Title: Community Recovery in a Preferential Attachment Graph,
Abstract: A message passing algorithm is derived for recovering communities within a
graph generated by a variation of the Barabási-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.