Attribute network models, stochastic approximation, and network sampling and ranking algorithms

TL;DR

This paper studies dynamic attribute-based random network models, showing their convergence to infinite trees and deriving asymptotics for network functionals like degree distribution and PageRank, with implications for sampling and ranking algorithms.

Contribution

It introduces a stochastic approximation framework for analyzing attribute-dependent network growth and derives explicit asymptotics for network functionals in the large network limit.

Findings

Degree tail exponents depend on attribute types.

PageRank scores have universal tail exponents across attributes.

PageRank and walk-based sampling are effective for rare minority detection.

Abstract

We analyze dynamic random network models where younger vertices connect to older ones with probabilities proportional to their degrees as well as a propensity kernel governed by their attribute types. Using stochastic approximation techniques we show that, in the large network limit, such networks converge in the local weak sense to limiting infinite random trees with an explicit description in terms of randomly stopped multi-type branching processes. This allows for the derivation of asymptotics for a wide class of network functionals implying, for example, that while degree distribution tail exponents depend on the attribute type (already derived by Jordan (2013)), PageRank centrality scores have the same tail exponent across attributes. The limit results also give explicit formulae for the performance of various network sampling mechanisms. One surprising consequence is the efficacy of PageRank and walk based network sampling schemes for directed networks in the setting of rare minorities.