Potential energy of complex networks: a novel perspective

TL;DR

This paper introduces a novel method for analyzing complex networks using a Schrödinger equation potential that captures network properties and detects phase transitions, including in real-world infrastructure networks.

Contribution

It proposes a new spectral approach linking the Schrödinger potential to the graph Laplacian, revealing critical phenomena and fractality in networks.

Findings

Potential profile diverges near critical connection probability

Fractal dimension peaks at percolation threshold

Real-world networks show criticality signatures in potential fractality

Abstract

We present a novel characterization of complex networks, based on the potential of an associated Schr\"odinger equation. The potential is designed so that the energy spectrum of the Schr\"odinger 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 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 median potential profile is quantified 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 median 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.