Simon's fundamental rich-get-richer model entails a dominant first-mover advantage

Abstract

Herbert Simon's classic rich-get-richer model is one of the simplest empirically supported mechanisms capable of generating heavy-tail size distributions for complex systems. Simon argued analytically that a population of flavored elements growing by either adding a novel element or randomly replicating an existing one would afford a distribution of group sizes with a power-law tail. Here, we show that, in fact, Simon's model does not produce a simple power law size distribution as the initial element has a dominant first-mover advantage, and will be overrepresented by a factor proportional to the inverse of the innovation probability. The first group's size discrepancy cannot be explained away as a transient of the model, and may therefore be many orders of magnitude greater than expected. We demonstrate how Simon's analysis was correct but incomplete, and expand our alternate analysis to quantify the variability of long term rankings for all groups. We find that the expected time for a first replication is infinite, and show how an incipient group must break the mechanism to improve their odds of success. We present an example of citation counts for a specific field that demonstrates a first-mover advantage consistent with our revised view of the rich-get-richer mechanism. Our findings call for a reexamination of preceding work invoking Simon's model and provide an expanded understanding going forward.

arXiv comment8 pages, 3 figures

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• Computational Paremiology: Charting the temporal, ecological dynamics of proverb use in books, news articles, and tweets
• Markov Chain Monte Carlo for generating ranked textual data
• Continuum rich-get-richer processes: Mean field analysis with an application to firm size
• Memory-cognizant generalization to Simon's random-copying neutral model