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 et al. [Science 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.