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Structure Properties of Generalized Farey graphs based on Dynamical Systems for Networks.

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.

SUBMITTER: Jiang W

PROVIDER: S-EPMC6093942 | biostudies-literature | 2018 Aug

REPOSITORIES: biostudies-literature

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Structure Properties of Generalized Farey graphs based on Dynamical Systems for Networks.

Jiang Wenchao W Zhai Yinhu Y Martin Paul P Zhao Zhiming Z

Scientific reports 20180815 1

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 gr ...[more]

PMID: 30111775

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Project description:BackgroundIn rule-based modeling, graphs are used to represent molecules: a colored vertex represents a component of a molecule, a vertex attribute represents the internal state of a component, and an edge represents a bond between components. Components of a molecule share the same color. Furthermore, graph-rewriting rules are used to represent molecular interactions. A rule that specifies addition (removal) of an edge represents a class of association (dissociation) reactions, and a rule that specifies a change of a vertex attribute represents a class of reactions that affect the internal state of a molecular component. A set of rules comprises an executable model that can be used to determine, through various means, the system-level dynamics of molecular interactions in a biochemical system.ResultsFor purposes of model annotation, we propose the use of hierarchical graphs to represent structural relationships among components and subcomponents of molecules. We illustrate how hierarchical graphs can be used to naturally document the structural organization of the functional components and subcomponents of two proteins: the protein tyrosine kinase Lck and the T cell receptor (TCR) complex. We also show that computational methods developed for regular graphs can be applied to hierarchical graphs. In particular, we describe a generalization of Nauty, a graph isomorphism and canonical labeling algorithm. The generalized version of the Nauty procedure, which we call HNauty, can be used to assign canonical labels to hierarchical graphs or more generally to graphs with multiple edge types. The difference between the Nauty and HNauty procedures is minor, but for completeness, we provide an explanation of the entire HNauty algorithm.ConclusionsHierarchical graphs provide more intuitive formal representations of proteins and other structured molecules with multiple functional components than do the regular graphs of current languages for specifying rule-based models, such as the BioNetGen language (BNGL). Thus, the proposed use of hierarchical graphs should promote clarity and better understanding of rule-based models.

Dataset Information

Structure Properties of Generalized Farey graphs based on Dynamical Systems for Networks.

Publications

Structure Properties of Generalized Farey graphs based on Dynamical Systems for Networks.

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