A version of Herbert A. Simon's model with slowly fading memory and its connections to branching processes

Abstract

Construct recursively a long string of words w1. .. wn, such that at each step k, w k+1 is a new word with a fixed probability p $\in$ (0, 1), and repeats some preceding word with complementary probability 1 -- p. More precisely, given a repetition occurs, w k+1 repeats the j-th word with probability proportional to j $\alpha$ for j = 1,. .. , k. We show that the proportion of distinct words occurring exactly times converges as the length n of the string goes to infinity to some probability mass function in the variable $\ge$ 1, whose tail decays as a power function when 1 -- p > $\alpha$/(1 + $\alpha$), and exponentially fast when 1 -- p < $\alpha$/(1 + $\alpha$).

References in corpus (8)

• Evolution of reference networks with aging
• World-Wide Web scaling exponent from Simon's 1955 model
• A Yule-Simon process with memory
• The dynamics of power laws: Fitness and aging in preferential attachment trees
• Random Graphs Associated to some Discrete and Continuous Time Preferential Attachment Models
• Decay of Relevance in Exponentially Growing Networks
• Generalized Nonlinear Yule Models
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Cited by in corpus (2)

• On a class of random walks with reinforced memory
• Memory-cognizant generalization to Simon's random-copying neutral model