91 research outputs found
Farey Graphs as Models for Complex Networks
Farey sequences of irreducible fractions between 0 and 1 can be related to
graph constructions known as Farey graphs. These graphs were first introduced
by Matula and Kornerup in 1979 and further studied by Colbourn in 1982 and they
have many interesting properties: they are minimally 3-colorable, uniquely
Hamiltonian, maximally outerplanar and perfect. In this paper we introduce a
simple generation method for a Farey graph family, and we study analytically
relevant topological properties: order, size, degree distribution and
correlation, clustering, transitivity, diameter and average distance. We show
that the graphs are a good model for networks associated with some complex
systems.Comment: Definitive version published in Theoretical Computer Scienc
Structure Properties of Generalized Farey graphs based on Dynamical Systems for Networks
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
Farey graphs as models for complex networks
Farey sequences of irreducible fractions between 0 and 1 can be related to graph constructions known as Farey graphs.
These graphs were first introduced by Matula and Kornerup in 1979 and further studied by Colbourn in 1982 and they have many interesting properties: they are minimally 3-colorable, uniquely Hamiltonian, maximally outerplanar and perfect.
In this paper we introduce a simple generation method for a Farey graph family, and we study analytically relevant topological properties: order, size, degree distribution and correlation, clustering, transitivity, diameter and average distance. We show that the graphs are a good model for networks associated with some complex systems.Peer Reviewe
Topological percolation on hyperbolic simplicial complexes
Simplicial complexes are increasingly used to understand the topology of
complex systems as different as brain networks and social interactions. It is
therefore of special interest to extend the study of percolation to simplicial
complexes. Here we propose a topological theory of percolation for discrete
hyperbolic simplicial complexes. Specifically we consider hyperbolic manifolds
in dimension and formed by simplicial complexes, and we investigate
their percolation properties in the presence of topological damage, i.e., when
nodes, links, triangles or tetrahedra are randomly removed. We show that in
simplicial complexes there are four topological percolation problems and
in , there are six. We demonstrate the presence of two percolation phase
transitions characteristic of hyperbolic spaces for the different variants of
topological percolation. While most of the known results on percolation in
hyperbolic manifolds are in , here we uncover the rich critical behavior
of hyperbolic manifolds, and show that triangle percolation displays a
Berezinskii-Kosterlitz-Thouless (BKT) transition. Finally we provide evidence
that topological percolation can display a critical behavior that is unexpected
if only node and link percolation are considered.Comment: (17 pages, 9 figures
Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models
In this work we consider a general class of 2-dimensional hybrid systems. Assuming that the system possesses an attracting equilibrium point, we show that, when periodically driven with a square-wave pulse, the system possesses a periodic orbit which may undergo smooth and nonsmooth grazing bifurcations. We perform a semi-rigorous study of the existence of periodic orbits for a particular model consisting of a leaky integrate-and-fire model with a dynamic threshold. We use the stroboscopic map, which in this context is a 2-dimensional piecewise-smooth discontinuous map. For some parameter values we are able to show that the map is a quasi-contraction possessing a (locally) unique maximin periodic orbit. We complement our analysis using advanced numerical techniques to provide a complete portrait of the dynamics as parameters are varied. We find that for some regions of the parameter space the model undergoes a cascade of gluing bifurcations, while for others the model shows multistability between orbits of different periodsPeer ReviewedPostprint (author's final draft
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