  We compare phase transition and critical phenomena of bond percolation on
Euclidean lattices, nonamenable graphs, and complex networks. On a Euclidean
lattice, percolation shows a phase transition between the nonpercolating phase
and percolating phase at the critical point. The critical point is stretched to
a finite region, called the critical phase, on nonamenable graphs. To
investigate the critical phase, we introduce a fractal exponent, which
characterizes a subextensive order of the system. We perform the Monte Carlo
simulations for percolation on two nonamenable graphs - the binary tree and the
enhanced binary tree. The former shows the nonpercolating phase and the
critical phase, whereas the latter shows all three phases. We also examine the
possibility of critical phase in complex networks. Our conjecture is that
networks with a growth mechanism have only the critical phase and the
percolating phase. We study percolation on a stochastically growing network
with and without a preferential attachment mechanism, and a deterministically
growing network, called the decorated flower, to show that the critical phase
appears in those models. We provide a finite-size scaling by using the fractal
exponent, which would be a powerful method for numerical analysis of the phase
transition involving the critical phase.
