Wilbert Samuel Rossi

Transient and limit performance of distributed relative localization

In this paper, we consider the problem of relative localization in a network of sensors, according to the formulation of Barooah and Hespanha. We introduce a distributed algorithm for its solution, and we study the algorithm performance by evaluating a suitable performance metric as a function of the network eigenvalues. Remarkably, the performance analysis indicates that it is preferable to stop the algorithm before convergence is reached: an estimate of the optimal stopping time is provided.

Average Resistance of Toroidal Graphs

Abstract. 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.

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 + ε factor, within a time which is independent of the graph and of the number of nodes. This bounded convergence time is closely related to the minimum exhibited by the previous algorithm and both facts lead to the following conclusion: in the presence of noisy data, cooperation is only useful till a certain limit.

Distributed Estimation From Relative and Absolute Measurements

This note defines the problem of least squares distributed estimation from relative and absolute measurements, by encoding the set of measurements in a weighted undirected graph. The role of its topology is studied by an electrical interpretation, which easily allows distinguishing between topologies that lead to “small” or “large” estimation errors. The least squares problem is solved by a distributed gradient algorithm: the computed solution is approximately optimal after a number of steps that does not depend on the size of the problem or on the graph-theoretic properties of its encoding. This fact indicates that only a limited cooperation between the sensors is necessary.

An index for the “local” influence in social networks

In this paper we define a novel index of node centrality in social networks that extends the recently proposed Harmonic Influence Centrality (HIC) and that we call Local-Harmonic Influence centrality (L-HIC). Indeed, when compared with the HIC, our index shows a local nature that rules out one pathological behavior of the HIC. Similarly to the HIC, the L-HIC can be approximated by a distributed message passing algorithm that is inspired by an analogy between electrical and social networks on tree graphs. We prove a result that guarantees convergence on graphs containing at most one cycle.

On the convergence of message passing computation of harmonic influence in social networks

The harmonic influence is a measure of node influence in social networks that quantifies the ability of a leader node to alter the average opinion of the network, acting against an adversary field node. The definition of harmonic influence assumes linear interactions between the nodes described by an undirected weighted graph; its computation is equivalent to solve a discrete Dirichlet problem associated to a grounded Laplacian for every node. This measure has been recently studied, under slightly more restrictive assumptions, by Vassio et al., IEEE Trans. Control Netw. Syst., 2014, who proposed a distributed message passing algorithm that concurrently computes the harmonic influence of all nodes. In this paper, we provide a convergence analysis for this algorithm, which largely extends upon previous results: we prove that the algorithm converges asymptotically, under the only assumption of the interaction Laplacian being symmetric. However, the convergence value does not in general coincide with the harmonic influence: by simulations, we show that when the network has a larger number of cycles, the algorithm becomes slower and less accurate, but nevertheless provides a useful approximation. Simulations also indicate that the symmetry condition is not necessary for convergence and that performance scales very well in the number of nodes of the graph.

Threshold Models of Cascades in Large-Scale Networks

The spread of new beliefs, behaviors, conventions, norms, and technologies in social and economic networks are often driven by cascading mechanisms, and so are contagion dynamics in financial networks. Global behaviors generally emerge from the interplay between the structure of the interconnection topology and the local agents’ interactions. We focus on the Threshold Model (TM) of cascades first introduced by Granovetter (1978). This can be interpreted as the best response dynamics in a network game whereby agents choose strategically between two actions. Each agent is equipped with an individual threshold representing the number of her neighbors who must have adopted a certain action for that to become the agents best response. We analyze the TM dynamics on large-scale networks with heterogeneous agents. Through a local mean-field approach, we obtain a nonlinear, one-dimensional, recursive equation that approximates the evolution of the TM dynamics on most of the networks of a given size and distribution of degrees and thresholds. We prove that, on all but a fraction of networks with given degree and threshold statistics that is vanishing as the network size grows large, the actual fraction of adopters of a given action is arbitrarily close to the output of the aforementioned recursion. Numerical simulations on some real network testbeds show good adherence to the theoretical predictions.