Abstract
We obtain the degree distribution for a class of growing network models on flat and curved spaces. These models evolve by preferential attachment weighted by a function of the distance between nodes. The degree distribution of these models is similar to that of the fitness model of Bianconi and Barabási, with a fitness distribution dependent on the metric and the density of nodes. We show that curvature singularities in these spaces can give rise to asymptotic Bose-Einstein condensation, but transient condensation can be observed also in smooth hyperbolic spaces with strong curvature. We provide numerical results for spaces of constant curvature (sphere, flat, and hyperbolic space) and we discuss the conditions for the breakdown of this approach and the critical points of the transition to distance-dominated attachment. Finally, we discuss the distribution of link lengths.
Highlights
Scale-free networks have attracted a wide interest as models for many systems
The fitness model of Bianconi and Barabasi [2] is the basic example of a network model where the attachment depends on the number of links and on another variable, namely the quality of the nodes
This model predicts multiple power-law scaling in the degree distribution and a phase with Bose-Einstein condensation on nodes with high fitness [3] and it has been applied to the WWW [12, 13], where Google could be an interesting example of an emerging condensate
Summary
Scale-free networks have attracted a wide interest as models for many systems Their degree distribution can be explained by the mechanism of preferential attachment in growing networks [1]. The fitness model of Bianconi and Barabasi [2] is the basic example of a network model where the attachment depends on the number of links and on another variable, namely the quality of the nodes This model predicts multiple power-law scaling in the degree distribution and a phase with Bose-Einstein condensation on nodes with high fitness [3] and it has been applied to the WWW [12, 13], where Google could be an interesting example of an emerging condensate
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