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

Nelson Antunes, Sayan Banerjee, Shankar Bhamidi, Vladas Pipiras

TL;DR

This work analyzes attributed evolving networks with preferential attachment modulated by node attributes and a kernel κ, establishing local weak convergence to stopping multi-type branching processes that describe the limiting local geometry. The authors develop a resolvability principle enabling transfer of asymptotics from a tractable U model to the original P model, and derive comprehensive asymptotics for both local (degree tails PageRank) and global functionals (maximal degree). They extend results to non-tree and uniform attachment regimes and apply the theory to network sampling, revealing how PageRank and walk-based sampling can mitigate minority bias in rare attribute settings. The results yield explicit tail exponents for PageRank independent of type, type-dependent degree tails, and detailed sampling bias formulas, with significant implications for fairness and inference in large-scale attributed networks.

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.

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

TL;DR

This work analyzes attributed evolving networks with preferential attachment modulated by node attributes and a kernel κ, establishing local weak convergence to stopping multi-type branching processes that describe the limiting local geometry. The authors develop a resolvability principle enabling transfer of asymptotics from a tractable U model to the original P model, and derive comprehensive asymptotics for both local (degree tails PageRank) and global functionals (maximal degree). They extend results to non-tree and uniform attachment regimes and apply the theory to network sampling, revealing how PageRank and walk-based sampling can mitigate minority bias in rare attribute settings. The results yield explicit tail exponents for PageRank independent of type, type-dependent degree tails, and detailed sampling bias formulas, with significant implications for fairness and inference in large-scale attributed networks.

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.
Paper Structure (40 sections, 32 theorems, 162 equations, 3 figures)

This paper contains 40 sections, 32 theorems, 162 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation for this work and analysis
  3. Organization of the paper
  4. Constructions and basic definitions
  5. The $\mathscrbf{U}$ Model
  6. Local weak convergence for trees
  7. Extended fringe decomposition for marked trees
  8. Convergence on the space of trees
  9. Local weak convergence for directed graphs
  10. Functionals of interest
  11. Main Results: Tree networks
  12. Resolvability
  13. Local weak convergence for finite attribute models
  14. Asymptotics for degree distribution, PageRank scores and homophily measures
  15. Asymptotics for global functionals
  16. ...and 25 more sections

Key Result

Lemma 2.2

Assume $\mathcal{S}$ is compact and $\kappa(\cdot, \cdot)$ is bounded. Then the above branching does not explode i.e. $T_n\stackrel{\mathrm{a.s.}}{\longrightarrow} \infty$ as $n\to\infty$. Let $\left\{\mathcal{G}_n:n\geq 0\right\} \equiv \left\{\mathop{\mathrm{BP}}\nolimits(T_n):n\geq 0\right\}$ den

Figures (3)

  • Figure 2.1: Fringe decomposition around vertex $v$ of a finite tree rooted at $\rho$. Here the blue colors represent roots of the respecitve trees.
  • Figure 2.2: A sin-tree $\mathcal{T}_\infty$, namely a tree rooted at $0$ with a single infinite path to infinity, and the corresponding extended fringe $F_3(0,\mathcal{T}_\infty)$ upto level three about $0$.
  • Figure 2.3:

Theorems & Definitions (63)

  • Definition 1.1: Attributed evolving network model class $\mathscrbf{P}$
  • Definition 2.1: Network model class $\mathscrbf{U}$
  • Lemma 2.2
  • Definition 2.3: Local weak convergence
  • Definition 2.4: Fringe distribution aldous-fringe
  • Lemma 2.5
  • Definition 2.6: Local weak convergence for directed graphs
  • Definition 2.7: PageRank scores with damping factor $c$
  • Definition 3.1: Resolvability
  • Lemma 3.3: jordan2013geometric
  • ...and 53 more