Quantum Transport Senses Community Structure in Networks

Quantum time evolution exhibits rich physics; however, unlike classical diffusion, the wave nature of quantum mechanics has not yet been extensively explored in modern data analysis. We propose that the Laplace transform of quantum transport (QT) generated by a graph Laplacian can be used to construct an ensemble of maps from a given undirected network to a circle S¹, such that closely-related nodes on the data network are grouped into sharply concentrated clusters on S¹. The resulting QT clustering (QTC) algorithm is shown to outperform the state-of-the-art spectral clustering method on synthetic and stock price data sets containing complex geometric patterns. With a consensus matrix computed from the ensemble of predictions, QTC can be extended to data exhibiting mixing and is demonstrated to depict 3D chromatin interactions based on cancer somatic copy number alteration data. The observed phenomenon of QTC can be interpreted as a collective behavior of the microscopic nodes that evolve as macroscopic cluster "orbitals" in an effective tight-binding model where the phase distribution of the effective two-point Green function (resolvent) facilitates the map from nodes to S¹.