The asymptotical error of broadcast gossip averaging algorithms

In problems of estimation and control which involve a network, efficient distributed computation of averages is a key issue. This paper presents theoretical and simulation results about the accumulation of errors during the computation of averages by means of iterative "broadcast gossip" algorithms. Using martingale theory, we prove that the expectation of the accumulated error can be bounded from above by a quantity which only depends on the mixing parameter of the algorithm and on few properties of the network: its size, its maximum degree and its spectral gap. Both analytical results and computer simulations show that in several network topologies of applicative interest the accumulated error goes to zero as the size of the network grows large.Comment: 10 pages, 3 figures. Based on a draft submitted to IFACWC201

On the mean square error of randomized averaging algorithms

This paper regards randomized discrete-time consensus systems that preserve the average "on average". As a main result, we provide an upper bound on the mean square deviation of the consensus value from the initial average. Then, we apply our result to systems where few or weakly correlated interactions take place: these assumptions cover several algorithms proposed in the literature. For such systems we show that, when the network size grows, the deviation tends to zero, and the speed of this decay is not slower than the inverse of the size. Our results are based on a new approach, which is unrelated to the convergence properties of the system.Comment: 11 pages. to appear as a journal publicatio

Effects of Network Communities and Topology Changes in Message-Passing Computation of Harmonic Influence in Social Networks

The harmonic influence is a measure of the importance of nodes in social networks, which can be approximately computed by a distributed message-passing algorithm. In this extended abstract we look at two open questions about this algorithm. How does it perform on real social networks, which have complex topologies structured in communities? How does it perform when the network topology changes while the algorithm is running? We answer these two questions by numerical experiments on a Facebook ego network and on synthetic networks, respectively. We find out that communities can introduce artefacts in the final approximation and cause the algorithm to overestimate the importance of "local leaders" within communities. We also observe that the algorithm is able to adapt smoothly to changes in the topology.Comment: 4 pages, 7 figures, submitted as conference extended abstrac

Asynchronous opinion dynamics on the kk-nearest-neighbors graph

This paper is about a new model of opinion dynamics with opinion-dependent connectivity. We assume that agents update their opinions asynchronously and that each agent's new opinion depends on the opinions of the $k$ agents that are closest to it. We show that the resulting dynamics is substantially different from comparable models in the literature, such as bounded-confidence models. We study the equilibria of the dynamics, observing that they are robust to perturbations caused by the introduction of new agents. We also prove that if the number of agents $n$ is smaller than $2k$, the dynamics converge to consensus. This condition is only sufficient.Comment: 17 pages, 4 figures, (to be) presented at the 57th IEEE Conference on Decision and Control, 201

Average resistance of toroidal graphs

The average effective resistance of a graph is a relevant performance index in many applications, including distributed estimation and control of network systems. In this paper, we study how the average resistance depends on the graph topology and specifically on the dimension of the graph. We concentrate on $d$-dimensional toroidal grids and we exploit the connection between resistance and Laplacian eigenvalues. Our analysis provides tight estimates of the average resistance, which are key to study its asymptotic behavior when the number of nodes grows to infinity. In dimension two, the average resistance diverges: in this case, we are able to capture its rate of growth when the sides of the grid grow at different rates. In higher dimensions, the average resistance is bounded uniformly in the number of nodes: in this case, we conjecture that its value is of order $1/d$ for large $d$. We prove this fact for hypercubes and when the side lengths go to infinity.Comment: 24 pages, 6 figures, to appear in SIAM Journal on Control and Optimization (SICON

Discontinuities and hysteresis in quantized average consensus

We consider continuous-time average consensus dynamics in which the agents' states are communicated through uniform quantizers. Solutions to the resulting system are defined in the Krasowskii sense and are proven to converge to conditions of "practical consensus". To cope with undesired chattering phenomena we introduce a hysteretic quantizer, and we study the convergence properties of the resulting dynamics by a hybrid system approach.Comment: 26 pages, 7 figures. Accepted for publication in Automatica. v4 is minor revision of v

Limited benefit of cooperation in distributed relative localization

Important applications in robotic and sensor networks require distributed algorithms to solve the so-called relative localization problem: a node-indexed vector has to be reconstructed from measurements of differences between neighbor nodes. In a recent note, we have studied the estimation error of a popular gradient descent algorithm showing that the mean square error has a minimum at a finite time, after which the performance worsens. This paper proposes a suitable modification of this algorithm incorporating more realistic "a priori" information on the position. The new algorithm presents a performance monotonically decreasing to the optimal one. Furthermore, we show that the optimal performance is approximated, up to a 1 + \eps factor, within a time which is independent of the graph and of the number of nodes. This convergence time is very much related to the minimum exhibited by the previous algorithm and both lead to the following conclusion: in the presence of noisy data, cooperation is only useful till a certain limit.Comment: 11 pages, 2 figures, submitted to conferenc