Community Recovery in a Preferential Attachment Graph

TL;DR

This paper introduces a message passing algorithm for community detection in preferential attachment graphs, leveraging vertex arrival times, and compares its performance with simpler heuristics through theoretical analysis and simulations.

Contribution

It develops a novel message passing algorithm for community recovery that uses arrival times, improving upon degree thresholding and child arrival-based methods.

Findings

Message passing algorithm outperforms degree thresholding.

Knowing arrival times improves classification accuracy.

Simulation results validate theoretical predictions.

Abstract

A message passing algorithm is derived for recovering communities within a graph generated by a variation of the Barab\'{a}si-Albert preferential attachment model. The estimator is assumed to know the arrival times, or order of attachment, of the vertices. The derivation of the algorithm is based on belief propagation under an independence assumption. Two precursors to the message passing algorithm are analyzed: the first is a degree thresholding (DT) algorithm and the second is an algorithm based on the arrival times of the children (C) of a given vertex, where the children of a given vertex are the vertices that attached to it. Comparison of the performance of the algorithms shows it is beneficial to know the arrival times, not just the number, of the children. The probability of correct classification of a vertex is asymptotically determined by the fraction of vertices arriving before it. Two extensions of Algorithm C are given: the first is based on joint likelihood of the children of a fixed set of vertices; it can sometimes be used to seed the message passing algorithm. The second is the message passing algorithm. Simulation results are given.