Ranking nodes in directed networks via continuous-time quantum walks

TL;DR

This paper introduces four novel centrality measures for directed networks based on continuous-time quantum walks, comparing their effectiveness with classical algorithms like HITS and PageRank on various graph sizes.

Contribution

The paper presents a new quantum walk-based framework for node centrality in directed networks, extending classical eigenvector methods with quantum dynamics.

Findings

All methods effectively identify top nodes in large graphs.

Some pathologies observed in small graphs do not affect large graph performance.

Quantum approaches show good agreement with classical centrality measures.

Abstract

Four new centrality measures for directed networks based on unitary, continuous-time quantum walks (CTQW) in $n$ dimensions -- where $n$ is the number of nodes -- are presented, tested and discussed. The main idea behind these methods consists in re-casting the classical HITS and PageRank algorithms as eigenvector problems for symmetric matrices, and using these symmetric matrices as Hamiltonians for CTQWs, in order to obtain a unitary evolution operator. The choice of the initial state is also crucial. Two options were tested: a vector with uniform occupation and a vector weighted w.r.t.~in- or out-degrees (for authority and hub centrality, respectively). Two methods are based on a HITS-derived Hamiltonian, and two use a PageRank-derived Hamiltonian. Centrality scores for the nodes are defined as the average occupation values. All the methods have been tested on a set of small, simple graphs in order to spot possible evident drawbacks, and then on a larger number of artificially generated larger-sized graphs, in order to draw a comparison with classical HITS and PageRank. Numerical results show that, despite some pathologies found in three of the methods when analyzing small graphs, all the methods are effective in finding the first and top ten nodes in larger graphs. We comment on the results and offer some insight into the good accordance between classical and quantum approaches.