Fractal dimensions of percolating networks

We use the generating function formalism to calculate the fractal dimensions for the percolating cluster at criticality in Erdos-Renyi (ER) and random scale free (SF) networks, with degree distribution P(k) = ck(-lambda). We show that the chemical dimension is d(1) = 2 for ER and SF networks with lambda > 4, as in percolation in d greater than or equal to d(c) = 6 dimensions. For 3 < lambda < 4 we show that d(1)=(lambda-2)/(lambda-3). The fractal dimension is d(f) = 6 (lambda > 4) and d(f) = 2(lambda-2)/(lambda-3) (3 < lambda < 4). and the embedding dimension is d(c) = 6 (lambda > 4) and d(c) = 2(lambda - 1)/(lambda - 3) (3 < lambda < 4). We discuss the meaning of these dimensions for networks. (C) 2004 Elsevier B.V. All rights reserved.