Average path length in uncorrelated random networks with hidden variables

Analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found. We apply the approach to classical random graphs of Erdös and Rényi (ER), evolving networks introduced by Barabási and Albert as well as random networks with asymptotic scale-free connectivity distributions characterized by an arbitrary scaling exponent α > 2. Our result for 2 < α < 3 shows that structural properties of asymptotic scale-free networks including numerous examples of real-world systems are even more intriguing then ultra-small world behavior noticed in pure scale-free structures and for large system sizes N → ∞ there is a saturation effect for the average path length. During the last few years random, evolving networks have become a very popular research domain among physicists [1, 2, 3, 4]. A lot of efforts were put into investigation of such systems, in order to recognize their structure and to analyze emerging complex properties. It was observed that despite network diversity, most of real web-like systems share three prominent structural features: small average path length (AP L), high clustering and scale-free (SF) degree distribution [1, 2, 3, 4, 5]. Several network topology generators have been proposed to embody the fundamental characteris-To find out how the small-world property (i.e. small AP L) arises, the idea of shortcuts has been proposed by Watts and Strogatz [13]. To understand where the ubiquity of scale-free distributions in real networks comes from, the concept of evolving networks basing on preferential attachment has been introduced by Barabási and Albert [6]. Recently Calderelli and coworkers [12] have presented another mechanism that accounts for origins of power-law connectivity distributions. It is interesting that the mechanism is neither related to dynamical properties nor to preferential attachment. Caldarelli et al. have studied a simple static network model in which each vertex i has assigned a tag h i (fitness, hidden variable) randomly drawn from a fixed probability distribution ρ(h). In their fitness model, edges are assigned to pairs of vertices with a given connection probability p ij , depending on the values of the tags h i and h j assigned at the edge end points. Similar models have been also analyzed in several other studies [14, 15, 16]. A generalization of the above-mentioned network models has been recently proposed by Boguñá and Pastor-Satorras [17]. In the cited paper, the authors have argued that such diverse networks like …