# A Quantum-inspired Similarity Measure for the Analysis of Complete Weighted Graphs

## TL;DR

This paper introduces a quantum-inspired method for measuring similarity between complete weighted graphs by leveraging commute-time minimum spanning trees and quantum walk-based probability distributions, enabling effective graph comparison.

## Contribution

The paper presents a novel quantum-inspired similarity measure for complete weighted graphs using commute-time spanning trees and quantum walks, which is positive definite and effective across various datasets.

## Key findings

- The proposed similarity measure is positive definite and acts as a kernel.
- Experimental results show the method's effectiveness on diverse graph datasets.
- The approach successfully compares time-varying financial networks.

## Abstract

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.