Density flow over networks: A mean-field game theoretic approach

Abstract

Abstract-A distributed routing control algorithm for dynamic networks has recently been presented in the literature. The networks were modeled using time evolution of density at network edges and the routing control algorithm allowed edge density to converge to a Wardrop equilibrium, which was characterized by an equal traffic density on all used paths. We borrow the idea and rearrange the density model to recast the problem within the framework of mean-field games. The contribution of this paper is three-fold. First, we provide a mean-field game formulation of the problem at hand. Second, we illustrate an extended state space solution approach. Third, we study the stochastic case where the density evolution is driven by a Brownian motion.

Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion

“…Many other stationary problems are examined in the literature, including obstacle problems [9], weaklycoupled systems [8], multi-populations [4], and logistic problems [13]. MFGs on networks, see [1], [3], [2], are important cases of one-dimensional MFGs.…”

Density Flow in Dynamical Networks via Mean-Field Games

“…Third, we study the stochastic case where the density evolution is driven by a Brownian motion. Additional results, that make this paper different from its conference version [8], are the following. We investigate the case where the density evolution is perturbed by a bounded adversarial disturbance.…”

“…Note that the transition rates depend on the routing policy/control α. This is then obtained as the minimizer in the computation of the Hamiltonian as expressed by (8). For the first part of the proof, note that the second equation is the boundary condition on the terminal penalty.…”

“…The first equation in (6) is the Hamilton-Jacobi-Bellman (HJB) equation with variable v(x, t) and parametrized in ρ(·). Given the boundary condition on final state (second equation in (6)), and assuming a given population behavior captured by ρ(·), the HJB equation is solved backwards and returns the value function and best-response behavior of the players given by (8). The HJB equation is coupled with a second ODE, which is the Fokker-Planck-Kolmogorov (FPK) (third equation in (6)), defined on variable ρ(·) and parametrized in α * (x, t).…”

Strategic thinking under social influence: Scalability, stability and robustness of allocations

“…Heterogeneity is studied in mean-field games with major and minor players [20,26]. Directions for further developments are i) the formulation, analysis and design of Stackelberg mean-field games where we have leaders and followers [24], ii) the analysis of mean-field games over structured environments by using networks [14], and iii) the applications to demand-side management and intelligent mobility [23].…”