On leaders and condensates in a growing network

Details On leaders and condensates in a growing network On leaders and condensates in a growing network

2010-06-03

arxiv.org/abs/1006.0587v2

J. Stat. Mech. (2010) P07031

Physics

Condensed Matter

Statistical Mechanics

29 pages, 13 figures, 3 tables. A few minor changes

Scientific paper

10.1088/1742-5468/2010/07/P07031

The Bianconi-Barabasi model of a growing network is revisited. This model, defined by a preferential attachment rule involving both the degrees of the nodes and their intrinsic fitnesses, has the fundamental property to undergo a phase transition to a condensed phase below some finite critical temperature, for an appropriate choice of the distribution of fitnesses. At high temperature it exhibits a crossover to the Barabasi-Albert model, and at low temperature, where the fitness landscape becomes very rugged, a crossover to the recently introduced record-driven growth process. We first present an analysis of the history of leaders, the leader being defined as the node with largest degree at a given time. In the generic finite-temperature regime, new leaders appear endlessly, albeit on a doubly logarithmic time scale, i.e., extremely slowly. We then give a novel picture for the dynamics in the condensed phase. The latter is characterized by an infinite hierarchy of condensates, whose sizes are non-self-averaging and keep fluctuating forever.

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