Network-Growth Rule Dependence of Fractal Dimension of Percolation Cluster on Square Lattice

TL;DR

This paper introduces a generalized network-growth rule with a parameter q to study how it affects the evolution, roughness, and fractal dimension of percolation clusters on a square lattice, revealing that higher q delays percolation.

Contribution

It proposes a new generalized network-growth rule incorporating a parameter q, enabling analysis of its impact on percolation cluster properties and dynamics.

Findings

Higher q delays the percolation transition.

The rule affects the roughness of the cluster boundary.

Fractal dimension varies with q.

Abstract

To investigate the network-growth rule dependence of certain geometric aspects of percolation clusters, we propose a generalized network-growth rule introducing a generalized parameter $q$ and we study the time evolution of the network. The rule we propose includes a rule in which elements are randomly connected step by step and the rule recently proposed by Achlioptas {\it et al.} [Science {\bf 323} (2009) 1453]. We consider the $q$-dependence of the dynamics of the number of elements in the largest cluster. As $q$ increases, the percolation step is delayed. Moreover, we also study the $q$-dependence of the roughness and the fractal dimension of the percolation cluster.