Convergence properties of optimal transport-based temporal networks

TL;DR

This paper investigates how networks evolving under optimal transport principles change their topological properties over time, revealing that transport costs decrease before network properties stabilize, with implications for designing efficient networks.

Contribution

It introduces a framework for analyzing the convergence of network properties in optimal transport-based evolution and demonstrates this behavior in both simulated and real biological networks.

Findings

Transport cost converges earlier than network properties.

Network properties monotonically decrease towards optimality.

Real biological networks exhibit similar convergence behavior.

Abstract

We study network properties of networks evolving in time based on optimal transport principles. These evolve from a structure covering uniformly a continuous space towards an optimal design in terms of optimal transport theory. At convergence, the networks should optimize the way resources are transported through it. As the network structure shapes in time towards optimality, its topological properties also change with it. The question is how do these change as we reach optimality. We study the behavior of various network properties on a number of network sequences evolving towards optimal design and find that the transport cost function converges earlier than network properties and that these monotonically decrease. This suggests a mechanism for designing optimal networks by compressing dense structures. We find a similar behavior in networks extracted from real images of the networks designed by the body shape of a slime mold evolving in time.