Approaching the thermodynamic limit in equilibrated scale-free networks

Area of Science:

• Network Science
• Statistical Physics
• Complex Systems

Background:

• Scale-free complex networks are ubiquitous in nature and technology.
• Understanding their limiting properties as network size (N) grows is crucial.
• Finite-size effects can deviate from idealized infinite-network models.

Purpose of the Study:

• To investigate how finite-size effects influence degree distributions in complex networks.
• To analyze the scaling behavior of the degree cutoff (k_cutoff) and maximal degree (k_max).
• To compare theoretical predictions with empirical observations in large-scale networks.

Main Methods:

• Analysis of degree distributions in equilibrated complex networks.
• Examination of finite-size scaling for k_cutoff and k_max.
• Identification of subleading and logarithmic corrections to scaling laws.

Main Results:

• k_cutoff scaling deviates from N;{alpha} predictions due to strong subleading corrections, even for N ~ 10;{9}.
• Logarithmic corrections to scaling are observed for networks with power-law degree distributions (pi(k) ~ k;{-3}).
• k_max distribution exhibits different scaling than k_cutoff and approaches the thermodynamic limit faster, with alpha;' = min[alpha, 1/(gamma-1)].

Conclusions:

• Standard scaling theories for infinite networks may not accurately describe finite-size complex networks.
• Subleading and logarithmic corrections are essential for accurate modeling of large-scale networks.
• The behavior of k_max provides a more robust indicator of network properties in finite systems.