Broadcast Gossip Algorithms for Consensus on Strongly Connected Digraphs

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.Comment: 30 pages, submitte

A Chemistry-Inspired Framework for Achieving Consensus in Wireless Sensor Networks

The aim of this paper is to show how simple interaction mechanisms, inspired by chemical systems, can provide the basic tools to design and analyze a mathematical model for achieving consensus in wireless sensor networks, characterized by balanced directed graphs. The convergence and stability of the model are first proven by using new mathematical tools, which are borrowed directly from chemical theory, and then validated by means of simulation results, for different network topologies and number of sensors. The underlying chemical theory is also used to derive simple interaction rules that may account for practical issues, such as the estimation of the number of neighbors and the robustness against perturbations. Finally, the proposed chemical solution is validated under real-world conditions by means of a four-node hardware implementation where the exchange of information among nodes takes place in a distributed manner (with no need for any admission control and synchronism procedure), simply relying on the transmission of a pulse whose rate is proportional to the state of each sensor.Comment: 12 pages, 10 figures, submitted to IEEE Sensors Journa

Push sum with transmission failures

The push-sum algorithm allows distributed computing of the average on a directed graph, and is particularly relevant when one is restricted to one-way and/or asynchronous communications. We investigate its behavior in the presence of unreliable communication channels where messages can be lost. We show that exponential convergence still holds and deduce fundamental properties that implicitly describe the distribution of the final value obtained. We analyze the error of the final common value we get for the essential case of two nodes, both theoretically and numerically. We provide performance comparison with a standard consensus algorithm

Cross-layer MAC Protocol for Unbiased Average Consensus under Random Interference

Wireless Sensor Networks have been revealed as a powerful technology to solve many different problems through sensor nodes cooperation. One important cooperative process is the so-called average gossip algorithm, which constitutes a building block to perform many inference tasks in an efficient and distributed manner. From the theoretical designs proposed in most previous work, this algorithm requires instantaneous symmetric links in order to reach average consensus. However, in a realistic scenario wireless communications are subject to interferences and other environmental factors, which results in random instantaneous topologies that are, in general, asymmetric. Consequently, the estimation of the average obtained by the gossip algorithm is a random variable, which its realizations may significantly differ from the average value. In the present work, we first derive a sufficient conditions for any MAC protocol to guarantee that the expected value of the obtained consensus random variable is the average of the initial values (unbiased estimator), while the variance of the estimator is minimum. Then, we propose a cross-layer and distributed link scheduling protocol based on carrier sense, which besides avoiding collisions, ensures both an unbiased estimation and close to minimum variance values. Extensive numerical results are presented to show the validity and efficiency of the proposed approach.Cross-layer MAC Protocol for Unbiased Average Consensus under Random InterferencesubmittedVersionNivå