A record-driven growth process

TL;DR

This paper introduces a new stochastic growth process where nodes connect to the highest quality node, revealing universal properties like the Golomb-Dickman constant and exhibiting self-similarity in network growth.

Contribution

The paper presents the record-driven growth process, linking network growth to record statistics and revealing universal asymptotic behaviors and connections to combinatorial problems.

Findings

Probability of a node being a leader equals the Golomb-Dickman constant

The process exhibits temporal self-similarity at late times

Connections established with permutation cycles and broken intervals

Abstract

We introduce a novel stochastic growth process, the record-driven growth process, which originates from the analysis of a class of growing networks in a universal limiting regime. Nodes are added one by one to a network, each node possessing a quality. The new incoming node connects to the preexisting node with best quality, that is, with record value for the quality. The emergent structure is that of a growing network, where groups are formed around record nodes (nodes endowed with the best intrinsic qualities). Special emphasis is put on the statistics of leaders (nodes whose degrees are the largest). The asymptotic probability for a node to be a leader is equal to the Golomb-Dickman constant omega=0.624329... which arises in problems of combinatorical nature. This outcome solves the problem of the determination of the record breaking rate for the sequence of correlated inter-record intervals. The process exhibits temporal self-similarity in the late-time regime. Connections with the statistics of the cycles of random permutations, the statistical properties of randomly broken intervals, and the Kesten variable are given.