arXiv subjects

Jean-Yves Le Boudec

Publications and source records attributed to Jean-Yves Le Boudec.

42 records · Page 3Linked to original sources

arXiv2011

On Mean Field Convergence and Stationary Regime

Assume that a family of stochastic processes on some Polish space $E$ converges to a deterministic process; the convergence is in distribution (hence in probability) at every fixed point in time. This assumption holds for a large family of processes, among which many mean field interaction models and is weaker than previously assumed. We show that any limit point of an invariant probability of the stochastic process is an invariant probability of the deterministic process. The results are valid in discrete and in continuous time.

math.DS↗

arXiv2011

Mean field for Markov Decision Processes: from Discrete to Continuous Optimization

We study the convergence of Markov Decision Processes made of a large number of objects to optimization problems on ordinary differential equations (ODE). We show that the optimal reward of such a Markov Decision Process, satisfying a Bellman equation, converges to the solution of a continuous Hamilton-Jacobi-Bellman (HJB) equation based on the mean field approximation of the Markov Decision Process. We give bounds on the difference of the rewards, and a constructive algorithm for deriving an approximating solution to the Markov Decision Process from a solution of the HJB equations. We illustrate the method on three examples pertaining respectively to investment strategies, population dynamics control and scheduling in queues are developed. They are used to illustrate and justify the construction of the controlled ODE and to show the gain obtained by solving a continuous HJB equation rather than a large discrete Bellman equation.

cs.AI↗

arXiv2011

The Bounded Confidence Model Of Opinion Dynamics

The bounded confidence model of opinion dynamics, introduced by Deffuant et al, is a stochastic model for the evolution of continuous-valued opinions within a finite group of peers. We prove that, as time goes to infinity, the opinions evolve globally into a random set of clusters too far apart to interact, and thereafter all opinions in every cluster converge to their barycenter. We then prove a mean-field limit result, propagation of chaos: as the number of peers goes to infinity in adequately started systems and time is rescaled accordingly, the opinion processes converge to i.i.d. nonlinear Markov (or McKean-Vlasov) processes; the limit opinion processes evolves as if under the influence of opinions drawn from its own instantaneous law, which are the unique solution of a nonlinear integro-differential equation of Kac type. This implies that the (random) empirical distribution processes converges to this (deterministic) solution. We then prove that, as time goes to infinity, this solution converges to a law concentrated on isolated opinions too far apart to interact, and identify sufficient conditions for the limit not to depend on the initial condition, and to be concentrated at a single opinion. Finally, we prove that if the equation has an initial condition with a density, then its solution has a density at all times, develop a numerical scheme for the corresponding functional equation, and show numerically that bifurcations may occur.

math.PR↗

arXiv2010

Selfish Response to Epidemic Propagation

An epidemic spreading in a network calls for a decision on the part of the network members: They should decide whether to protect themselves or not. Their decision depends on the trade-off between their perceived risk of being infected and the cost of being protected. The network members can make decisions repeatedly, based on information that they receive about the changing infection level in the network. We study the equilibrium states reached by a network whose members increase (resp. decrease) their security deployment when learning that the network infection is widespread (resp. limited). Our main finding is that the equilibrium level of infection increases as the learning rate of the members increases. We confirm this result in three scenarios for the behavior of the members: strictly rational cost minimizers, not strictly rational, and strictly rational but split into two response classes. In the first two cases, we completely characterize the stability and the domains of attraction of the equilibrium points, even though the first case leads to a differential inclusion. We validate our conclusions with simulations on human mobility traces.

eess.SY↗

arXiv2007

Performance of Ultra-Wideband Impulse Radio in Presence of Impulsive Interference

We analyze the performance of coherent impulsive-radio (IR) ultra-wideband (UWB) channel in presence of the interference generated by concurrent transmissions of the systems with the same impulsive radio. We derive a novel algorithm, using Monte-Carlo method, to calculate a lower bound on the rate that can be achieved using maximum-likelihood estimator. Using this bound we show that such a channel is very robust to interference, in contrast to the nearest-neighbor detector.

cs.IT↗

arXiv2006

Analysis of a Reputation System for Mobile Ad-Hoc Networks with Liars

The application of decentralized reputation systems is a promising approach to ensure cooperation and fairness, as well as to address random failures and malicious attacks in Mobile Ad-Hoc Networks. However, they are potentially vulnerable to liars. With our work, we provide a first step to analyzing robustness of a reputation system based on a deviation test. Using a mean-field approach to our stochastic process model, we show that liars have no impact unless their number exceeds a certain threshold (phase transition). We give precise formulae for the critical values and thus provide guidelines for an optimal choice of parameters.

cs.PF↗