We design the weights in consensus algorithms for spatially correlated random topologies. These arise with 1) networks with spatially correlated random link failures and 2) networks with randomized averaging protocols. We show that the weight optimization problem is convex for both symmetric and asymmetric random graphs. With symmetric random networks, we choose the consensus mean-square error (MSE) convergence rate as the optimization criterion and explicitly express this rate as a function of the link formation probabilities, the link formation spatial correlations, and the consensus weights. We prove that the MSE convergence rate is a convex, nonsmooth function of the weights, enabling global optimization of the weights for arbitrary link formation probabilities and link correlation structures. We extend our results to the case of asymmetric random links. We adopt as optimization criterion the mean-square deviation (MSdev) of the nodes' states from the current average state. We prove that MSdev is a convex function of the weights. Simulations show that significant performance gain is achieved with our weight design method when compared with other methods available in the literature.
We provide comprehensive simulation experiments to demonstrate the effectiveness of our approach. We provide two different models of random networks with correlated link failures; in addition, we study the broadcast gossip algorithm @cite_28 , as an example of randomized protocol with asymmetric links. In all cases, simulations confirm that our method shows significant gain compared to the methods available in the literature. Also, we show that the gain increases with the network size.
@cite_28
2132612558 2110100895 2101517602 2161305486
In this paper, we study two related iterative randomized algorithms for distributed computation of averages. The first algorithm is the Broadcast Gossip Algorithm, in which at each iteration one randomly selected node broadcasts its own state to its neighbors. The second algorithm is a novel variation of the former, in which at each iteration every node is allowed to broadcast: hence, this algorithm, which we call Collision Broadcast Gossip Algorithm (CBGA), is affected by interference among messages. The performance of both algorithms is evaluated in terms of rate of convergence and asymptotical error: focusing on large Abelian Cayley networks, we highlight the role of topology and of design parameters. We show that on fully connected graphs the rate of convergence is bounded away from one, whereas the asymptotical error is bounded away from zero. On the contrary, on sparse graphs the rate of convergence goes to one and the asymptotical error goes to zero, as the size of the network grows larger. Our results also show that the performance of the CBGA is close to the performance of the BGA: this indicates the robustness of broadcast gossip algorithms to interferences. Over the last decade, we have seen a revolution in connectivity between computers, and a resulting paradigm shift from centralized to highly distributed systems. With massive scale also comes massive instability, as node and link failures become the norm rather than the exception. For such highly volatile systems, decentralized gossip-based protocols are emerging as an approach to maintaining simplicity and scalability while achieving fault-tolerant information dissemination. In this paper, we study the problem of computing aggregates with gossip-style protocols. Our first contribution is an analysis of simple gossip-based protocols for the computation of sums, averages, random samples, quantiles, and other aggregate functions, and we show that our protocols converge exponentially fast to the true answer when using uniform gossip. Our second contribution is the definition of a precise notion of the speed with which a node's data diffuses through the network. We show that this diffusion speed is at the heart of the approximation guarantees for all of the above problems. We analyze the diffusion speed of uniform gossip in the presence of node and link failures, as well as for flooding-based mechanisms. The latter expose interesting connections to random walks on graphs. Motivated by applications to sensor, peer-to-peer and ad hoc networks, we study distributed asynchronous algorithms, also known as gossip algorithms, for computation and information exchange in an arbitrarily connected network of nodes. Nodes in such networks operate under limited computational, communication and energy resources. These constraints naturally give rise to "gossip" algorithms: schemes which distribute the computational burden and in which a node communicates with a randomly chosen neighbor. We analyze the averaging problem under the gossip constraint for arbitrary network, and find that the averaging time of a gossip algorithm depends on the second largest eigenvalue of a doubly stochastic matrix characterizing the algorithm. Using recent results of Boyd, Diaconis and Xiao (2003), we show that minimizing this quantity to design the fastest averaging algorithm on the network is a semi-definite program (SDP). In general, SDPs cannot be solved distributedly; however, exploiting problem structure, we propose a subgradient method that distributedly solves the optimization problem over the network. The relation of averaging time to the second largest eigenvalue naturally relates it to the mixing time of a random walk with transition probabilities that are derived from the gossip algorithm. We use this connection to study the performance of gossip algorithm on two popular networks: wireless sensor networks, which are modeled as geometric random graphs, and the Internet graph under the so-called preferential connectivity model. Motivated by applications to wireless sensor, peer-to-peer, and ad hoc networks, we study distributed broadcasting algorithms for exchanging information and computing in an arbitrarily connected network of nodes. Specifically, we study a broadcasting-based gossiping algorithm to compute the (possibly weighted) average of the initial measurements of the nodes at every node in the network. We show that the broadcast gossip algorithm converges almost surely to a consensus. We prove that the random consensus value is, in expectation, the average of initial node measurements and that it can be made arbitrarily close to this value in mean squared error sense, under a balanced connectivity model and by trading off convergence speed with accuracy of the computation. We provide theoretical and numerical results on the mean square error performance, on the convergence rate and study the effect of the ldquomixing parameterrdquo on the convergence rate of the broadcast gossip algorithm. The results indicate that the mean squared error strictly decreases through iterations until the consensus is achieved. Finally, we assess and compare the communication cost of the broadcast gossip algorithm to achieve a given distance to consensus through theoretical and numerical results.