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Edge Exchangeable Graphs: Connectedness, Gaussianity and Completeness

Edward Eriksson

TL;DR

This paper analyzes edge-exchangeable graphs constructed by sampling edges from a fixed measure μ on unordered vertex pairs, using both discrete-time and continuous-time formalisms. It delivers a complete criterion for eventual forever connectedness, showing that connectedness holds a.s. when μ has connected support and ∑_{e} μ_e/M_e < ∞, and fails otherwise, while also proving asymptotic Gaussianity of the vertex count in the connected regime via urn-based couplings. It resolves Janson’s open problems on when the graphs become eventually essentially complete and provides a nuanced treatment of when the graph growth can be considered essentially complete through a partition-respecting exponential-arrival analysis. Collectively, the results illuminate how the edge-measure μ governs sparsity, connectivity, and growth in edge-exchangeable graphs, with implications for modeling large sparse networks via edge arrivals and informing prior choices over μ.

Abstract

We characterize some asymptotic properties of edge exchangeable random graphs in terms of the measure used to generate them. In particular, we give a necessary and sufficient condition for eventual forever connectedness, a sufficient condition for asymptotic normality of the vertex count, and a necessary and sufficient condition for the produced graph to be eventually forever almost complete.

Edge Exchangeable Graphs: Connectedness, Gaussianity and Completeness

TL;DR

This paper analyzes edge-exchangeable graphs constructed by sampling edges from a fixed measure μ on unordered vertex pairs, using both discrete-time and continuous-time formalisms. It delivers a complete criterion for eventual forever connectedness, showing that connectedness holds a.s. when μ has connected support and ∑_{e} μ_e/M_e < ∞, and fails otherwise, while also proving asymptotic Gaussianity of the vertex count in the connected regime via urn-based couplings. It resolves Janson’s open problems on when the graphs become eventually essentially complete and provides a nuanced treatment of when the graph growth can be considered essentially complete through a partition-respecting exponential-arrival analysis. Collectively, the results illuminate how the edge-measure μ governs sparsity, connectivity, and growth in edge-exchangeable graphs, with implications for modeling large sparse networks via edge arrivals and informing prior choices over μ.

Abstract

We characterize some asymptotic properties of edge exchangeable random graphs in terms of the measure used to generate them. In particular, we give a necessary and sufficient condition for eventual forever connectedness, a sufficient condition for asymptotic normality of the vertex count, and a necessary and sufficient condition for the produced graph to be eventually forever almost complete.
Paper Structure (19 sections, 18 theorems, 55 equations, 2 figures)

This paper contains 19 sections, 18 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model (Informal)
  3. Contributions
  4. Structure
  5. Motivation
  6. Motivating Edge Exchangeable Models
  7. Motivating our Contribution
  8. Related Work
  9. Previous Work on Edge Exchangeable Models
  10. Comparisons to other Models
  11. Model (Formal)
  12. Discrete Time
  13. Continuous Time
  14. Preliminaries and Notation
  15. Connectedness
  16. ...and 4 more sections

Key Result

Lemma 5

If $A$ is a property that depends only on the order of appearance of edges then $G_t$ has $A$ eventually forever iff $G_n$ has $A$ eventually forever.

Figures (2)

  • Figure 1: Edge Notation
  • Figure 2: Examples and non-examples of essentially complete graphs. The first two are not essentially complete as they fail the induced subgraph condition. The former is missing edges, the latter a vertex (and its associated edges). The third is an essentially complete graph because the subgraph induced by vertices $1,2,3,4$ is the complete graph on four vertices, the fifth vertex is the only extra vertex, and that vertex is not isolated.

Theorems & Definitions (58)

  • Remark 1
  • Remark 2
  • Definition 3
  • Remark 4
  • Lemma 5
  • proof
  • Definition 6
  • Remark 7
  • Definition 8
  • Remark 9
  • ...and 48 more