Fluid-Spatiotemporal Stochastic Geometry: Information Flow in Non-Stationary Fields

2026-07-01

AI Summary

This work addresses the limitations of traditional spatial network models that rely on stationary point processes and thus fail to capture the non-stationary spatiotemporal evolution of node density fields. The authors propose a fluid–spatiotemporal stochastic geometry framework, interpreting dynamic network topology as the hydrodynamic limit of discrete node configurations. By solving an inverse boundary-value problem, they identify latent dynamics and construct a scalar potential field grounded in the minimum kinetic energy principle from optimal transport theory, thereby unifying Lagrangian continuous transport with Eulerian discrete interference geometry. Their key contribution lies in establishing, for the first time, information flux vectors and material derivatives as sufficient statistics for macroscopic convection and kinematic predictors of topological deformation. This reveals intrinsic connections among coordination overhead, control signaling, and topological kinematic entropy, and yields analytical expressions for information flux and energy–density scaling laws in non-stationary networks, laying a theoretical foundation for capacity limits and control mechanism design in dynamic networks.

Abstract

The fundamental limits of information flow in spatial networks are usually characterized under stationary spatial point processes, but this assumption cannot capture non-stationary regimes where the node intensity field evolves continuously in space and time. This paper develops Fluid-Spatiotemporal Stochastic Geometry (F-STSG), treating dynamic network topology as a hydrodynamic limit of the discrete node constellation. We formulate the identification of latent network dynamics as an inverse boundary value problem and, using the minimum kinetic energy principle from optimal transport, establish the existence and uniqueness of a scalar potential field governing the compressive evolution of network load. The resulting field-theoretic formulation couples continuous Lagrangian transport with discrete Eulerian interference geometry. Based on this model, we derive the information flux vector as a sufficient statistic for macroscopic advection and the material derivative as a kinematic predictor of topological divergence. We further characterize non-stationary network limits through energy-density scaling and source-channel interpretation, showing how coordination overhead, topology deformation, and control signaling requirements are linked to the kinematic entropy of the evolving network topology.

Problem

• non-stationary networks
• information flow
• spatiotemporal dynamics
• network topology evolution
• stochastic geometry

Innovation

• Fluid-Spatiotemporal Stochastic Geometry
• optimal transport
• scalar potential field
• information flux vector
• kinematic entropy