Pseudofractal scale-free web

We find that scale-free random networks are excellently modeled by simple deterministic graphs. Our graph has a discrete degree distribution (degree is the number of connections of a vertex), which is characterized by a power law with exponent γ=1+ln3/ln2. Properties of this compact structure are surprisingly close to those of growing random scale-free networks with γ in the most interesting region, between 2 and 3. We succeed to find exactly and numerically with high precision all main characteristics of the graph. In particular, we obtain the exact shortest-path-length distribution. For a large network (lnN≫1) the distribution tends to a Gaussian of width ∼√lnN centered at l¯∼lnN. We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent 2+γ.