## Section: New Results

### Network systems and graph analysis

#### Distributed estimation of graph Laplacian eigenvalues

Participants : A. Kibangou [Contact person] , T.-M. D. Tran.

Linear average-consensus is a well-known iterative protocol allowing agents to converge to the average of initial values by taking suitable convex combinations of the messages received from neighbors. From the recent literature, it is known that, after a finite time, some consecutive measurements of a state of the consensus dynamical system can be used to compute the exact average of the initial condition. In [23] , we have shown that these measurements can also be used for estimating the Laplacian eigenvalues of the graph representing the network. As recently shown in the literature, by solving the factorization of the averaging matrix, the Laplacian eigenvalues can be inferred. In our paper, the problem is posed as a constrained consensus problem. A first formulation (direct approach) yields a non-convex optimization problem, which we solve in a distributed way using Lagrange multipliers. A second formulation (indirect approach) is obtained after a suitable re-parameterization. The problem is then convex and is solved by using the distributed subgradient algorithm and the alternating direction method of multipliers (ADMM). The proposed algorithms allow estimating the actual Laplacian eigenvalues with high accuracy. However, they face numerical instability when considering very large graphs.

#### Distributed solution to the network reconstruction problem

Participants : A. Kibangou [Contact person] , T.-M. D. Tran.

We address the problem of reconstructing the network topology from data propagated through the network by means of a linear average-consensus protocol. In [34] , we propose a new method based on the distributed estimation of graph Laplacian spectral properties. Precisely, the identification of the network topology is implemented by estimating both eigenvalues and eigenvectors of the consensus matrix, which is related to the graph Laplacian matrix. Having already solved in [23] the problem of estimating the eigenvalues (see paragraph above), in this paper we focus on the eigenvectors. We show how the topology can be reconstructed in presence of anonymous nodes, i.e., nodes that do not disclose their ID. Actually, in presence of anonymous nodes, eigenvectors are estimated up to a permutation of rows; the obtained graph is then isomorphic to the original one. Moreoved, under some observability assumption on the consensus dynamical system (if the graph is node-observable or neighborhood-observable from the node of interest) and if all the entries of the initial condition of the network state are distinct, then the node can exactly reconstruct the network topology. If the entries of the initial condition of the network state are independently generated from a continuous probability distribution, then the node can reconstruct the network topology almost surely. The main assumption in this work is: all eigenvalues are distinct, that is the case of most random graphs. Future works encompass the design of the network reconstruction protocol that deals with spectrums in which the multiplicities of the eigenvalues can be higher than 1 and also directed graphs. In addition, numerical issues for large graphs are to be considered for making the proposed method scalable.