Beyond the shortest path: the path length index as a distribution

TL;DR

This paper introduces a novel approach to analyze the distribution of all possible paths between nodes in a network, extending beyond shortest path analysis to better capture network connectivity and dynamics.

Contribution

It presents an analytical solution for counting paths in complete graphs and a depth-limited method for general graphs, enabling distribution-based path analysis.

Findings

Distribution of path lengths provides richer network insights.

Interactive and early stopping procedures improve computational feasibility.

Impact on traditional distance-based graph indices demonstrated.

Abstract

The traditional complex network approach considers only the shortest paths from one node to another, not taking into account several other possible paths. This limitation is significant, for example, in urban mobility studies. In this short report, as the first steps, we present an exhaustive approach to address that problem and show we can go beyond the shortest path, but we do not need to go so far: we present an interactive procedure and an early stop possibility. After presenting some fundamental concepts in graph theory, we presented an analytical solution for the problem of counting the number of possible paths between two nodes in complete graphs, and a depth-limited approach to get all possible paths between each pair of nodes in a general graph (an NP-hard problem). We do not collapse the distribution of path lengths between a pair of nodes into a scalar number, we look at the distribution itself - taking all paths up to a pre-defined path length (considering a truncated distribution), and show the impact of that approach on the most straightforward distance-based graph index: the walk/path length.