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-getricher mechanism. Our findings call for a reexamination of preceding work invoking Simon's model and provide an expanded understanding going forward.

Early life imprints the hierarchy of T cell clone sizes

“…We exactly simulated a system of reduced size to validate the mean-field approximation. The distributions of the exact and mean-field simulations agree to within stochasticity ( Figure 5), with the exception of the largest clone, which is larger in the exact simulations as has been discussed elsewhere (Dodds et al, 2017).…”

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

“…One final point we comment on regarding this moment is the average number of appearances for the variant describing the introductory element seeded at time τ = 1, which is given by (1/Ω) − (1−µ) . This expression is in general agreement with the analysis found in [19] (assuming that µ 1 such that 1/Γ(2−µ) ≈ 1) demonstrating the intrinsic advantage offered to the first variant to appear in a realization of Simon's model.…”

“…The fundamental property of this model is that the likelihood of the next element being of a certain variant is dependent upon the number of previous occurrences of this variant. Notwithstanding the model's apparent simplicity, it has been shown to accurately describe the distribution of abundances within a number of complex systems, including the citation dynamics of scientific literature [18,19], family-names [20], and the growth of both the world wide web [21] and open-source software developments [22].…”

“…Remarkably, in spite of the model's supposed straightforwardness, the research community is still providing new insights into its behaviors. For example, analytical quantification of the first-mover advantage offered towards the initial variant in Simon's model was demonstrated in [19], while the effect of averaging across multiple realizations of a closely related model upon the distribution of larger abundances, denoted as peloton dynamics, have also been considered [27]. There have also been a number of studies which have incorporated memory-effects within Simon's model such that the likelihood of a variant being copied is no longer uniform across all previous elements but rather is dependent upon how long ago the variant was last used.…”

Calibration of a density-based model of urban morphogenesis

“…The two contradictory processes of urban concentration and urban sprawl are captured by the model, what allows to reproduce with a good precision a large number of existing morphologies. We can expect aggregation mechanisms such as preferential attachment to be good candidates in urban growth explanation, as it was shown that the Simon model based on these generates power-laws typical of urban systems such as scaling laws [ 28 ]. The question at which scale it is possible and relevant to define and try to simulate urban form is rather open, and will in fact depend on which issues are being tackled.…”