Complex networks with complex weights

Abstract

In many studies, it is common to use binary (i.e., unweighted) edges to examine networks of entities that are either adjacent or not adjacent. Researchers have generalized such binary networks to incorporate edge weights, which allow one to encode node--node interactions with heterogeneous intensities or frequencies (e.g., in transportation networks, supply chains, and social networks). Most such studies have considered real-valued weights, despite the fact that networks with complex weights arise in fields as diverse as quantum information, quantum chemistry, electrodynamics, rheology, and machine learning. Many of the standard network-science approaches in the study of classical systems rely on the real-valued nature of edge weights, so it is necessary to generalize them if one seeks to use them to analyze networks with complex edge weights. In this paper, we examine how standard network-analysis methods fail to capture structural features of networks with complex edge weights. We then generalize several network measures to the complex domain and show that random-walk centralities provide a useful approach to examine node importances in networks with complex weights.

17 pages, 7 figures, 1 table

References in corpus (16)

• Spatial search by quantum walk
• Fractional statistics in anyon collisions
• Testing real quantum theory in an optical quantum network
• Ruling out real-valued standard formalism of quantum theory
• Alignment and integration of complex networks by hypergraph-based spectral clustering
• Simulating quantum systems using real Hilbert spaces
• The topological Dirac equation of networks and simplicial complexes
• Generalized quantum-classical correspondence for random walks on graphs
• Synchrony for weak coupling in the complexified Kuramoto model
• Eigenvalue spectra and stability of directed complex networks
• Periodicity of quantum walks defined by mixed paths and mixed cycles
• Dirac gauge theory for topological spinors in 3+1 dimensional networks
• Characterization and comparison of large directed graphs through the spectra of the magnetic Laplacian
• Modeling Deformed Transmission Lines for Continuous Strain Sensing Applications
• Networks with complex weights: Green function and power series
• Schrödinger-Lohe type models of quantum synchronization with nonidentical oscillators

Cited by in corpus (9)

• Complex Quantum Networks: a Topical Review
• Higher-order Connection Laplacians for Directed Simplicial Complexes
• Complexified Synchrony
• Quantum entropy couples matter with geometry
• Synchronization in the complexified Kuramoto model
• Spectral statistics of interpolating random circulant matrix and its applications to random circulant graphs
• Clustering-induced localization of quantum walks on networks
• Liouville Fock state lattices and potential simulators
• Optimal array geometries for kinetic magnetism and Nagaoka polarons