Memory-cognizant generalization to Simon's random-copying neutral model

TL;DR

This paper introduces a generalized version of Simon's random-copying model that incorporates memory effects with arbitrary age-dependent kernels, enabling more flexible modeling of sequence structures beyond uniform copying.

Contribution

It extends Simon's classical model by allowing non-uniform, memory-dependent copying, providing a framework to analyze and simulate more complex sequence dynamics.

Findings

Demonstrates how memory effects influence sequence structure

Shows the model's flexibility with arbitrary memory kernels

Links previous memory-dependent models as special cases

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.