## Section: New Results

### Multi-agent systems and network games

#### 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 first results of this thesis concerned the complexity of the best response algorithm under round-robin revision sequence, a classical centralized iterative algorithm to find a Nash Equilibrium. In a more recent work, submitted for publication, and described in the report [40], we focus on distributed versions of the same algorithm. We compute the average complexity over all potential games of best response dynamics under a random i.i.d. revision sequence, since it can be implemented in a distributed way using Poisson clocks. We obtain a distributed algorithm whose execution time is within a constant factor of the optimal centralized one. We then 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.

#### Using a linear gain to accelerate 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 some recent work, we have devised a possible way to compensate for the errors caused by packet collisions, by modifying the updating weights. Such a modification compensates for the loss of information in an unreliable network, but results in a reduced convergence speed. In [30], we propose a faster method - based on a suitable gain in the consensus dynamics - to solve the unreliable average consensus problem. We find a sufficient condition for the gain to preserve stability of the network. Simulations are used to discuss the choice of the gain, and to compare our method with the literature.

#### Mean-field analysis of the convergence time of 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, and proved to converge on general graphs by Rossi and Frasca, 2016. In [36], we presented an want to evaluate the convergence time of this algorithm by using a mean-field approach. The mean-field dynamics is first introduced in a “homogeneous” setting, where it is exact, then heuristically extended to a non-homogeneous setting. The rigorous analysis of the mean-field dynamics is complemented by numerical examples and simulations that demonstrate the validity of the approach.

#### Modeling birds on wires

Participants : A. Aydogdu, P. Frasca [Contact person] , C. d'Apice, R. Manzo, J. M. Thornton, B. Gachomo, T. Wilson, B. Cheung, U. Tariq, W.m. Saidel, B. Piccoli.

The paper [13] introduces a mathematical model to study the group dynamics of birds resting on wires. The model is agent-based and postulates attraction-repulsion forces between the interacting birds: the interactions are “topological”, in the sense that they involve a given number of neighbors irrespective of their distance. The main properties of the model are investigated by combining rigorous mathematical analysis and simulations. This analysis gives indications about the total length of a group and the inter-animal spacings within it: in particular, the model predicts birds to be more widely spaced near the borders of each group. We compare these insights from the model with new experimental data, derived from the analysis of pictures of pigeons and starlings taken by the team in New Jersey. We have used two different image elaboration protocols to derive the data for the statistical analysis, which allowed us to establish a good agreement with the model and to quantify its main parameters. Our data also seem to indicate potential handedness of the birds: we investigated this issue by analyzing the group organization features and the group dynamics at the arrival of new birds. However, data are still insufficient to draw a definite conclusion on this matter. Finally, arrivals and departures of birds from the group are included in a refined version of the model, by means of suitable stochastic processes

#### Network Games: Condensation of the Graph as a Hierarchical interpretation of the Game

Participants : G. Casadei, C. Canudas de Wit [Contact person] .

Control and optimization over large population networks have become a popular topic within the control community. The main reason is that modern applications re- quire multiple systems to communicate and interact with each other to fulfill the desired task. For instance power networks, sensor networks and social networks are solid examples in which is fundamental to control different parts of the network to achieve a global desired behavior. In the recent years, the control community has largely focused on cooperative approaches to networks. In this framework the agents in the network are willing to collaborate and find an agreement between each other in such a way that they coordinate their motion.

However, not in all the frameworks and not in all the situations, it is possible to consider a cooperative approach. In several scenarios, the nodes are selfish and in competition with the others to pursue their goal. This leads to a non-cooperative interaction between the agents and thus to games played over networks. When the number of nodes in the network is large, it becomes analytically impossible to use conventional game theoretic tools to find a solution to the problem. This motivated researchers to define a new type of games, named aggregative, where the response of an agent depends, rather than on each other players decision, on the aggregation of all the other agents action.

We considered a refined typology of networks games in which the aggregate information is depending on a directed communication graph and showed that under a certain number of conditions the players reach a Nash Equilibrium. Then we study the influence of this graph topology on the structure of the game and show that the condensation of the graph leads to a hierarchical interpretation of the game and thus to a quasi-sequential architecture of optimization. Then, we introduce the concept of physical graph and control graph in flow networks, and show that the condensation of the control graph helps in determining the equilibrium the agents will reach.