Quantum Motif Clustering

Chris Cade, Farrokh Labib, and Ido Niesen

QuSoft & CWI, Amsterdam, the Netherlands.

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We present three quantum algorithms for clustering graphs based on higher-order patterns, known as motif clustering. One uses a straightforward application of Grover search, the other two make use of quantum approximate counting, and all of them obtain square-root like speedups over the fastest classical algorithms in various settings. In order to use approximate counting in the context of clustering, we show that for general weighted graphs the performance of spectral clustering is mostly left unchanged by the presence of constant (relative) errors on the edge weights. Finally, we extend the original analysis of motif clustering in order to better understand the role of multiple `anchor nodes' in motifs and the types of relationships that this method of clustering can and cannot capture.

The study of complex networks has impacted many fields of science, including biology, sociology, neuroscience, and finance. It is commonplace to study the connectivity patterns of networks and graphs in order to uncover important structures in the underlying data. One method that provides insight into the connectivity structure of a network is graph clustering: finding groups of vertices in a network that are highly connected to each other, but less so to the rest of the network. This is usually done at the edge and vertex level of the network, but more recently, it is becoming popular to study more sophisticated connectivity patterns, for example by studying multi-vertex relationships using small subgraphs, or patterns of connectivity, known as 'motifs'. This allows clustering to be performed at a higher level, grouping together vertices that participate together in many common motifs, and is becoming a useful tool for providing deeper insight into a network's function and structure. However, the detection of motifs in a network is often computationally challenging, leading to a computational bottleneck in the application of these algorithms. In this paper, we present three quantum algorithms that help to speed up motif-based clustering. The speedups rely on two commonly-used quantum subroutines: Grover search and quantum counting, and provide a speedup that is at most quadratic in the input size.

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Cited by

[1] Junpeng Zhan, "Variational Quantum Search with Shallow Depth for Unstructured Database Search", arXiv:2212.09505, (2022).

[2] Simon Apers and Ronald de Wolf, "Quantum Speedup for Graph Sparsification, Cut Approximation and Laplacian Solving", arXiv:1911.07306, (2019).

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