Abstract
We propose a characterization of complex networks, based on the potential of an associated Schrödinger equation. The potential is designed so that the energy spectrum of the Schrödinger equation coincides with the graph spectrum of the normalized Laplacian. Crucial information is retained in the reconstructed potential, which provides a compact representation of the properties of the network structure. The median potential over several random network realizations, which we call ensemble potential, is fitted via a Landau-like function, and its length scale is found to diverge as the critical connection probability is approached from above. The ruggedness of the ensemble potential profile is quantified by using the Higuchi fractal dimension, which displays a maximum at the critical connection probability. This demonstrates that this technique can be successfully employed in the study of random networks, as an alternative indicator of the percolation phase transition. We apply the proposed approach to the investigation of real-world networks describing infrastructures (US power grid). Curiously, although no notion of phase transition can be given for such networks, the fractality of the ensemble potential displays signatures of criticality. We also show that standard techniques (such as the scaling features of the largest connected component) do not detect any signature or remnant of criticality.
Highlights
We propose a characterization of complex networks, based on the potential of an associated Schrödinger equation
We shall find that the fractality of the ensemble potential displays signatures of criticality. The content of this Article is organized as follows: in “Setting up the problem: from graph spectra to reconstructed potentials” section we present the properties of the Laplacian spectrum and discuss the method for reconstructing the associated potentials via dressing transformations; in “Erdös-Rényi phase transition in the reconstructed potential framework” section we examine the description of the Erdös and Rényi (ER) network provided by the reconstructed potential framework, focusing on the critical behavior of its length scale, depth and Higuchi Fractal dimension (HFD) at the phase transition; in “Analysis of a real-world network” section we use this approach to investigate a real-world network, the US power grid
We have proposed a quantum-inspired approach to investigate complex networks: by using the mathematical framework provided by dressing transformations, we have developed a technique to uniquely associate a Schrödinger-like potential to the graph spectrum of a given network
Summary
We propose a characterization of complex networks, based on the potential of an associated Schrödinger equation. The roughness of the Vm profile represents a signature of fractality, a property that emerges around pc as a result of the median operation, but is not detected in single-realization reconstructed potentials, characterized instead by a smooth profile at all connection probabilities.
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