Abstract
Simon's classical random-copying model, introduced in 1955, has garnered much attention for its ability, in spite of an apparent simplicity, to produce characteristics similar to those observed across the spectrum of complex systems. Through a discrete-time mechanism in which items are added to a sequence based upon rich-gets-richer dynamics, Simon demonstrated that the resulting size distributions of such sequences exhibit power-law tails. The simplicity of this model arises from the approach by which copying occurs uniformly over all previous elements in the sequence. Here we propose a generalization of this model which moves away from this uniform assumption, instead incorporating memory effects that allow the copying event to occur via an arbitrary age-dependent kernel. Through this approach we first demonstrate the potential to determine further information regarding the structure of sequences from the classical model before illustrating, via analytical study and numeric simulation, the flexibility offered by the arbitrary choice of memory. Furthermore we demonstrate how previously proposed memory-dependent models can be further studied as specific cases of the proposed framework.
Highlights
Within natural systems from an assortment of domains there are underlying properties which are found to consistently appear
The original representation describes an author creating a body of text that is the population of interest, with every word used representing an element:each unique word corresponds to a variant in the neutral model framework
How the variant of this element is chosen occurs through a probabilistic framework whereby there may be a mutation with probability (w.p.) μ such that this new element is a new variant in the population
Summary
Within natural systems from an assortment of domains there are underlying properties which are found to consistently appear. Such techniques have previously proven extremely useful in representing similar randomcopying phenomena in online diffusion scenarios [35,36,37,38,39,40] We take this approach in describing the classical Simon’s model and demonstrate how the interpretation allows numerous quantities describing the distribution of abundances within the process to be obtained in an analytically tractable manner. We proceed to use a similar framework to study a general memory function which depends upon the time t − τ elapsed since an element appeared to produce a generalized Simon’s model (GSM) and demonstrate how statistical properties of abundances arising through such a process may be obtained for an arbitrary choice of said memory function.
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