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 $\gamma =1+\ln 3/\ln 2.$ 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 $\left(\ln N\gg 1\right)$ the distribution tends to a Gaussian of width $\sim \sqrt{\ln N}$ centered at $\mathcal{l}¯\sim \ln N.\mathrm{}$ We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent $2+\gamma .$

• Received 8 December 2001

DOI:https://doi.org/10.1103/PhysRevE.65.066122