Fractal dimensions of percolating networks

We use the generating function formalism to calculate the fractal dimensions for the percolating cluster at criticality in Erd s–Rényi (ER) and random scale free (SF) networks, with degree distribution P(k)=ck−λ. We show that the chemical dimension is dl=2 for ER and SF networks with λ>4, as in percolation in d dc=6 dimensions. For 3<λ<4 we show that dl=(λ−2)/(λ−3). The fractal dimension is df=4 (λ>4) and df=2(λ−2)/(λ−3) (3<λ<4), and the embedding dimension is dc=6 (λ>4) and dc=2(λ−1)/(λ−3) (3<λ<4). We discuss the meaning of these dimensions for networks.