A quantum dynamics approach to spectral analysis in small-world complex networks

Spectra and eigenstates of the adjacency matrix of a complex network provide many important information about its structure and dynamics. By mapping the adjacency matrix of a small-world network to the Hamiltonian of a quantum system, the statistical properties of the spectra and the localization of eigenstates are analyzed. Theoretical analysis and numerical results show that the spectrum corresponding to small-world networks with small rewiring probabilities have properties consistent with those of quantum integrable systems. When the rewiring probability in small-world networks is higher than a certain threshold, the spectral statistics agree with those of quantum chaotic systems. These findings may lead to further understanding of both complex networks and complex quantum systems.