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

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.