2004/02/12 by Gregory Surdutovich, G. I. Surdutovich, Vladimir Gol’dshtein +6
Biochemistry, Genetics and Molecular Biology · Computer Science · Physics and Astronomy · #Complex Network Analysis Techniques #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Gene Regulatory Network Analysis #Topological and Geometric Data Analysis #cond-mat.dis-nn
paper · pdf · doi:10.48550/arxiv.cond-mat/0402338
11 pages, 8 figures
arxiv created 2004/02/12 · openalex publication_date 2004/02/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.