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Marked random graphs with given degree sequence: large deviations on the local topology

Rangel Baldasso, Alan Pereira, Guilherme Reis

Abstract

We investigate the behavior of the empirical neighbourhood distribution of marked graphs in the framework of local weak convergence. We establish a large deviation principle for such families of empirical measures. The proof builds on Bordenave and Caputo's seminal 2015 paper, and Delgosha and Anantharam's 2019 introduction of BC entropy, relying on combinatorial lemmas that allow one to construct suitable approximations of measures supported on marked trees.

Marked random graphs with given degree sequence: large deviations on the local topology

Abstract

We investigate the behavior of the empirical neighbourhood distribution of marked graphs in the framework of local weak convergence. We establish a large deviation principle for such families of empirical measures. The proof builds on Bordenave and Caputo's seminal 2015 paper, and Delgosha and Anantharam's 2019 introduction of BC entropy, relying on combinatorial lemmas that allow one to construct suitable approximations of measures supported on marked trees.
Paper Structure (13 sections, 12 theorems, 69 equations)

This paper contains 13 sections, 12 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Motivation.
  3. Overview of the proof.
  4. Related works
  5. Preliminaries
  6. Marked graphs
  7. Review of BC entropy and LDP
  8. Main result and proof
  9. Uniform marked graphs with given degree sequence.
  10. Proof of Theorem \ref{['theorem:main']}.
  11. Colored Configuration Model
  12. Proof of Lemma \ref{['lemma:modification']}
  13. "Mass transport" on matrices of colored degree

Key Result

Theorem 1.1

Consider the sequence of marked graphs $\bar{G}_n$ as above. The sequence of empirical neighborhood measures $U(\bar{G}_n)$ satisfies a large deviation principle with rate function presented in Theorem theorem:main.

Theorems & Definitions (25)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Definition 2.2: Local weak convergence
  • Proposition 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6: da, Theorem 2
  • Remark 3.1
  • Theorem 3.2
  • ...and 15 more