Potential energy of complex networks: a quantum mechanical perspective

Area of Science:

• Complex systems analysis
• Network science
• Mathematical physics

Background:

• Complex networks are ubiquitous in nature and technology.
• Understanding their structural properties and phase transitions is crucial.
• Existing methods may not capture subtle critical phenomena in all network types.

Purpose of the Study:

• To develop a new characterization of complex networks using quantum mechanics.
• To identify critical behaviors and phase transitions in random and real-world networks.
• To provide an alternative indicator for percolation phase transitions.

Main Methods:

• Associating a Schrödinger equation with a complex network.
• Designing a potential such that its energy spectrum matches the graph spectrum of the normalized Laplacian.
• Analyzing the ensemble potential using Landau-like functions and Higuchi fractal dimension.
• Investigating random networks and the US power grid infrastructure network.

Main Results:

• The reconstructed potential compactly represents network structure properties.
• The length scale of the ensemble potential diverges near the critical connection probability.
• The Higuchi fractal dimension peaks at the critical connection probability, indicating a percolation phase transition.
• The ensemble potential's fractality reveals signatures of criticality in the US power grid, undetected by standard methods.

Conclusions:

• The proposed Schrödinger equation-based method effectively characterizes complex networks.
• This technique serves as an alternative indicator for percolation phase transitions in random networks.
• The method uncovers hidden critical signatures in real-world networks, suggesting broader applicability in network science.