A wave packet signature for complex networks. (5th September 2018) - Aziz, Furqan; Wilson, Richard C; Hancock, Edwin R
Abstract: This article explores the possibility of using a quantum graph representation to investigate information flow across a complex network in the form of wave propagation. The term quantum graph refers to a representation of a network with differential operators acting on functions defined over real valued intervals associated with the edges. These provide a convenient representation which allows differential operators from calculus to be generalized to graph or network structures, the simplest of which is the metric Laplacian. We present complete solutions of a wave equation on a quantum graph, obtained using the quantum graph Laplacian. The solutions obtained using the Laplacian are more complex than those obtained from the traditional vertex-based graph Laplacian counterpart due to the more complex structure of the eigenfunctions. Specifically, we provide explicit solutions for the wave equation where the initial condition is a Gaussian wave packet confined to a single edge of the graph. We use the new solutions to develop methods for characterizing complex networks and demonstrate their advantages over those obtained using the discrete graph Laplacian. The proposed method is highly robust in distinguishing networks that are co-spectral with respect to different graph representations. Moreover, the partial differential equations defined using the quantum graph Laplacian may be of great interest in the study of complex networks where distance, speed of propagation and connectivity structure of the networks are important.