub-components for a more detailed diagnosis suitable in the SDN and NFV context. In addition, we specify and validate a self-modeling based diagnosis using Bayesian Networks. This approach differs from the state of the art in the discovery of network and service dependencies at run-time and the building of the diagnosis model of any SDN infrastructure using our templates