20 research outputs found
Mean-field limit of collective dynamics with time-varying weights
In this paper, we derive the mean-field limit of a collective dynamics model
with time-varying weights, for weight dynamics that preserve the total mass of
the system as well as indistinguishability of the agents. The limit equation is
a transport equation with source, where the (non-local) transport term
corresponds to the position dynamics, and the (non-local) source term comes
from the weight redistribution among the agents. We show existence and
uniqueness of the solution for both microscopic and macroscopic models and
introduce a new empirical measure taking into account the weights. We obtain
the convergence of the microscopic model to the macroscopic one by showing
continuity of the macroscopic solution with respect to the initial data, in the
Wasserstein and Bounded Lipschitz topologies
Graph Limit for Interacting Particle Systems on Weighted Random Graphs
In this article, we study the large-population limit of interacting particle
systems posed on weighted random graphs. In that aim, we introduce a general
framework for the construction of weighted random graphs, generalizing the
concept of graphons. We prove that as the number of particles tends to
infinity, the finite-dimensional particle system converges in probability to
the solution of a deterministic graph-limit equation, in which the graphon
prescribing the interaction is given by the first moment of the weighted random
graph law. We also study interacting particle systems posed on switching
weighted random graphs, which are obtained by resetting the weighted random
graph at regular time intervals. We show that these systems converge to the
same graph-limit equation, in which the interaction is prescribed by a
constant-in-time graphon
Mean-field limit of collective dynamics with time-varying weights
In this paper, we derive the mean-field limit of a collective dynamics model with time-varying weights, for weight dynamics that preserve the total mass of the system as well as indistinguishability of the agents. The limit equation is a transport equation with source, where the (non-local) transport term corresponds to the position dynamics, and the (non-local) source term comes from the weight redistribution among the agents. We show existence and uniqueness of the solution for both microscopic and macroscopic models and introduce a new empirical measure taking into account the weights. We obtain the convergence of the microscopic model to the macroscopic one by showing continuity of the macroscopic solution with respect to the initial data, in the Wasserstein and Bounded Lipschitz topologies
Mean-field and graph limits for collective dynamics models with time-varying weights
In this paper, we study a model for opinion dynamics where the influence
weights of agents evolve in time via an equation which is coupled with the
opinions' evolution. We explore the natural question of the large population
limit with two approaches: the now classical mean-field limit and the more
recent graph limit. After establishing the existence and uniqueness of
solutions to the models that we will consider, we provide a rigorous
mathematical justification for taking the graph limit in a general context.
Then, establishing the key notion of indistinguishability, which is a necessary
framework to consider the mean-field limit, we prove the subordination of the
mean-field limit to the graph one in that context. This actually provides an
alternative (but weaker) proof for the mean-field limit. We conclude by showing
some numerical simulations to illustrate our results
Control of reaction-diffusion equations on time-evolving manifolds
Among the main actors of organism development there are morphogens, which are
signaling molecules diffusing in the developing organism and acting on cells to
produce local responses. Growth is thus determined by the distribution of such
signal. Meanwhile, the diffusion of the signal is itself affected by the
changes in shape and size of the organism. In other words, there is a complete
coupling between the diffusion of the signal and the change of the shapes. In
this paper, we introduce a mathematical model to investigate such coupling. The
shape is given by a manifold, that varies in time as the result of a
deformation given by a transport equation. The signal is represented by a
density, diffusing on the manifold via a diffusion equation. We show the
non-commutativity of the transport and diffusion evolution by introducing a new
concept of Lie bracket between the diffusion and the transport operator. We
also provide numerical simulations showing this phenomenon
Social Dynamics Models with Time-Varying Influence
This paper introduces an augmented model for first-order opinion dynamics, in which a weight of influence is attributed to each agent. Each agent's influence on another agent's opinion is then proportional not only to the classical interaction function, but also to its weight. The weights evolve in time and their equations are coupled with the opinions' evolution. We show that the well-known conditions for convergence to consensus can be generalized to this framework. In the case of interaction functions with bounded support, we show that constant weights lead to clustering with conditions similar to those of the classical model. Four specific models are designed by prescribing a specific weight dynamics, then the convergence of the opinions and the evolution of the weights for each of them is studied. We prove the existence of different long-term behaviors , such as emergence of a single leader and emergence of two co-leaders. The we illustrate them via numerical simulations. Lastly, a statistical analysis is provided for the speed of convergence to consensus and for the clustering behavior of each model, together with a comparison to the classical opinion dynamics with constant equal weights
Control of collective dynamics with time-varying weights
This paper focuses on a model for opinion dynamics, where the influence weights of agents evolve in time. We formulate a control problem of consensus type, in which the objective is to drive all agents to a final target point under suitable control constraints. Controllability is discussed for the corresponding problem with and without constraints on the total mass of the system, and control strategies are designed with the steepest descent approach. The mean-field limit is described both for the opinion dynamics and the control problem. Numerical simulations illustrate the control strategies for the finite-dimensional system
Consensus Formation in First-Order Graphon Models with Time-Varying Topologies
In this article, we investigate the asymptotic formation of consensus for
several classes of time-dependent cooperative graphon dynamics. After
motivating the use of this type of macroscopic models to describe multi-agent
systems, we adapt the classical notion of scrambling coefficient to this
setting, leverage it to establish sufficient conditions ensuring the
exponential convergence to consensus with respect to the -norm
topology. We then shift our attention to consensus formation expressed in terms
of the -norm, and prove three different consensus result for symmetric,
balanced and strongly connected topologies, which involve a suitable
generalisation of the notion of algebraic connectivity to this
infinite-dimensional framework. We then show that, just as in the
finite-dimensional setting, the notion of algebraic connectivity that we
propose encodes information about the connectivity properties of the underlying
interaction topology. We finally use the corresponding results to shed some
light on the relation between - and -consensus formation, and
illustrate our contributions by a series of numerical simulations.Comment: 48 pages, 16 figure
Sparse control of Hegselmann-Krause models: Black hole and declustering
International audienceThis paper elaborates control strategies to prevent clustering effects in opinion formation models. This is the exact opposite of numerous situations encountered in the literature where, on the contrary, one seeks controls promoting consensus. In order to promote declustering, instead of using the classical variance that does not capture well the phenomenon of dispersion, we introduce an entropy-type functional that is adapted to measuring pairwise distances between agents. We then focus on a Hegselmann-Krause-type system and design declustering sparse controls both in finite-dimensional and kinetic models. We provide general conditions characterizing whether clustering can be avoided as function of the initial data. Such results include the description of black holes (where complete collapse to consensus is not avoidable), safety zones (where the control can keep the system far from clustering), basins of attraction (attractive zones around the clustering set) and collapse prevention (when convergence to the clustering set can be avoided)