Graph sequences sampled from Robinson graphons

The function $\Gamma$ 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 $\Gamma$ is compatible with the cut-norm $\|\cdot \|_\Box$, in the sense that graphons close in cut-norm have similar $\Gamma$-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 $\Gamma (w)$. 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 $\Gamma (G_n)\rightarrow 0$. Finally, using probabilistic techniques we show that the rate of convergence of $\Gamma$ for graph sequences sampled from a Robinson graphon can differ substantially depending on how strongly $w$ exhibits the Robinson property.