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- Mean-field Games for Bio-inspired Collective Decision-making in Dynamical Networks(arXiv)
Author : Leonardo Stella, Dario Bauso
Abstract : Given a large number of homogeneous players that are distributed across three possible states, we consider the problem in which these players have to control their transition rates, while minimizing a cost. The optimal transition rates are based on the players’ knowledge of their current state and of the distribution of all the other players, and this introduces mean-field terms in the running and the terminal cost. The first contribution involves a mean-field game model that brings together macroscopic and microscopic dynamics. We obtain the mean-field equilibrium associated with this model, by solving the corresponding initial-terminal value problem. We perform an asymptotic analysis to obtain a stationary equilibrium for the system. The second contribution involves the study of the microscopic dynamics of the system for a finite number of players that interact in a structured environment modeled by an interaction topology. The third contribution is the specialization of the model to describe honeybee swarms, virus propagation, and cascading failures in interconnected smart-grids. A numerical analysis is conducted which involves two types of cyber-attacks. We simulate in which ways failures propagate across the interconnected smart grids and the impact on the grids frequencies. We reframe our analysis within the context of Lyapunov’s linearisation method and stability theory of nonlinear systems and Kuramoto coupled oscillators model.
2. Multiequilibria analysis for a class of collective decision-making networked systems(arXiv)
Author : Angela Fontan, Claudio Altafini
Abstract : The models of collective decision-making considered in this paper are nonlinear interconnected cooperative systems with saturating interactions. These systems encode the possible outcomes of a decision process into different steady states of the dynamics. In particular, they are characterized by two main attractors in the positive and negative orthant, representing two choices of agreement among the agents, associated to the Perron-Frobenius eigenvector of the system. In this paper we give conditions for the appearance of other equilibria of mixed sign. The conditions are inspired by Perron-Frobenius theory and are related to the algebraic connectivity of the network. We also show how all these equilibria must be contained in a solid disk of radius given by the norm of the equilibrium point which is located in the positive orthant