Information Ranking and Power Laws on Trees

TL;DR

This paper investigates conditions under which solutions to weighted stochastic recursions on trees exhibit power law tails, introducing new analytical methods that extend existing theorems and enable large deviations analysis.

Contribution

It develops two novel approaches—extending Goldie's implicit renewal theorem and applying large deviations—to analyze power law tails in recursive tree processes.

Findings

Extended Goldie's theorem to recursions on trees

Established conditions for power law tails in weighted recursive processes

Provided new tools for analyzing weighted branching processes

Abstract

We study the situations when the solution to a weighted stochastic recursion has a power law tail. To this end, we develop two complementary approaches, the first one extends Goldie's (1991) implicit renewal theorem to cover recursions on trees; and the second one is based on a direct sample path large deviations analysis of weighted recursive random sums. We believe that these methods may be of independent interest in the analysis of more general weighted branching processes as well as in the analysis of algorithms.