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The connectivity and phase transition in inhomogeneous random graphs of finite types

Hamin Jung

Abstract

A significant generalization of the Erdös-Rényi random graph model is an `inhomogeneous' random graph where the edge probabilities vary according to vertex types. We identify the threshold value for this random graph with a finite number of vertex types to be connected and examine the model's behavior near this threshold value. In particular, we show that the threshold value is $c \frac{\log n }{n}$ for some $c>0$ which is explicitly determined, where $n$ denotes the number of vertices. Furthermore, we prove that near the threshold, the graph consists of a giant component and isolated vertices. We also investigate the phase transition and provide an alternative proof of the results by Bollobás et al. [Random Struct. Algorithms, 31, 3-122 (2007)]. Our proofs are based on an exploration process that corresponds to the graph, and instead of relying heavily on branching processes, we employ a random walk constructed from the exploration process. We then apply a large deviations theory to show that a reasonably large component is always significantly larger, a strategy used in both connectivity and phase transition analysis.

The connectivity and phase transition in inhomogeneous random graphs of finite types

Abstract

A significant generalization of the Erdös-Rényi random graph model is an `inhomogeneous' random graph where the edge probabilities vary according to vertex types. We identify the threshold value for this random graph with a finite number of vertex types to be connected and examine the model's behavior near this threshold value. In particular, we show that the threshold value is for some which is explicitly determined, where denotes the number of vertices. Furthermore, we prove that near the threshold, the graph consists of a giant component and isolated vertices. We also investigate the phase transition and provide an alternative proof of the results by Bollobás et al. [Random Struct. Algorithms, 31, 3-122 (2007)]. Our proofs are based on an exploration process that corresponds to the graph, and instead of relying heavily on branching processes, we employ a random walk constructed from the exploration process. We then apply a large deviations theory to show that a reasonably large component is always significantly larger, a strategy used in both connectivity and phase transition analysis.

Paper Structure

This paper contains 16 sections, 19 theorems, 130 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Multi-type branching processes
  5. The exploration process
  6. The upper exploration process
  7. The lower exploration process
  8. The largest component
  9. The subcritical case: $\mu \lambda<1$
  10. The supercritical case: $\mu \lambda> 1$
  11. Connectivity
  12. The subcritical case
  13. The supercritical case
  14. The critical case
  15. Connectivity for random graphs with multiple blocks
  16. ...and 1 more sections

Key Result

Theorem 1.1

Consider $\mathbb{G} := \mathbb{G}_{n,c/n}$ where $c>0$ is a constant.

Figures (3)

  • Figure 1: Model with $m=3$, $0.2\bm{c}$
  • Figure 2: Model with $m=3$, $0.5\bm{c}$
  • Figure 3: Model with $m=3$, $\bm{c}$

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 33 more