Graph sequences sampled from Robinson graphons

Abstract: The function $\Gamma$ on the space of graphons, introduced in [CGH$<sup>+$15], aims to measure the extent to which a graphon $w$ exhibits the Robinson property: for all $x&lt;y&lt;z$, $w(x,z)\leq \min{ w(x,y),w(y,z)}$. 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 $|\cdot |<em>\Box$, 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 $|w-R_w|</em>\Box$ 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.