Explosive percolation via control of the largest cluster

Area of Science:

• Physics
• Statistical Mechanics
• Complex Systems

Background:

• Percolation theory describes phase transitions in random networks.
• Explosive percolation models exhibit abrupt transitions but often lack expected characteristics.
• Understanding cluster properties is key to characterizing transition orders.

Purpose of the Study:

• To investigate a simplified approach to achieve first-order percolation transitions.
• To analyze the statistical properties and morphology of clusters in this model.
• To determine if this simplified model aligns with theoretical expectations for first-order transitions.

Main Methods:

• Simulating percolation on networks by considering only the largest cluster.
• Analyzing cluster size distributions and cluster compactness.
• Calculating the fractal dimension of cluster perimeters at the transition point.

Main Results:

• A first-order percolation transition is achieved by focusing solely on the largest cluster.
• The model produces Gaussian cluster distributions and compact clusters.
• Cluster perimeters exhibit fractal behavior with a dimension of 1.23 ± 0.03 at the transition.

Conclusions:

• Considering only the largest cluster is sufficient for a first-order percolation transition.
• This simplified model reproduces key characteristics of first-order transitions, including Gaussian distributions and compact clusters.
• The fractal nature of cluster perimeters at the transition point provides further evidence for the model's validity.