Abstract
Preferential attachment drives the evolution of many complex networks. Its analytical studies mostly consider the simplest case of a network that grows uniformly in time despite the accelerating growth of many real networks. Motivated by the observation that the average degree growth of nodes is time invariant in empirical network data, we study the degree dynamics in the relevant class of network models where preferential attachment is combined with heterogeneous node fitness and aging. We propose an analytical framework based on the time invariance of the studied systems and show that it is self-consistent only for two special network growth forms: the uniform and the exponential network growth. Conversely, the breaking of such time invariance explains the winner-takes-all effect in some model settings, revealing the connection between the Bose-Einstein condensation in the Bianconi-Barabási model and similar gelation in superlinear preferential attachment. Aging is necessary to reproduce realistic node degree growth curves and can prevent the winner-takes-all effect under weak conditions. Our results are verified by extensive numerical simulations.
Highlights
The original work on the preferential attachment network growth mechanism [1] has importantly contributed to the formation of the interdisciplinary field of network science [2,3]
We show that an accelerated network growth, a feature that is common in real networks [14,15,16] yet usually overlooked by network modeling, is an important part of the interplay between preferential attachment and the macroscopic degree growth patterns
We introduce a mathematical formalism for growing networks with the timeinvariant degree growth as a fundamental assumption, but without an assumption on the network growth form in the first place
Summary
The original work on the preferential attachment network growth mechanism [1] has importantly contributed to the formation of the interdisciplinary field of network science [2,3]. We build our work on the observation that in many realworld networks, using citation networks as an example here, the degree growth is time invariant: the average degree of nodes of different ages has the same functional dependency on node age regardless of when the nodes have entered the network. This seemingly minor observation is not trivial. Of the different growth forms that can be considered, the exponential network growth is consistent with the time-invariant degree growth
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