  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.
