The Dynamics of Power laws: Fitness and Aging in Preferential Attachment Trees

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.

Does the hα-index reinforce the Matthew effect in science? The introduction of agent-based simulations into scientometrics

“…Meyer, Lorscheid, and Troitzsch (2009) provide a bibliometric analysis of the first decade of the Journal of Artificial Societies and Social Simulations (JASSS). The Matthews effect itself has extensively been simulated (for example, in physics) under the heading of preferential attachment (Abbasi, Hossain, & Leydesdorff, 2012;Barabási, 2002;Barabási et al, 2002;Bonitz et al, 1999;Garavaglia, van der Hofstad, & Woeginger, 2017;Newman, 2001;Petersen et al, 2014).…”

Studies on generalized Yule models

“…with density Figure 1 shows the shapes of the distribution function (17) and the density function (18) for different values of the characterizing parameter ν. Notice the rather different behaviour for values of ν strictly less than 1.…”

“…Distribution function(17) (top) and density function(18) in linear plot (middle) and loglog plot (bottom). The parameter ν is set to ν = (1/4, 1/2, 3/4, 1) = (blue, orange, green, red) and t = 1.…”

From trees to graphs: collapsing continuous-time branching processes

“…An extension of the PAM has been proposed by us in [11], where we introduce fitness and aging in preferential attachment trees. The methodology used in the present work is applicable to the case with aging only.…”

“…that can be rewritten as in (1.4) using Γ functions, since in this case α * = 1 + δ/m (see [18,Section 4.2], [11,Proposition 3.15]). For the random recursive graph, calculations are easier.…”