Number of paths in a graph

TL;DR

This paper introduces methods to analytically count paths of specific lengths in graphs, compares computational complexities, and proposes a recursive algorithm applicable to directed and weighted networks.

Contribution

It presents a novel analytic framework for counting paths, introduces three walk types for efficient computation, and offers a recursive algorithm for all-path enumeration.

Findings

Analytic solutions for path counts using matrix methods.

Different approaches optimized for various path lengths.

A recursive algorithm for enumerating all paths in directed, weighted graphs.

Abstract

The $k$-th power of the adjacency matrix of a simple undirected graph represents the number of walks with length $k$ between pairs of nodes. As a walk where no node repeats, a path is a walk where each node is only visited once. The set of paths constitutes a relatively small subset of all possible walks. We introduce three types of walks, representing subsets of all possible walks. Considered types of walks allow for deriving an analytic solution for the number of paths of a certain length between node pairs in a matrix form. Depending on the path length, different approaches possess the lowest computational complexity. We also propose a recursive algorithm for determining all paths in a graph, which can be generalised to directed (un)weighted networks.