Mesoscale Building Blocks of Pedestrian Mobility: a Discrete Vector Field Approach

Abstract

Understanding and characterising pedestrian mobility is crucial to develop sustainable cities. While classical statistical analysis and diffusion models are commonly used to analyze human trajectories either at the microscopic (e.g. sidewalk flows) or macroscopic scale (e.g. origindestination matrices), they may not be suitable for capturing the nuances and intricacies of mobility patterns at the mesoscale. To overcome these limitations, the problem is approached by leveraging on vector field theory with the aim to describe how the urban geometry and structure of sidewalk networks affect pedestrian mobility flows. Considering the particularities of pedestrian movement (e.g. limited travel range) the discrete- (DTRW) and continuous-time (CTRW) random walk dynamics have been implemented to retrieve a baseline agent-based net flow along the edges of pedestrian networks with a temporal budget of mobility. These flows are subsequently interpreted as discrete vector fields. The Helmholtz-Hodge decomposition (HHD) allows the partition of vector fields into three well-defined patterns: cyclic (solenoidal and harmonic) and divergent/convergent (gradient) components. Results show that when mobility is agnostic to edge lengths (DTRW), that is, when the time budget is spent equivalently along the edges (steps), high-density regions with larger degree nodes show attractiveness, as existing literature already describes. However, when the time budget is spent proportionally to the edge lengths (CTRWs), the same regions show a repulsive effect. Intermediate regimes arise as well in the continuum between these two processes. An analytical description of both DTRWs and CTRWs has been developed to accurately estimate the gradient components of the vector fields. However, the presented deterministic developments do not predict the presence of the cyclic components as they seem to emerge from the stochasticity of the process. To validate this idea, the variance of the cyclic component, or its mean squared flow (MSF), has been analysed. Results show that the MSF grows linearly with the temporal budget of the walkers. This behaviour is similar to the characteristic linear temporal evolution of the Mean-Squared Displacement (MSD) in random walks and Brownian motion. Ultimately, this work contributes to the existing description and understanding of the behaviour of different random walk dynamics on spatially embedded graphs, providing a baseline to understand and analyse pedestrian mobility on sidewalk networks in future works.