## Section: New Results

### Control of multi-agent systems and opinion dynamics

#### Open multi-agent systems: Dynamic consensus

Participants : W. S. Rossi, P. Frasca [Contact person] .

In [53] we investigate a dynamic consensus problem for an open multi-agent system. Open multi-agent systems are characterized by a time-varying set of agents connected by a network: agents may leave and new agents may join the network at any time, thus the term “open”. The dynamic consensus problem consists in achieving agreement about the time-varying average of a set of reference signals that are assumed to be the agents’ inputs. Dynamic consensus has recently found application in the context of distributed estimation for electric demand-side management, where a large population of connected domestic appliances needs to estimate its future average power consumption. Since the considered network of devices changes as new appliances log in and out, there is a need to develop and characterize dynamic consensus algorithms for these open scenarios. In this paper we give several initial contributions both to a general theory of open multi-agent systems and to the specific problem of dynamic consensus within this context. On the theoretical side, we propose a formal definition of open multi-agent system, a suitable notion of stability, and some sufficient conditions to establish it. On the applied side, we design a novel dynamic consensus algorithm, the Open Proportional Dynamic Consensus algorithm. We characterize some of its convergence properties in the proposed open-multi-agent systems framework and we illustrate its evolution by numerical simulations.

#### Robust average consensus over unreliable networks

Participants : F. Acciani, P. Frasca [Contact person] , G. Heijenk, A. Stoorvogel.

Packet loss is a serious issue in wireless consensus networks, as even few failures might prevent a network to converge to the desired consensus value. In the last four years, we have devised some possible ways to compensate for the errors caused by packet collisions, by modifying the updating weights. Since these modifications may result in a reduced convergence speed, a gain parameter is used to increase the convergence speed, and an analysis of the stability of the network is performed, leading to a criterion to choose such gain to guarantee network stability. For the implementation of the compensation method, we propose a new communication algorithm, which uses both synchronous and asynchronous mechanisms to achieve average consensus and to deal with uncertainty in packet delivery. The paper [14] provides a complete account of our results.

#### Asynchronous opinion dynamics on the $k$-nearest-neighbors graph

Participants : W. S. Rossi, P. Frasca [Contact person] .

This work is about a new model of opinion dynamics with opinion-dependent connectivity. We assume that agents update their opinions asynchronously and that each agent’s new opinion depends on the opinions of the k agents that are closest to it. In the paper [63], we show that the resulting dynamics is substantially different from comparable models in the literature, such as bounded-confidence models. We study the equilibria of the dynamics, observing that they are robust to perturbations caused by the introduction of new agents. We also prove that if the number of agents n is smaller than 2k, the dynamics converge to consensus. This condition is only sufficient.

#### Quantization effects in opinion dynamics

Participants : F. Ceragioli, P. Frasca [Contact person] .

This work deals with continuous-time opinion dynamics that feature the interplay of continuous opinions and discrete behaviors. In our model, the opinion of one individual is only influenced by the behaviors of fellow individuals. The key technical difficulty in the study of these dynamics is that the right-hand sides of the equations are discontinuous and thus their solutions must be intended in some generalized sense: in our analysis, we consider both Carathéodory and Krasovskii solutions. We first prove the existence and completeness of Carathéodory solutions from every initial condition and we highlight a pathological behavior of Carathéodory solutions, which can converge to points that are not (Carathéodory) equilibria. Notably, such points can be arbitrarily far from consensus and indeed simulations show that convergence to nonconsensus configurations is common. In order to cope with these pathological attractors, we study Krasovskii solutions. We give an estimate of the asymptotic distance of all Krasovskii solutions from consensus and we prove its tightness by an example of equilibrium such that this distance is quadratic in the number of agents. This fact implies that quantization can drastically destroy consensus. However, consensus is guaranteed in some special cases, for instance, when the communication among the individuals is described by either a complete or a complete bipartite graph. These results are reported in details in [19], whereas the book chapter [66] puts them in the broader context of consensus-seeking dynamics with discontinuous right-hand side.

#### Message-passing computation of harmonic influence in social networks

Participants : W. S. Rossi, P. Frasca [Contact person] .

In the study of networks, identifying the most important nodes is of capital importance. The concept of Harmonic Influence has been recently proposed as a metric for the importance of nodes in a social network. This metric evaluates the ability for one node to sway the opinions of the other nodes in the network, under the assumption of a linear diffusion of opinions in the network. A distributed message passing algorithm for its computation has been proposed by Vassio et al., 2014, but its convergence guarantees were limited to trees and regular graphs. In [36], we prove that the algorithm converges on general graphs. In [64], we offer two additional contributions to its study. We evaluate how the presence of communities in the network impacts the algorithm performance, and how the algorithm performs on networks which change topology during the execution of the algorithm.

#### Distributed control and game theory: self-optimizing systems

Participants : F. Garin [Contact person] , B. Gaujal [POLARIS] , S. Durand.

The design of distributed algorithms for a networked control system composed of multiple interacting agents, in order to drive the global system towards a desired optimal functioning, can benefit from tools and algorithms from game theory. This is the motivation of the Ph.D. thesis of Stéphane Durand, a collaboration between POLARIS and NECS teams.

The focus of this thesis is on the complexity of the best response algorithm to find a Nash equilibrium for potential games. Best response is a simple greedy algorithm, known to converge to a Nash equilibrium if players play one after the other in a round-robin way, but with a worst-case complexity which is exponential in the number of players. We consider instead its average complexity over the ensemble of random potential games, showing that such average complexity is surprisingly low, only linear in the number of players. Then we focus on removing the need of a centralised scheduler enforcing the round robin order of play. In [52], [21] we consider agents activated according to independent local Poisson clocks, and we show that (despite the possible overlaps of the computations of some players), we can still obtain convergence, with an average complexity of order $nlogn/loglogn$, where $n$ is the number of players. In [51] we show how to take advantage of the structure of the interactions between players in a network game: noninteracting players can play simultaneously. This improves best response algorithm, both in the centralized and in the distributed case.

#### Control of switched interconnected large-scale systems

Participants : H. Fourati [Contact person] , D. Belkhiat, D. Jabri.

We proposed in [27] a new design of a decentralized output-feedback tracking control for a class of switched large-scale systems with external bounded disturbances. The controller proposed herein is synthesized to satisfy the robust ${H}_{\infty}$ tracking performance with local disturbance attenuation levels. Based on multiple switched Lyapunov functions, sufficient conditions proving the existence of the proposed controller are formulated in terms of Linear Matrix Inequalities (LMI).