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Entropy

Can a Quantum Walk Tell Which Is Which?A Study of Quantum Walk-Based Graph Similarity

Giorgia Minello, Luca Rossi, Andrea Torsello A

Published Mar 26, 2019• 14 citations
Publisher

Abstract

We consider the problem of measuring the similarity between two graphs using continuous-time quantum walks and comparing their time-evolution by means of the quantum Jensen-Shannon divergence. Contrary to previous works that focused solely on undirected graphs, here we consider the case of both directed and undirected graphs. We also consider the use of alternative Hamiltonians as well as the possibility of integrating additional node-level topological information into the proposed framework. We set up a graph classification task and we provide empirical evidence that: (1) our similarity measure can effectively incorporate the edge directionality information, leading to a significant improvement in classification accuracy; (2) the choice of the quantum walk Hamiltonian does not have a significant effect on the classification accuracy; (3) the addition of node-level topological information improves the classification accuracy in some but not all cases. We also theoretically prove that under certain constraints, the proposed similarity measure is positive definite and thus a valid kernel measure. Finally, we describe a fully quantum procedure to compute the kernel.

DOI
10.3390/e21030328

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We consider the problem of measuring the similarity between two graphs using continuous-time quantum walks and comparing their time-evolution by means of the quantum Jensen-Shannon divergence. Contrary to previous works that focused solely on undirected graphs, here we consider the case of both directed and undirected graphs. We also consider the use of alternative Hamiltonians as well as the possibility of integrating additional node-level topological information into the proposed framework. We set up a graph classification task and we provide empirical evidence that: (1) our similarity measure can effectively incorporate the edge directionality information, leading to a significant improvement in classification accuracy; (2) the choice of the quantum walk Hamiltonian does not have a significant effect on the classification accuracy; (3) the addition of node-level topological information improves the classification accuracy in some but not all cases. We also theoretically prove that under certain constraints, the proposed similarity measure is positive definite and thus a valid kernel measure. Finally, we describe a fully quantum procedure to compute the kernel.
In this paper we propose a quantum algorithm to measure the similarity between a pair of unattributed graphs. We design an experiment where the two graphs are merged by establishing a complete set of connections between their nodes and the resulting structure is probed through the evolution of continuous-time quantum walks. In order to analyze the behavior of the walks without causing wave function collapse, we base our analysis on the recently introduced quantum Jensen-Shannon divergence. In particular, we show that the divergence between the evolution of two suitably initialized quantum walks over this structure is maximum when the original pair of graphs is isomorphic. We also prove that under special conditions the divergence is minimum when the sets of eigenvalues of the Hamiltonians associated with the two original graphs have an empty intersection.
We develop a novel method for measuring the similarity between complete weighted graphs, which are probed by means of the discrete-time quantum walks. Directly probing complete graphs using discrete-time quantum walks is intractable due to the cost of simulating the quantum walk. We overcome this problem by extracting a commute time minimum spanning tree from the complete weighted graph. The spanning tree is probed by a discrete-time quantum walk which is initialized using a weighted version of the Perron-Frobenius operator. This naturally encapsulates the edge weight information for the spanning tree extracted from the original graph. For each pair of complete weighted graphs to be compared, we simulate a discrete-time quantum walk on each of the corresponding commute time minimum spanning trees and, then, compute the associated density matrices for the quantum walks. The probability of the walk visiting each edge of the spanning tree is given by the diagonal elements of the density matrices. The similarity between each pair of graphs is then computed using either: 1) the inner product or 2) the negative exponential of the Jensen-Shannon divergence between the probability distributions. We show that in both cases the resulting similarity measure is positive definite and, therefore, corresponds to a kernel on the graphs. We perform a series of experiments on publicly available graph datasets from a variety of different domains, together with time-varying financial networks extracted from data for the New York Stock Exchange. Our experiments demonstrate the effectiveness of the proposed similarity measures.
We develop a novel method for measuring the similarity between complete weighted graphs, which are probed by means of discrete-time quantum walks. Directly probing complete graphs using discrete-time quantum walks is intractable due to the cost of simulating the quantum walk. We overcome this problem by extracting a commute-time minimum spanning tree from the complete weighted graph. The spanning tree is probed by a discrete time quantum walk which is initialised using a weighted version of the Perron-Frobenius operator. This naturally encapsulates the edge weight information for the spanning tree extracted from the original graph. For each pair of complete weighted graphs to be compared, we simulate a discrete-time quantum walk on each of the corresponding commute time minimum spanning trees, and then compute the associated density matrices for the quantum walks. The probability of the walk visiting each edge of the spanning tree is given by the diagonal elements of the density matrices. The similarity between each pair of graphs is then computed using either a) the inner product or b) the negative exponential of the Jensen-Shannon divergence between the probability distributions. We show that in both cases the resulting similarity measure is positive definite and therefore corresponds to a kernel on the graphs. We perform a series of experiments on publicly available graph datasets from a variety of different domains, together with time-varying financial networks extracted from data for the New York Stock Exchange. Our experiments demonstrate the effectiveness of the proposed similarity measures.
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Various quantum-walk-based algorithms have been proposed to analyze and rank the centrality of graph vertices. However, issues arise when working with directed graphs: the resulting non-Hermitian Hamiltonian leads to nonunitary dynamics, and the total probability of the quantum walker is no longer conserved. In this paper, we discuss a method for simulating directed graphs using PTPTPT-symmetric quantum walks, allowing probability-conserving nonunitary evolution. This method is equivalent to mapping the directed graph to an undirected, yet weighted, complete graph over the same vertex set, and can be extended to cover interdependent networks of directed graphs. Previous work has shown centrality measures based on the continuous-time quantum walk provide an eigenvectorlike quantum centrality; using the PTPTPT-symmetric framework, we extend these centrality algorithms to directed graphs with a significantly reduced Hilbert space compared to previous proposals. In certain cases, this centrality measure provides an advantage over classical algorithms used in network analysis, for example, by breaking vertex rank degeneracy. Finally, we perform a statistical analysis over ensembles of random graphs, and show strong agreement with the classical PageRank measure on directed acyclic graphs.