Scaling of the spanning threshold in gradient percolation.

Area of Science:

• Computational Physics
• Statistical Mechanics
• Network Science

Background:

• Percolation theory is crucial for understanding connectivity in random systems.
• Simulating correlated percolation often requires complex methods.
• Applying gradients offers a simplified approach to introducing correlations.

Purpose of the Study:

• To present a simple and fast method for applying correlations in percolation simulations.
• To determine exact spanning thresholds for small networks under a gradient.
• To investigate the behavior of thresholds in larger networks with varying gradients.

Main Methods:

• Applying a uniform gradient to occupancy probabilities in site percolation.
• Exact calculations for small 2D (up to 4x4) and 3D (up to 2x2x2) networks.
• Numerical simulations for larger network sizes.

Main Results:

• Exact spanning thresholds were determined for small, gradient-percolated networks.
• Numerical results indicate a linear relationship between the threshold and gradient for moderate gradient values.
• The gradient method provides a computationally efficient way to model correlations.

Conclusions:

• The uniform gradient method is an effective and efficient technique for simulating correlated percolation.
• The observed linear modification of the threshold is a key finding for larger systems.
• This approach simplifies the study of connectivity in complex networks.