Surfacic networks

TL;DR

This paper introduces new tools for analyzing surfacic networks, which are built on two-dimensional surfaces, focusing on elevation effects on shortest paths and network centrality, with empirical and toy model illustrations.

Contribution

It presents novel measures like lazy paths, graph arduousness, and excess effort to understand elevation impacts on surfacic networks, supported by empirical pedestrian network analysis.

Findings

Excess effort follows a non-trivial power law distribution.

Elevation fluctuations significantly influence network centrality.

Steep slopes are common along shortest paths regardless of elevation differences.

Abstract

Surfacic networks are structures built upon a two-dimensional manifold. Many systems, including transportation networks and various urban networks, fall into this category. The fluctuations of node elevations imply significant deviations from typical plane networks and require specific tools to understand their impact. Here, we present such tools, including lazy paths that minimize elevation differences, graph arduousness which measures the tiring nature of shortest paths, and the excess effort, which characterizes positive elevation variations along shortest paths. We illustrate these measures using toy models of surfacic networks and empirically examine pedestrian networks in selected cities. Specifically, we examine how changes in elevation affect the spatial distribution of betweenness centrality. We also demonstrate that the excess effort follows a non-trivial power law distribution, with an exponent that is not universal, which illustrates that there is a significant probability of encountering steep slopes along shortest paths, regardless of the elevation difference between the starting point and the destination. These findings highlight the significance of elevation fluctuations in shaping network characteristics. Surfacic networks offer a promising framework for comprehensively analyzing and modeling complex systems that are situated on or constrained to a surface environment.