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Limiting Behavior of Degree-Degree Metrics under Local Convergence in Probability

Andrei-Eugeniu Patularu, Pim van der Hoorn

TL;DR

This work studies the limiting behavior of degree-degree metrics under local convergence in probability for sequences of sparse random graphs. It develops general limit theorems for Pearson's $r$, Spearman's $\\rho$, Kendall's $\\tau$, the degree-distance $\\delta$, and the two local measures ANND and ANNR, expressing limits in terms of the limiting rooted graph $(G,o)$ with law $\\mu$ and a uniformly chosen neighbor $V$. The results are then applied to rank-1 inhomogeneous random graphs and random geometric graphs, yielding explicit ANND limits and, in the geometric case, a closed-form limit for Pearson's correlation, highlighting neutral vs. assortative mixing in these models. Overall, the paper provides a unified framework to analyze local-degree correlations across a broad class of sparse graph models with practical implications for network analysis and modeling.

Abstract

This paper investigates the limiting behaviour of degree-degree correlation metrics for sequences of random graphs under a general assumption of local convergence in probability. We establish convergence results for Pearson's correlation coefficient r, Spearman's rho, Kendall's tau, average nearest neighbour degree (ANND), and average nearest neighbour rank (ANNR). Our results explicitly show how the limits of these degree-degree correlation metrics depend on the local structure of the graph. We then apply our general results to study degree-degree correlations in rank-1 inhomogeneous random graphs and random geometric graphs, deriving explicit expressions for ANND in both models and for Pearson's correlation coefficient in the latter one. Keywords: random graphs, degree-degree metrics, neutral mixing

Limiting Behavior of Degree-Degree Metrics under Local Convergence in Probability

TL;DR

This work studies the limiting behavior of degree-degree metrics under local convergence in probability for sequences of sparse random graphs. It develops general limit theorems for Pearson's , Spearman's , Kendall's , the degree-distance , and the two local measures ANND and ANNR, expressing limits in terms of the limiting rooted graph with law and a uniformly chosen neighbor . The results are then applied to rank-1 inhomogeneous random graphs and random geometric graphs, yielding explicit ANND limits and, in the geometric case, a closed-form limit for Pearson's correlation, highlighting neutral vs. assortative mixing in these models. Overall, the paper provides a unified framework to analyze local-degree correlations across a broad class of sparse graph models with practical implications for network analysis and modeling.

Abstract

This paper investigates the limiting behaviour of degree-degree correlation metrics for sequences of random graphs under a general assumption of local convergence in probability. We establish convergence results for Pearson's correlation coefficient r, Spearman's rho, Kendall's tau, average nearest neighbour degree (ANND), and average nearest neighbour rank (ANNR). Our results explicitly show how the limits of these degree-degree correlation metrics depend on the local structure of the graph. We then apply our general results to study degree-degree correlations in rank-1 inhomogeneous random graphs and random geometric graphs, deriving explicit expressions for ANND in both models and for Pearson's correlation coefficient in the latter one. Keywords: random graphs, degree-degree metrics, neutral mixing
Paper Structure (26 sections, 15 theorems, 108 equations, 1 figure)

This paper contains 26 sections, 15 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Local convergence
  4. Statement of main results
  5. Known result for Pearson's correlation coefficient
  6. Spearman's rho
  7. Kendall's tau
  8. Degree-degree distance
  9. Average nearest neighbor degree and rank
  10. Applications
  11. Rank-1 Inhomogeneous Random Graphs
  12. Random Geometric Graphs
  13. Proofs of main results
  14. General idea and approach
  15. Spearman's rho
  16. ...and 11 more sections

Key Result

Theorem 2.2

Let $(G_{n})_{n \ge 1}$ be a sequence of graphs, where $|V(G_{n})|$ tends to infinity and $(d_{o_n}^3)_{n \ge 1}$ is uniformly integrable. Let $(G,o)$ be a random variable in $\mathcal{G}^\ast$ with law $\mu$, such that $\mathbb{P}\left(d_o \ge 1\right) > 0$. Assume that $G_n$ converges in probabili where $V$ is a uniformly chosen neighbour of root $o$ in graph $(G, o)$.

Figures (1)

  • Figure 1: Joint neighborhood of the root $o$ and uniform neighbor $V$ at distance $r$ for a $2$-dimensional Random Geometric Graph with radius $R$. The green areas contain the first type of neighbors and the purple area contains the joint neighbors (second type).

Theorems & Definitions (38)

  • Definition 2.1: Rooted graphs
  • Definition 2.2: Neighborhood of the root
  • Definition 2.3: Rooted isomorphisms
  • Definition 2.4
  • Remark 2.1
  • Theorem 2.2: hofstad2024random Theorem 2.26
  • Theorem 2.3
  • Theorem 2.4
  • Definition 2.5
  • Theorem 2.5
  • ...and 28 more