Scaling properties of multilayer random networks.

Area of Science:

• Complex systems
• Network science
• Statistical physics

Background:

• Multilayer networks are prevalent in various natural and man-made systems.
• Spectral and eigenfunction properties are crucial for understanding dynamics and critical phenomena on these networks.

Purpose of the Study:

• To numerically demonstrate a scaling law for the normalized localization length (β) of eigenfunctions in multilayer random networks.
• To validate this scaling law on real-world network data.

Main Methods:

• Numerical simulations of multilayer random networks.
• Analysis of spectral and eigenfunction characteristics.
• Derivation and validation of a scaling law for β.

Main Results:

• The normalized localization length (β) follows the scaling law β=x*/(1+x*).
• The parameter x* is defined as x*=γ(b_eff²/L)δ, with δ approximately 1.
• b_eff represents the effective bandwidth of the network's adjacency matrix of size L.

Conclusions:

• The derived scaling law provides a simplified model for eigenfunction localization in multilayer networks.
• This finding can enhance the understanding of criticality in complex network systems.
• The law aids in predicting eigenfunction localization properties for both synthetic and real-world networks.