Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Graph sequences sampled from Robinson graphons

Mahya Ghandehari, Jeannette Janssen

Abstract

The function $Γ$ on the space of graphons, introduced in [CGH$^+$15], aims to measure the extent to which a graphon $w$ exhibits the Robinson property: for all $x<y<z$, $w(x,z)\leq \min\{ w(x,y),w(y,z)\}$. Robinson graphons form a model for graphs with a natural line embedding so that most edges are local. Function $Γ$ is compatible with the cut-norm $\|\cdot \|_\Box$, in the sense that graphons close in cut-norm have similar $Γ$-values. Here we show the converse, by proving that every graphon $w$ can be approximated by a Robinson graphon $R_w$ so that $\|w-R_w\|_\Box$ is bounded in terms of $Γ(w)$. We then use classical techniques from functional analysis to show that a converging graph sequence $\{G_n\}$ converges to a Robinson graphon if and only if $Γ(G_n)\rightarrow 0$. Finally, using probabilistic techniques we show that the rate of convergence of $Γ$ for graph sequences sampled from a Robinson graphon can differ substantially depending on how strongly $w$ exhibits the Robinson property.

Graph sequences sampled from Robinson graphons

Abstract

The function on the space of graphons, introduced in [CGH15], aims to measure the extent to which a graphon exhibits the Robinson property: for all , . Robinson graphons form a model for graphs with a natural line embedding so that most edges are local. Function is compatible with the cut-norm , in the sense that graphons close in cut-norm have similar -values. Here we show the converse, by proving that every graphon can be approximated by a Robinson graphon so that is bounded in terms of . We then use classical techniques from functional analysis to show that a converging graph sequence converges to a Robinson graphon if and only if . Finally, using probabilistic techniques we show that the rate of convergence of for graph sequences sampled from a Robinson graphon can differ substantially depending on how strongly exhibits the Robinson property.

Paper Structure

This paper contains 19 sections, 105 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Contribution of this paper
  3. Similar graph parameters
  4. Definitions, notations and background
  5. Cut-norm, cut-distance, graph limits, and $w$-random graphs
  6. Functions $\Gamma$ and $\widetilde{\Gamma}$
  7. Robinson approximation of graphons
  8. Properties of $R_w$ and proof of Theorem \ref{['thm:main']}
  9. A lower bound on $\Gamma$ in terms of the "upper left" and "lower right" regions
  10. Black, white and grey regions in $\Delta$
  11. Proof of Theorem \ref{['thm:main']}.
  12. Recognition of graph sequences sampled from Robinson graphons
  13. Stepping operator
  14. Robinsonian graphons
  15. Rate of decay for $\widetilde{\Gamma}$ of samples from Robinson graphons
  16. ...and 4 more sections

Figures (4)

  • Figure 1: Regions ${\rm UL} (a,b)$ (blue) and ${\rm LR} (a,b)$ (red)
  • Figure 2: We let $w_u=\overline{w}(S_u\times T_u)$, $w_l=\overline{w}(S_l\times T_l)$. Lemma \ref{['lem:Gamma-lower-bound']} bounds $\Gamma$ if $w_u>w_l$.
  • Figure 3: $w$ as in Remark \ref{['remark:tight-example']}.
  • Figure 4: Black and white regions; example for $m=2$.

Theorems & Definitions (16)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 6 more