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. ∗Research supported by the Secretaria de Estado de Universidades e Investigación (Ministerio de Educación y Ciencia), Spain, and the European Regional Development Fund (ERDF) under project TEC2005-03575