Dynamical fitness models: evidence of universality classes for preferential attachment graphs

Abstract

In this paper we define a family of preferential attachment models for random graphs with fitness in the following way: independently for each node, at each time step a random fitness is drawn according to the position of a moving average process with positive increments. We will define two regimes in which our graph reproduces some features of two wellknown preferential attachment models: the Bianconi-Barabási and the Barabási-Albert models. We will discuss a few conjectures on these models, including the convergence of the degree sequence and the appearance of Bose-Einstein condensation in the network when the drift of the fitness process has order comparable to the graph size.

Distance evolutions in growing preferential attachment graphs

“…[22,23] consider a version where the attachment function can be sublinear in the degree. In [18,20,24,28] vertices are equipped with a fitness and in [38] the arriving vertices have a power of choice. Spatial variants where vertices have a location in an underlying Euclidean space are studied in [2,35,36].…”

“…We have initiated work in this direction and hope to communicate results on it soon. Some frequently studied global properties are the size of the giant component and its robustness against site or edge percolation [23,26,30,36], and condensation phenomena [10,18,20,28,38]. 1.3.…”