Pseudofractal Scale-free Web

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We find that scale-free random networks are excellently modeled by a deterministic graph. This 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+3/2. Properties of this simple 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 the large network (N ≫ 1) the distribution tends to a Gaussian of width √ N centered at ℓ N. We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent 2+γ.