Graphicality conditions for general scale-free complex networks and their application to visibility graphs

Area of Science:

• Network Science
• Statistical Physics
• Complex Systems

Background:

• Scale-free networks are ubiquitous in nature and technology.
• Understanding their structural properties, like graphicality conditions, is crucial.
• Existing research often focuses on uncorrelated networks, leaving general cases less explored.

Purpose of the Study:

• To derive general graphicality conditions for scale-free networks.
• To establish an upper bound relating the degree distribution exponent (γ) and the cutoff exponent (κ).
• To investigate visibility networks from fractional Brownian motion (fBm) and propose a new relationship for their exponents.

Main Methods:

• Derivation of graphicality conditions for general scale-free networks.
• Mathematical analysis to establish an upper bound for κ in relation to γ.
• Review of numerical research on visibility networks and fBm processes.

Main Results:

• The graphicality conditions for general scale-free networks are identical to those for uncorrelated networks.
• An upper bound κ < 1/γ is established for networks with a power-law degree distribution (γ ≤ 2).
• A new relationship γ ⪅ 1/H is postulated for visibility networks, connecting the degree exponent (γ) and Hurst exponent (H).

Conclusions:

• The findings extend the understanding of scale-free network properties to more general cases.
• The established bound provides a useful constraint for analyzing network structures.
• The postulated relationship for visibility networks opens new avenues for research in complex systems and time series analysis.