Self-similar planar graphs as models for complex networks

In this paper we introduce a family of planar, modular and self-similar graphs which have small-world and scale-free properties. The main parameters of this family are comparable to those of networks associated to complex systems, and therefore the graphs are of interest as mathematical models for these systems. As the clustering coefficient of the graphs is zero, this family is an explicit construction that does not match the usual characterization of hierarchical modular networks, namely that vertices have clustering values inversely proportional to their degrees.Comment: 10 pages, submitted to 19th International Workshop on Combinatorial Algorithms (IWOCA 2008

Models deterministes de xarxes complexes

En estudis recents s'ha observat que moltes xarxes associades a sistemes complexos pertanyen a una nova categoria que s'ha volgut anomenar petit-món amb invariància d'escala. Molts dels models que s'han desenvolupat per a la descripció matemàtica d'aquestes xarxes es basen en construccions probabilístiques. Tanmateix, la consideració de models deterministes és útil per completar i millorar les tècniques probabilístiques i d'altres basades en simulacions. En aquest article introduïm els conceptes i models bàsics que s'han considerat en l'estudi de xarxes complexes i es presenten diversos models deterministes que es generen a partir de grafs complets.Recent studies have shown that a number of networks associated to complex systems belong to the new category of “scale–free small–world networks”. The mathematical description of these networks is often based in probabilistic models. However, deterministic models are useful to improve or complete the analysis of these networks obtained by probabilsitic techniques or by simulation. In this paper we introduce the concepts and basic models which have been considered to analyze complex networks and we describe several deterministic models which are obtained from complete graphs

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Planar unclustered scale-free graphs as models for technological and biological networks

Many real life networks present an average path length logarithmic with the number of nodes and a degree distribution which follows a power law. Often these networks have also a modular and self-similar structure and, in some cases - usually associated with topological restrictions- their clustering is low and they are almost planar. In this paper we introduce a family of graphs which share all these properties and are defined by two parameters. As their construction is deterministic, we obtain exact analytic expressions for relevant properties of the graphs including the degree distribution, degree correlation, diameter, and average distance, as a function of the two defining parameters. Thus, the graphs are useful to model some complex networks, in particular several families of technological and biological networks, and in the design of new practical communication algorithms in relation to their dynamical processes. They can also help understanding the underlying mechanisms that have produced their particular structure.Comment: Accepted for publication in Physica