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Ranking nodes in directed networks via continuous-time quantum walks

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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.

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Data availability statement

The datasets generated during the current study are available from the corresponding author on reasonable request.

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Funding

This work was partially supported by the project PRA_2020_92 “Quantum computing, technologies and application” funded by the University of Pisa, and by the INdAM—GNCS Project CUP_E53C22001930001.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127, Pisa, Italy

    Paola Boito

  2. C. R. ENEA Casaccia, Via Anguillarese 301, 00123, Roma, Italy

    Roberto Grena

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  1. Paola Boito
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  2. Roberto Grena
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Correspondence to Roberto Grena.

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Boito, P., Grena, R. Ranking nodes in directed networks via continuous-time quantum walks. Quantum Inf Process 22, 246 (2023). https://doi.org/10.1007/s11128-023-03975-6

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