Abstract
Farey graphs are simultaneously small-world, uniquely Hamiltonian, minimally 3-colorable, maximally outerplanar and perfect. Farey graphs are therefore famous in deterministic models for complex networks. By lacking of the most important characteristics of scale-free, Farey graphs are not a good model for networks associated with some empirical complex systems. We discuss here a category of graphs which are extension of the well-known Farey graphs. These new models are named generalized Farey graphs here. We focus on the analysis of the topological characteristics of the new models and deduce the complicated and graceful analytical results from the growth mechanism used in generalized Farey graphs. The conclusions show that the new models not only possess the properties of being small-world and highly clustered, but also possess the quality of being scale-free. We also find that it is precisely because of the exponential increase of nodes’ degrees in generalized Farey graphs as they grow that caused the new networks to have scale-free characteristics. In contrast, the linear incrementation of nodes’ degrees in Farey graphs can only cause an exponential degree distribution.
Highlights
The first deterministic model is deduced by Comellas[3]; it presents the property of being small-world
We find that our generalized Farey graphs are different from Farey graphs in that they possess the new property of being scale-free
We generalize the construction method of Farey graphs, to construct generalized Farey graphs, and we focus on analytic solutions to derive their topological properties
Summary
Farey graphs are simultaneously small-world, uniquely Hamiltonian, minimally 3-colorable, maximally outerplanar and perfect. By lacking of the most important characteristics of scale-free, Farey graphs are not a good model for networks associated with some empirical complex systems. We find that it is precisely because of the exponential increase of nodes’ degrees in generalized Farey graphs as they grow that caused the new networks to have scale-free characteristics. Deterministic models can be designed that have the same important properties similar as random models, such as being scale-free, small-world and highly clustered, and can be used to imitate empirical networks appropriately. Several papers show that Farey graphs are a good model for networks associated with some complex systems[16,17,18,19], but they have not the important characteristic of being scale-free. The reason for this is that the largest proportion of nodes in two graphs Ft and Gm,t have only a degree of 2 (being newly added)
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