The simplicity of planar networks

TL;DR

This paper introduces the simplicity index and profile to compare shortest and simplest paths in planar networks, revealing fundamental organizational differences between biological and urban systems and tracking structural changes over time.

Contribution

It defines new metrics for analyzing path simplicity in planar networks and applies them to diverse real-world and artificial systems, uncovering key organizational insights.

Findings

Biological networks show hierarchical organization of straight paths.

Urban networks have random lengths and locations of simple paths.

Simplicity metrics reveal structural changes in evolving networks.

Abstract

Shortest paths are not always simple. In planar networks, they can be very different from those with the smallest number of turns - the simplest paths. The statistical comparison of the lengths of the shortest and simplest paths provides a non trivial and non local information about the spatial organization of these graphs. We define the simplicity index as the average ratio of these lengths and the simplicity profile characterizes the simplicity at different scales. We measure these metrics on artificial (roads, highways, railways) and natural networks (leaves, slime mould, insect wings) and show that there are fundamental differences in the organization of urban and biological systems, related to their function, navigation or distribution: straight lines are organized hierarchically in biological cases, and have random lengths and locations in urban systems. In the case of time evolving networks, the simplicity is able to reveal important structural changes during their evolution.