James P Bagrow et al 2008 J. Phys. A: Math. Theor. 41 185001 doi:10.1088/1751-8113/41/18/185001
Phase transition in the rich-get-richer mechanism due to finite-size effects
James P Bagrow1, Jie Sun2 and Daniel ben-Avraham1
Show affiliationsThe rich-get-richer mechanism (agents increase their 'wealth' randomly at a rate proportional to their holdings) is often invoked to explain the Pareto power-law distribution observed in many physical situations, such as the degree distribution of growing scale-free nets. We use two different analytical approaches, as well as numerical simulations, to study the case where the number of agents is fixed and finite (but large), and the rich-get-richer mechanism is invoked a fraction r of the time (the remainder of the time wealth is disbursed by a homogeneous process). At short times, we recover the Pareto law observed for an unbounded number of agents. In later times, the (moving) distribution can be scaled to reveal a phase transition with a Gaussian asymptotic form for 
, and a Pareto-like tail (on the positive side) and a novel stretched exponential decay (on the negative side) for 
.
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Issue 18 (9 May 2008)
Received 13 December 2007, in final form 18 March 2008
Published 18 April 2008
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Phase transition in the rich-get-richer mechanism due to finite-size effects
James P Bagrow et al 2008 J. Phys. A: Math. Theor. 41 185001
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