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.

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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 graphtheoretic properties of its encoding. This fact indicates that only a limited cooperation between the sensors is necessary.

This paper introduces a general model of opinion dynamics with opinion-dependent connectivity. Agents update their opinions asynchronously: for the updating agent, the new opinion is the average of the k closest opinions within a subset of m agents that are sampled from the population of size n. Depending on k and m with respect to n, the dynamics can have a variety of equilibria, which include consensus and clustered configurations. The model covers as special cases a classical gossip update (if m = n) and a deterministic update defined by the k nearest neighbors (if m = k). We prove that the dynamics converges to consensus if n > 2(m − k). Before convergence, however, the dynamics can remain for long time in the vicinity of metastable clustered configurations.