Section: New Results

Networked systems and graph analysis

It has been recently shown in [45] that by collecting noise-contaminated time series generated by a coupled-oscillator system at each node of a network, it is possible to robustly reconstruct its topology, i.e. determine the graph Laplacian. Restricting ourselves to linear consensus dynamics over undirected communication networks. In [18] , we have introduced a new dynamic average consensus least-squares algorithm to locally estimate these time series at each node, thus making the reconstruction process fully distributed and more easily applicable in the real world. We have also proposed a novel efficient method for separating the off-diagonal entries of the reconstructed Laplacian, and examined several concepts related to the trace of the dynamic correlation matrix of the coupled single integrators, which is a distinctive element of our network reconstruction method.

As recently shown, Laplacian eigenvalues can be estimated by solving the factorization of the average consensus Matrix [46] . The problem was viewed as a constrained consensus optimization one. The main assumption was about the knowledge of the final consensus value. Indeed, estimation of the Laplacian eigenvalues can be carried out using measurements of the transient of the consensus protocol and the steady state (consensus value). In [34] , we relaxed the assumptions by considering that the consensus value is only approximately known. We formulated a convex optimization, which allowed us to make use of recent well-known techniques and results dealing with convex optimization problem proposed in the literature (the Alternating Direction of Multipliers Method, ADMM), [40] , [42] . Recently, we assumed that the consensus value is completely unknown and has to be found simultaneously with Laplacian eigenvalues. In such a case the problem becomes a convex combination problem where the cost function comprises two terms, one that is average consensus problem, and the rest is the consensus problem to estimate the Laplacian eigenvalues. The simulations indicate that the proposed algorithm is efficient enough to provide the nonzero distinct Laplacian eigenvalues with high accuracy. These eigenvalues are then used to assess the robustness of the graph by means of some spectral metrics, the number of spanning trees and the Kirchoff index precisely.

In [16] , we have studied of observability in consensus networks modeled with strongly regular graphs or distance regular graphs. The first result consists in a Kalman-like simple algebraic criterion for observability in distance regular graphs. This criterion consists in evaluating the rank of a matrix built with the components of the Bose-Mesner algebra associated with the considered graph. Then, we have defined some bipartite graphs that capture the observability properties of the graph to be studied. In particular, we showed that necessary and sufficient observability conditions are given by the nullity of the so-called local bipartite observability graph (resp. local unfolded bipartite observability graph) for strongly regular graphs (resp. distance regular graphs). When the nullity cannot be derived directly from the structure of these bipartite graphs, the rank of the associated bi-adjacency matrix allows evaluating observability. Eventually, as a by-product of the main results we have shown that non-observability can be stated just by comparing the valency of the graph to be studied with a bound computed from the number of vertices of the graph and its diameter. Similarly non-observability can also be stated by evaluating the size of the maximum matching in the above mentioned bipartite graphs. Non-observability is strongly linked to privacy preserving feature of a given network. Indeed, when a node is neighborhood non-observable, it means that the data of the other nodes (excluding those of its neighborhood) cannot be retrieve from such a node. Therefore security efforts in order to preserve privacy of the entire network must be focused on nodes that are neighborhood-observable.

We have studied average consensus in wireless sensor networks with aim of providing a way to reach consensus in a finite number of steps [17] . In particular, we investigate the design of consensus protocols when, for security reasons for instance, the underlying graph is constrained to be strongly regular or distance regular. The proposed design method is based on parameters of the intersection array characterizing the underlying graph. With this protocol, at execution time, average consensus is achieved in a number of steps equal to the diameter of the graph, i.e. the smallest possible number of steps to achieve consensus. We have extended the parametric consensus protocol recently introduced by F. Morbidi, to more realistic agents modeled as double integrators and interacting over an undirected communication network. The stability properties of the new protocol in terms of the real parameter “s” are studied for some relevant graph topologies, and the connection with the notion of bipartite consensus is highlighted. The theory is illustrated with the help of two worked examples, dealing with the coordination of a team of quadrotor UAVs and with cooperative temperature measurement in an indoor environment [32] .