ИСТИНА

Node degree distribution is probably the most commonly studied characteristics of complex networks. There is a variety of simple models of networks, which generate power law degree distributions commonly observed in experimental datasets, including different variations of preferential attachment \cite{litlink1}, hyperbolic embedding \cite{litlink2}, etc. One useful example is the so-called Apollonian network \cite{litlink3}. This is a planar graph produced by a deterministic hierarchical procedure, namely by iterative separation of a triangle into three adjacent triangles, then each of the resulting triangles into smaller once, etc. Apollonian network is a deterministic object with a fixed degree distribution: a discrete power law with log-periodic support and exponent $\ln 3/\ln 2$. It is interesting to ask if it is possible to modify the Apollonian algorithm to get a power law degree distribution with a different exponent, while preserving other important properties of the Apollonian network. Here we suggest an algorithm, which allows to build a hierarchical network similar to the Apollonian network, but based on iteratively dividing different polygons, e.g., tetragon or hexagon. For both cases we obtain the recurrence equation for the degree distribution and solve it using the generating function technique. We obtain the moments of degree distribution and show that for the large number of iterations the distribution converges to a power law with exponent $\alpha_4 = \ln 3/(\ln 3 - \ln 2)$ for tetragon and $\alpha_6 = \frac{\ln8-\ln3}{\ln 4 - \ln 3}$ for hexagon. We have also studied the degree distribution numerically. The numerically observed exponent is in perfect agreement with analytic theory. We also obtain the scaling function describing finite-iteration corrections to the power law, as well as the average network diameter as a function of number of iterations.