A Yule - Simon process with memory

– The Yule-Simon model has been used as a tool to describe the growth of diverse systems, acquiring a paradigmatic character in many fields of research. Here we study a modified Yule-Simon model that takes into account the full history of the system by means of an hyperbolic memory kernel. We show how the memory kernel changes the properties of preferential attachment and provide an approximate analytical solution for the frequency distribution density as well as for the frequency-rank distribution. In 1925 Yule [1] proposed a model to explain experimental data on the abundances of biological genera [2]. Thirty years later, Simon introduced an elegant copy and growth model [3], in spirit equivalent to Yule’s model, to explain the observed power-law distribution of word frequencies in texts [4–6]. In Simon’s growth model, new words are added to a text (more generally a stream) with constant probability p at each time step, whereas with complementary probability p̄ = 1 − p an already occurred word is chosen uniformly from within the already formed text (stream). This model yields a power-law distribution density for word frequencies P (k) ∼ k with β = 1 + 1/p̄ . The same mechanism is at play in the preferential attachment (PA) model for growing networks proposed, in their pioneering article, by Barabási and Albert [7]. In that case, a network is constructed by progressively adding new nodes and linking them to existing nodes with a probability proportional to their current connectivity. Yule-Simon processes and PA schemes are closely related to each other and a mapping between them has been provided by Bornholdt and Ebel [8]. In the original Yule-Simon process, the metaphor of text construction is somehow misleading because in that process there is no notion of temporal ordering. All existing words are equivalent and in many respects everything goes as in a Polya urn model [9]. However, the notion of temporal ordering may play an important role in determining the dynamics of many real systems. In this perspective it is interesting to investigate models where temporal ordering is explicitly taken into account. A first attempt in this direction has been provided by Dorogovtsev and Mendes (DM) [10], who studied a generalization of the Barabási-Albert