Pseudofractal scale-free web.

TLDR: It is found that scale-free random networks are excellently modeled by simple deterministic graphs and exactly and numerically with high precision all main characteristics of the graph are found.

Abstract: 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 $\ensuremath{\gamma}=1+\mathrm{ln}3/\mathrm{ln}2.$ Properties of this compact structure are surprisingly close to those of growing random scale-free networks with \ensuremath{\gamma} 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 $(\mathrm{ln}N\ensuremath{\gg}1)$ the distribution tends to a Gaussian of width $\ensuremath{\sim}\sqrt{\mathrm{ln}N}$ centered at $\mathcal{l}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\sim}\mathrm{ln}N.$ We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent $2+\ensuremath{\gamma}.$