Random walk hierarchy measure: What is more hierarchical, a chain, a tree or a star?

TL;DR

This paper introduces a novel hierarchy measure based on random walks that effectively distinguishes different hierarchical structures like chains, trees, and stars, and is computationally efficient for real networks.

Contribution

The paper proposes a new hierarchy measure using random walks that assigns higher scores to multi-level pyramidal structures and converges for regular trees, improving upon existing methods.

Findings

Directed trees score higher than chains and stars.

The measure converges for regular trees based on branching number.

Applied to real networks, it produces intuitive and ecologically consistent results.

Abstract

Signs of hierarchy are prevalent in a wide range of systems in nature and society. One of the key problems is quantifying the importance of hierarchical organisation in the structure of the network representing the interactions or connections between the fundamental units of the studied system. Although a number of notable methods are already available, their vast majority is treating all directed acyclic graphs as already maximally hierarchical. Here we propose a hierarchy measure based on random walks on the network. The novelty of our approach is that directed trees corresponding to multi level pyramidal structures obtain higher hierarchy scores compared to directed chains and directed stars. Furthermore, in the thermodynamic limit the hierarchy measure of regular trees is converging to a well defined limit depending only on the branching number. When applied to real networks, our method is computationally very effective, as the result can be evaluated with arbitrary precision by subsequent multiplications of the transition matrix describing the random walk process. In addition, the tests on real world networks provided very intuitive results, e.g., the trophic levels obtained from our approach on a food web were highly consistent with former results from ecology.