Abstract
Continuous-time branching processes describe the evolution of a population whose individuals generate a random number of children according to a birth process. Such branching processes can be used to understand preferential attachment models in which the birth rates are linear functions. We are motivated by citation networks, where power-law citation counts are observed as well as aging in the citation patterns. To model this, we introduce fitness and age-dependence in these birth processes. The multiplicative fitness moderates the rate at which children are born, while the aging is integrable, so that individuals receives a finite number of children in their lifetime. We show the existence of a limiting degree distribution for such processes. In the preferential attachment case, where fitness and aging are absent, this limiting degree distribution is known to have power-law tails. We show that the limiting degree distribution has exponential tails for bounded fitnesses in the presence of integrable aging, while the power-law tail is restored when integrable aging is combined with fitness with unbounded support with at most exponential tails. In the absence of integrable aging, such processes are explosive.
Highlights
Preferential attachment models (PAMs) aim to describe dynamical networks
While we do not know precisely what the necessary and sufficient conditions are on the aging and the fitness distribution to assure a power-law degree distribution, our results suggests that affine PA weights with integrable aging and fitnesses with at most an exponential tail in general do so, a feature that was not observed before
2.4 Dynamical Power-Laws for Exponential Fitness and Integrable Aging In Sect. 5.3 we introduce three different classes of fitness distributions, for which we give the asymptotics for the limiting degree distribution of the corresponding continuous-time branching processes (CTBP)
Summary
Preferential attachment models (PAMs) aim to describe dynamical networks. As for many real-world networks, PAMs present power-law degree distributions that arise directly from the dynamics, and are not artificially imposed as, for instance, in configuration models or inhomogeneous random graphs. These models show the so-called old-get-richer effect, meaning that the vertices of highest degrees are the vertices present early in the network formation An extension of this model is called preferential attachment models with a random number of edges [8], where new vertices are added to the graph with a different number of edges according to a fixed distribution, and again power-law degree sequences arise. The idea is that old papers are less likely to be cited than new papers Such aging has been observed in many citation network datasets and makes PAMs with weight functions depending only on the degree ill-suited for them. Motivated by this and the wish to understand the qualitative behavior of PAMs with general aging and fitness, the starting point of our model is the CTBP or tree setting. Because of its motivating role in this paper, let us discuss the empirical properties of citation networks in detail
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