A Random Walk Approach to Broadcasting on Random Recursive Trees

2024-05-07

arXiv.org

AI Summary

This paper investigates the noisy broadcast problem on random recursive trees: a ±1 message emitted from an unknown root propagates along edges with flip probability $q$, and only the leaf-node values are observable, with propagation order unknown. Addressing this robust inference task, we extend the analysis of the majority estimator—previously limited to uniform or linear preferential attachment trees—to two broad generalizations: minimal-growth trees and shape-exchangeable trees. We uncover deep connections between the broadcast process and non-homogeneous random walks, memory-augmented Pólya’s urn schemes, and branching Markov chains. Using asymptotic probabilistic analysis and tree-generation models, we derive precise asymptotics for the success probability of the majority estimator. Crucially, we prove that it achieves constant-recovery performance across a wide class of tree structures, significantly broadening the theoretical scope beyond prior restrictive assumptions on tree growth mechanisms.

Technology Category

• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Probabilistic Circuits and Graphical Models
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

In the broadcasting problem on trees, a ${0,1}$-message originating in an unknown node is passed along the tree with a certain error probability $q$. The goal is to estimate the original message without knowing the order in which the nodes were informed. A variation of the problem is considering this broadcasting process on a randomly growing tree, which Addario-Berry et al. have investigated for uniform and linear preferential attachment recursive trees. We extend their studies of the majority estimator to the entire group of very simple increasing trees as well as shape exchangeable trees using the connection to inhomogeneous random walks and other stochastic processes with memory effects such as P'olya Urns.

Problem

• Estimating original message from noisy tree broadcasts
• Analyzing majority estimator on random recursive trees
• Extending analysis to simple increasing and shape-exchangeable trees

Innovation

• Random walk with memory effects
• Majority estimator on trees
• Analyzing simple increasing trees