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

The normalized Laplacian spectrum of subdivisions of a graph

Determining and analyzing the spectra of graphs is an important and exciting research topic in mathematics science and theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to random walks. In this paper, we give the spectra of the normalized Laplacian of iterated subdivisions of simple connected graphs. As an example of application of these results we find the exact values of their multiplicative degree-Kirchhoff index, Kemeny's constant and number of spanning trees.Postprint (published version

On the spectra of Markov matrices for weighted Sierpinski graphs

Relevant information from networked systems can be obtained by analyzing the spectra of matrices associated to their graph representations. In particular, the eigenvalues and eigenvectors of the Markov matrix and related Laplacian and normalized Laplacian matrices allow the study of structural and dynamical aspects of a network, like its synchronizability and random walks properties. In this study we obtain, in a recursive way, the spectra of Markov matrices of a family of rotationally invariant weighted Sierpinski graphs. From them we find analytic expressions for the weighted count of spanning trees and the random target access time for random walks on this family of weighted graphs.Postprint (published version

Bumblebees: a multiagent combinatorial optimization algorithm inspired by social insect behaviour

This paper introduces a multiagent optimization algorithm inspired by the collective behavior of social insects. In our method, each agent encodes a possible solution of the problem to solve, and evolves in a way similar to real life insects. We test the algorithm on a classical difficult problem, the $k$-coloring of a graph, and we compare its performance in relation to a standard genetic algorithm and another multiagent system. The results show that this algorithm is faster and outperforms the other methods for a range of random graphs with different orders and densities. Moreover, the method is easy to adapt to solve different NP-complete problems

Vertex labeling and routing in self-similar outerplanar unclustered graphs modeling complex networks

Es publicarà a Journal of Physics A: Mathematical and TheoreticalThis paper introduces a labeling and optimal routing algorithm for a family of modular, self-similar, small-world graphs with clustering zero. Many properties of this family are comparable to those of networks associated with technological and biological systems with a low clustering, like the power grid, some electronic circuits and protein networks. For these systems, the existence of models with an efficient routing protocol is of interest to design practical communication algorithms in relation to dynamical processes (including synchronization) and also to understand the underlying mechanisms that have shaped their particular structures.Peer Reviewe

Mean first-passage time for random walks on generalized deterministic recursive trees

Preprin

Modeling complex networks with self-similar outerplanar unclustered graphs

This paper introduces a family of modular, self-similar, small-world graphs with clustering zero. Relevant properties of this family are comparable to those of some networks associated with technological systems with a low clustering, like the power grid or some electronic circuits. Moreover, the graphs are outerplanar and it is know that many algorithms that are NP-complete for general graphs perform polynomial in outerplanar graphs. Therefore the graphs constitute a good mathematical model for these systems

A fast and efficient algorithm to identify clusters in networks

A characteristic feature of many relevant real life networks, like the WWW, Internet, transportation and communication networks, or even biological and social networks, is their clustering structure. We discuss in this paper a novel algorithm to identify clusters -sets of densely interconnected nodes- in a network. The algorithm is based on local information and therefore it is very fast with respect other proposed methods, while it keeps a similar performance in detecting the clusters

The number and degree distribution of spanning trees in the Tower of Hanoi graph

The number of spanning trees of a graph is an important invariant related to topological and dynamic properties of the graph, such as its reliability, communication aspects, synchronization, and so on. However, the practical enumeration of spanning trees and the study of their properties remain a challenge, particularly for large networks. In this paper, we study the number and degree distribution of the spanning trees in the Hanoi graph. We first establish recursion relations between the number of spanning trees and other spanning subgraphs of the Hanoi graph, from which we find an exact analytical expression for the number of spanning trees of the n-disc Hanoi graph. This result allows the calculation of the spanning tree entropy which is then compared with those for other graphs with the same average degree. Then, we introduce a vertex labeling which allows to find, for each vertex of the graph, its degree distribution among all possible spanning trees.Postprint (author's final draft

Deterministic hierarchical networks

It has recently been shown that many networks associated with complex systems are small-world (they have both a large local clustering and a small average distance and diameter) and they are also scale-free (the degrees are distributed according to a power-law). Moreover, these networks are very often hierarchical, as they describe the modularity of the systems which are modeled. While most of the studies for complex networks are based on stochastic methods, a deterministic approach, with an exact determination of the main relevant parameters of the networks, has proven useful to complement and enhance the probabilistic and simulation techniques and therefore to provide a better understanding of the systems modeled. In this paper we find the diameter, clustering and degree distribution of a generic family of deterministic hierarchical small-world scale-free networks which has been considered for modeling real life complex systems