A general model of hierarchical fractal scale-free networks

Abstract

We propose a general model of unweighted and undirected networks having the scale-free property and fractal nature. Unlike the existing models of fractal scale-free networks (FSFNs), the present model can systematically and widely change the network structure. In this model, an FSFN is iteratively formed by replacing each edge in the previous generation network with a small graph called a generator. The choice of generators enables us to control the scale-free property, fractality, and other structural properties of hierarchical FSFNs. We calculate theoretically various characteristic quantities of networks, such as the exponent of the power-law degree distribution, fractal dimension, average clustering coefficient, global clustering coefficient, and joint probability describing the nearest-neighbor degree correlation. As an example of analyses of phenomena occurring on FSFNs, we also present the critical point and critical exponents of the bond-percolation transition on infinite FSFNs, which is related to the robustness of networks against edge removal. By comparing the percolation critical points of FSFNs whose structural properties are the same as each other except for the clustering nature, we clarify the effect of the clustering on the robustness of FSFNs. As demonstrated by this example, the present model makes it possible to elucidate how a specific structural property influences a phenomenon occurring on FSFNs by varying systematically the structures of FSFNs. Finally, we extend our model for deterministic FSFNs to a model of non-deterministic ones by introducing asymmetric generators and reexamine all characteristic quantities and the percolation problem for such non-deterministic FSFNs.

Scaling theory of fractal complex networks

“…However, these efforts did not lead to consensus. Thus, there is a lack of realistic (and not just deterministic 20 , 21 , or reflecting the renormalization procedure 2 , 22 , 23 ) fractal network models that would allow testing the role of fractality in the context of geometry-involving issues 24 , such as navigability, localization of information sources, prediction of hidden network connections, etc. These are of particular importance when faced with the confirmed fractal properties of different information, biological, and even social networks (see e.g.…”

Bifractality of fractal scale-free networks

“…(2) [and thus Eqs. (3) and ( 4)] for a wide range of FSFNs, we first investigate a class of deterministic hierarchical FSFNs formed by the single-generator model [37]. In this model, the tth generation network G t is recursively constructed by replacing every edge in G t−1 with a small graph G gen called a generator, so that two root nodes predefined in G gen coincide with the two terminal nodes of the replaced edge, as illustrated by Fig.…”

“…Let us consider a generator G gen with m gen edges, in which the two root nodes have degree κ and they are separated by distance λ. The number of nodes in the tth generation network G t by the single-generator model is [37]…”

“…In addition to γ and D f , various properties of G t , such as the clustering coefficient, degree correlation, and percolation properties, can be analytically obtained by using the quantities determined by the structure of the generator [37]. We can obtain a wide range of deterministic hierarchical FSFNs, including the (u, v)-flower and those by the SHM model, by choosing from a variety of generators.…”

“…In this work we investigate the bifractal property of broader classes of FSFNs. To this end we analyze four types of FSFNs, namely deterministic hierarchical FSFNs formed by the single-generator model [37], stochastic hierarchical FSFNs formed by the multigenerator model, nonhierarchical FSFNs at the percolation critical points, and real-world FSFNs. The combination of these types of networks covers a wide range of FSFNs with a variety of fractal, scale-free, and clustering properties.…”