Some properties of hierarchic regular networks (a peep into fractal structures)

TL;DR

This paper analyzes the properties of hierarchic regular fractal networks, highlighting their logarithmic path length dependence, zero clustering coefficient, and differences from small-world networks, including network unification and randomness models.

Contribution

It provides exact relations for hierarchic fractal structures and introduces models for unifying small-world and tree networks with new algebraic insights.

Findings

Hierarchic fractal networks have logarithmic path length dependence.

Clustering coefficient in these networks is zero and slightly increases with randomization.

Different models of randomness affect network properties distinctly.

Abstract

We give exact relations for certain types of the hierarchic fractal structures. In the blatant distinction from regular networks of the "small world" (SW) topology [1], regular fractal networks manifests the logarithmic dependence of the characteristic path length on number of vertices N which is typically just for random networks. Due to this fact, for the hierarchic networks there is no effect of an abrupt drop of the mean path length caused by the introduction into a regular structure only a few random edges. The clustering coefficient of the hierarchic structures is equal zero and in a process of randomization it slightly grows up to the random networks value k/N, where k is mean number of the nearest neighbors, in vivid contrast with the small world networks behavior. We considered also the problem of unification of two isolated SW and Caley tree networks in the limits of weak (through one point) and strong (many points) connection and deduced a specific algebra for the mean path length of the unified network in terms of the isolated networks path lengths. The sewing together of the tree-like networks may be performed in the one-bottom point or in the crown top to crown top manners. In such a way we come to the concept of bi-trees of two types having different topologies. We demonstrated the difference between two possible probabilistic models of introducing randomness: the model with substitution of a certain number of cut off links of the regular lattice by the equal number of edges stochastically distributed over the lattice (totally random model) and the artillery model, when one end of the broken link remains connected with the origin lattice.