Optimal transport by a Lagrangian dynamics of population distribution

TL;DR

This paper addresses modeling human mobility as a coarse-grained dynamical system on networks using a Lagrangian formalism. It introduces an interpretable model with inertia $I_{aa}$, dissipation $\Gamma_{ab}$, and a harmonic potential $V$ whose minimum follows the target distribution $\boldsymbol{\mu}(t)$, parameterized by $I_{aa}$, $\Gamma_{ab}$, and $\Lambda_{ab}$, and learns the dynamics via $q_a(t)=\ln M_a(t)$ and $m_a(t)=\frac{e^{q_a(t)}}{\sum_b e^{q_b(t)}}$. The method is validated on synthetic and real-world datasets (Japan, Finland, Madrid), showing that inertia and dissipation are both important and that interactions and route optimization shape the dynamics. Dynamical susceptibilities reveal how initial perturbations propagate and identify regions of high sensitivity, enabling scenario design and stress testing.

Abstract

Human mobility, enabled by diverse transportation modes, is fundamental to urban functionality. Studying these movements across scales-from microscopic to macroscopic-yields valuable insights into urban dynamics. Local adaptation and (self-)organization in such systems are expected to result in dynamical behaviors that are represented by stationary trajectories of an appropriate effective action. In this study we develop a Lagrangian dynamical model for movement processes, using local population functions as the coordinate variables. An efficient gradient descent algorithm is introduced to estimate the optimal Lagrangian parameters minimizing a local error function of the dynamical process. We show that even a quadratic Lagrangian, incorporating dissipation, effectively captures the dynamics of synthetic and empirical movement data. The inferred models reveal that inertia and dissipation are of comparable importance, while interactions and randomness in the movements induce significant qualitative changes in model parameters. Our results provide an interpretable and generative model for human mobility, with potential applications in movement prediction.

Figures (7)

• Figure 1: Learning a Lagrangian dynamics by a local gradient descent algorithm.
• Figure 2: Comparing the synthetic and inferred model dynamics in a movement process of $T=100$ time steps. The linear size of network is $L=10$ and total population is $M=10^4$.
• Figure 3: Probability distribution of the model parameters ($\Gamma_{ab}$ and $\Lambda_{ab}$). ((a1),(b1)) From modeling of the synthetic dynamics. ((a2),(b2)) From modeling of the real dynamics.
• Figure 4: Probability distribution of the local ratio of inertia to dissipation ($I_{a}$). (a) From modeling of the synthetic dynamics. (b) From modeling of the real dynamics.
• Figure 5: Probability distribution of the local fluxes ($\Phi_a$), dissipations ($\gamma_a$), and characteristic times ($\tau_a$). ((a1),(b1),(c1)) From modeling of the synthetic dynamics. ((a2),(b2),(c2)) From modeling of the real dynamics.