## Section: New Results

### Networked systems and graph analysis

#### Distributed graph-discovery

Participants : A. Kibangou [Contact person] , T.-M. D. Tran, F. Garin, A. de Almeida [UFC Brazil] .

The availability of information on the communication topology of a wireless sensor network is essential for the design of the estimation algorithms. In the context of distributed self-organized sensor networks, there is no central unit with the knowledge of the network, and the agents must run some distributed network discovery algorithm.

We have studied the problem of estimating the eigenvalues of the Laplacian matrix associated with a graph modeling the interconnections between the nodes of a given network. Our approach is based on properties resulting from the factorization of the average consensus matrix. Indeed, as recently shown [45] , the average consensus matrix can be written as a product of Laplacian based consensus matrices whose stepsizes are given by the inverse of the nonzero Laplacian eigenvalues. By distributively solving the factorization of the average consensus matrix, we have shown that the Laplacian eigenvalues can be computed as the inverse of the stepsizes in each estimated factor, where these factors are constrained to be structured as Laplacian based consensus matrices. A constrained optimization problem was formulated and distributed gradient descent methods have been formulated. As formulated, the problem can be viewed as a consensus problem with equality constraints. In contrast to the state-of-the-art, the proposed algorithm does not require great resources in both computation and storage. This algorithm can also be viewed as a way for decentralizing the design of finite-time average consensus protocol recently proposed in the literature.

Laplacian eigenvalues have several interesting properties that can help to study networks, however they cannot uniquely characterize the topology of the network. Therefore, we have directly studied the problem of topology identification in [20] . The considered set-up concerns a collaborative wireless sensor network where nodes locally exchange coded informative data before transmitting the combined data towards a remote fusion center equipped with an antenna array. For this communication scenario, a new blind estimation algorithm was developed for jointly recovering network transmitted data and connection topology at the fusion center. The proposed algorithm is based on a two-stage approach. The first stage is concerned with the estimation of the channel gains linking the nodes to the fusion center antennas. The second stage performs a joint estimation of network data and connection topology matrices by exploiting a constrained (PARALIND) tensor model for the collected data at the fusion center.

Distributed network-discovery algorithms become even more challenging in the case where the algorithm must be anonymous, namely in the case when the agents do not have or do not want to disclose their identifiers (id.s), either for technological reasons (in time-varying self-organized networks, assigning unique identifiers to agents is a challenge) or for privacy concerns. In anonymous networks, even simple tasks such as counting the number of agents are challenging problems. In [24] we have proposed an algorithm for node-counting in anoymous networks. It is based on a graph-constrained LTI system similar to linear consensus, and on system identification: the idea is that the order of the system is the number of agents, and based on local observations each agent tries to identify the order of the system, testing the rank of the Hankel matrix from the output data.

#### Observability in consensus networks

Participants : A. Kibangou [Contact person] , C. Commault [Grenoble INP] .

Studying the observability problem of a system consists in answering the question: is it possible, for a given node, to reconstruct the entire network state just from its own measurements and those of its neighbors? Studying observability for arbitrary graphs is particularly a tough task, therefore, studies are generally restricted to some families of graphs; for instance, recently, observability has been studied for paths and circular graphs and also grids where the study was carried out based on rules on number theory. We have considered families of graphs admitting an association scheme such that strongly regular graphs and distance regular graphs. The regularity properties of these kinds of graphs can particularly be useful for robustifying the network as for cryptographic systems. Based on the so-called Bose-Mesner algebra, we have stated observability conditions on consensus networks modeled with graphs modeled with strongly regular graphs and distance regular graphs; for this purpose, we have introduced the notion of local observability bipartite graph that allows characterizing the observability in consensus networks. We have shown that the observability condition is given by the nullity of the so-called "local bipartite observability graph"; when the nullity of the graph cannot be derived directly from the structure of the local bipartite observability graph, the rank of the associated bi-adjacency matrix allows evaluating the observability. The bi-adjacency matrix of the local bipartite observability graph must be full column rank for guaranteeing observability. From this general necessary and sufficient condition, we have deduced sufficient conditions for strongly regular graphs and distance regular graphs [25] .