We study a general framework for broadcast gossip algorithms which use companion variables to solve the average consensus problem. Each node maintains an initial state and a companion variable. Iterative updates are performed asynchronously whereby one random node broadcasts its current state and companion variable and all other nodes receiving the broadcast update their state and companion variable. We provide conditions under which this scheme is guaranteed to converge to a consensus solution, where all nodes have the same limiting values, on any strongly connected directed graph. Under stronger conditions, which are reasonable when the underlying communication graph is undirected, we guarantee that the consensus value is equal to the average, both in expectation and in the mean-squared sense. Our analysis uses tools from non-negative matrix theory and perturbation theory. The perturbation results rely on a parameter being sufficiently small. We characterize the allowable upper bound as well as the optimal setting for the perturbation parameter as a function of the network topology, and this allows us to characterize the worst-case rate of convergence. Simulations illustrate that, in comparison to existing broadcast gossip algorithms, the approaches proposed in this paper have the advantage that they simultaneously can be guaranteed to converge to the average consensus and they converge in a small number of broadcasts.
Recent work of Cai and Ishii @cite_11 @cite_19 analyzes related distributed averaging algorithms on directed graphs that use companion variables. The two types of algorithms analyzed in @cite_11 @cite_19 involve asynchronous pairwise updates and synchronous updates. They make use of tools from matrix perturbation theory, and the work in the present article can be seen as generalizing the results in @cite_11 @cite_19 for broadcast gossip updates.
@cite_19 @cite_11
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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. This paper studies joint beamforming and power control in a coordinated multicell downlink system that serves multiple users per cell to maximize the minimum weighted signal-to-interference-plus-noise ratio. The optimal solution and distributed algorithm with geometrically fast convergence rate are derived by employing the nonlinear Perron-Frobenius theory and the multicell network duality. The iterative algorithm, though operating in a distributed manner, still requires instantaneous power update within the coordinated cluster through the backhaul. The backhaul information exchange and message passing may become prohibitive with increasing number of transmit antennas and increasing number of users. In order to derive asymptotically optimal solution, random matrix theory is leveraged to design a distributed algorithm that only requires statistical information. The advantage of our approach is that there is no instantaneous power update through backhaul. Moreover, by using nonlinear Perron-Frobenius theory and random matrix theory, an effective primal network and an effective dual network are proposed to characterize and interpret the asymptotic solution. Distributed averaging describes a class of network algorithms for the decentralized computation of aggregate statistics. Initially, each node has a scalar data value, and the goal is to compute the average of these values at every node (the so-called average consensus problem). Nodes iteratively exchange information with their neighbors and perform local updates until the value at every node converges to the initial network average. Much previous work has focused on algorithms where each node maintains and updates a single value; every time an update is performed, the previous value is forgotten. Convergence to the average consensus is achieved asymptotically. The convergence rate is fundamentally limited by network connectivity, and it can be prohibitively slow on topologies such as grids and random geometric graphs, even if the update rules are optimized. In this paper, we provide the first theoretical demonstration that adding a local prediction component to the update rule can significantly improve the convergence rate of distributed averaging algorithms. We focus on the case where the local predictor is a linear combination of the node's current and previous values (i.e., two memory taps), and our update rule computes a combination of the predictor and the usual weighted linear combination of values received from neighboring nodes. We derive the optimal mixing parameter for combining the predictor with the neighbors' values, and conduct a theoretical analysis of the improvement in convergence rate that can be achieved using this acceleration methodology. For a chain topology on N nodes, this leads to a factor of N improvement over standard consensus, and for a two-dimensional grid, our approach achieves a factor of ?N improvement. 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.