Critical line in random-threshold networks with inhomogeneous thresholds.

Area of Science:

• Complex systems
• Network science
• Statistical physics

Background:

• Random-threshold networks (RTNs) are crucial models for understanding system dynamics.
• Characterizing critical connectivity (K_c) is essential for predicting network stability and behavior.

Purpose of the Study:

• To analytically and numerically determine the critical connectivity (K_c) in RTNs.
• To investigate the impact of homogeneous and inhomogeneous thresholds on K_c.
• To explore the influence of local correlations on network dynamics.

Main Methods:

• Analytical calculations of critical connectivity (K_c).
• Numerical simulations to confirm analytical findings.
• Analysis of threshold distributions and connectivity patterns.

Main Results:

• A superlinear increase of K_c with average absolute threshold magnitude (R) was found, approaching K_c(R) ≈ R²(2lnR) for large R.
• Universal asymptotic scaling was demonstrated for RTNs with Poissonian connectivity and specific threshold distributions.
• Inhomogeneous thresholds altered perturbation propagation: increasing it in sparse networks and decreasing it in dense networks, with a crossover point K_d.

Conclusions:

• The study provides a precise analytical framework for K_c in RTNs.
• Network behavior transitions from ordered to chaotic dynamics with weak correlations between thresholds and in-degree.
• Findings offer insights into the robustness and dynamics of complex networks.