Effect of disorder strength on optimal paths in complex networks

Area of Science:

• Network science
• Statistical physics
• Complex systems

Background:

• Disordered networks exhibit complex scaling properties.
• Understanding transitions between disorder regimes is crucial for network analysis.
• Erdos-Rényi (ER) and scale-free (SF) networks represent fundamental network models.

Purpose of the Study:

• To investigate the transition between strong and weak disorder regimes in the scaling of optimal path lengths.
• To analyze this transition in both ER and SF networks.
• To characterize the crossover network size and its dependence on disorder strength.

Main Methods:

• Analyzing the average optimal path length (l(opt)) in networks with tunable disorder.
• Defining a measure to quantify proximity to the strong disorder limit.
• Developing and validating a scaling ansatz for this measure.
• Deriving scaling relations for the crossover network size (N*(a)).

Main Results:

• A crossover network size N*(a) dictates the transition from strong to weak disorder regimes.
• In the strong disorder regime, l(opt) scales as N(1/3) for ER and SF (λ≥4) networks, and N^((λ-3)/(λ-1)) for SF (3<λ<4) networks.
• In the weak disorder regime, l(opt) scales as ln N for ER and SF (λ>3) networks.
• The crossover size N*(a) scales as a^3 for ER and SF (λ≥4) networks, and a^((λ-1)/(λ-3)) for SF (3<λ<4) networks.

Conclusions:

• The study successfully characterizes the transition between strong and weak disorder regimes in complex networks.
• Scaling behaviors of optimal path lengths are distinctly different in the two regimes.
• The crossover network size provides a critical parameter for understanding disorder effects and scales predictably with disorder strength.