Density Flow in Dynamical Networks via Mean-Field Games

Abstract

Current distributed routing control algorithms for dynamic networks model networks using the time evolution of density at network edges, while the routing control algorithm ensures edge density to converge to a Wardrop equilibrium, which was characterized by an equal traffic density on all used paths. We rearrange the density model to recast the problem within the framework of mean-field games. In doing that, we illustrate an extended state-space solution approach and we study the stochastic case where the density evolution is driven by a Brownian motion. Further, we investigate the case where the density evolution is perturbed by a bounded adversarial disturbance. For both the stochastic and the worst-case scenarios, we provide conditions for the density to converge to a preassigned set. Moreover, we analyze such conditions from two different perspectives, repeated games with vector payoffs and inclusion theory.

Game Theoretic Decentralized Feedback Controls in Markov Jump Processes

“…We solve Problem 1 and the related mean-field game (8) through state space extension, in spirit with [4]; namely we review ρ as an additional state variable.…”

“…In [4] for the infinite horizon problem, the authors take the value functions as V (ρ) = dist(ρ, M ), where M is the global equilibrium manifold. Therefore in our finite horizon problem we assume that…”

“…To prove such a convergence result, we recast the problem within the framework of optimal control theory. A similar problem is studied in [4], in which the authors consider a centralized control and a density flow for each edge dependent on the density of the whole population. This implies that each player minimizes a common cost functional which depends on the whole population's density distribution.…”

“…Differently from [4], we consider a decentralized control (as in [5][6][7]), in which the density of each node is controlled locally. We highlight next three distinct approaches relating to routing/jump problems.…”

“…We highlight next three distinct approaches relating to routing/jump problems. The first one consists in controlling the probability to jump from a node to another one (or to flow along the edges) [4]. The second one consists in controlling the transition rate from nodes (or edges) [8], and the last one in assigning the product among the probability and the relative transition rate.…”