A mean-field game model for homogeneous flocking

Abstract

Empirically derived continuum models of collective behavior among large populations of dynamic agents are a subject of intense study in several fields, including biology, engineering, and finance. We formulate and study a mean-field game model whose behavior mimics an empirically derived nonlocal homogeneous flocking model for agents with gradient self-propulsion dynamics. The mean-field game framework provides a non-cooperative optimal control description of the behavior of a population of agents in a distributed setting. In this description, each agent's state is driven by optimally controlled dynamics that result in a Nash equilibrium between itself and the population. The optimal control is computed by minimizing a cost that depends only on its own state and a mean-field term. The agent distribution in phase space evolves under the optimal feedback control policy. We exploit the low-rank perturbative nature of the nonlocal term in the forward-backward system of equations governing the state and control distributions and provide a closed-loop linear stability analysis demonstrating that our model exhibits bifurcations similar to those found in the empirical model. The present work is a step towards developing a set of tools for systematic analysis, and eventually design, of collective behavior of non-cooperative dynamic agents via an inverse modeling approach.

Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences

“…In chemistry, coupling function methods have been used for understanding, effecting, and predicting interactions between oscillatory electrochemical reactions [28,32,37,56,57]. In biology the new methods have been applied to characterize genetic oscillators and homogeneous flocking [51,52], and in cardiorespiratory physiology they have been used for reconstruction of the human cardiorespiratory coupling function and phase resetting curve [35,36,59,62]. In social sciences, the function underlying the interactions between different social and economic dynamical dependences has been determined [50,51,60].…”

“…In biology the new methods have been applied to characterize genetic oscillators and homogeneous flocking [51,52], and in cardiorespiratory physiology they have been used for reconstruction of the human cardiorespiratory coupling function and phase resetting curve [35,36,59,62]. In social sciences, the function underlying the interactions between different social and economic dynamical dependences has been determined [50,51,60]. The mechanical coupling functions between coupled metronomes were reconstructed in a similar way [47].…”

“…Applications have now been reported in, for example: chemistry, neuroscience, cardiorespiratory physiology, biology, social sciences, mechanics, ferromagnetism, secure encryption and ecology [28,32,[35][36][37]46,47,[50][51][52][53][54][55][56][57][58][59][60][61]. In chemistry, coupling function methods have been used for understanding, effecting, and predicting interactions between oscillatory electrochemical reactions [28,32,37,56,57].…”

Self-Organised Collision-Free Flocking Mechanism in Heterogeneous Robot Swarms

“…To achieve the collective behaviours and to develop collective mechanisms, a large and growing body of literature has investigated modelling and designing the swarm systems. Considerable works have been undertaken from various angles such as distribution [14], communication [15], sensing [16] and coordination [17]. As a significant aspect of swarm systems, cooperation strategy particularly plays an important role in successfully achieving flocking behaviour.…”

“…Hence, different strategies have been developed based on various disciplines to present collective motion as a connected swarm behaviour, for example, in the study conducted by He et al [18], disk graph and Delaunay graph methods were used to present connectivity with various distances. In [14], mean-field game model was presented by partial differential equations to describe the swarm dynamics based on state and distributions. The performed study by Thrun et al [19] proposed a cluster analysis that used the projection method based on the topographic map.…”