A Partial Characterization of Robinsonian $L^p$ Graphons

Abstract

We present a characterization of Robinsonian $L^p$ graphons for $p > 5$. Each $L^p$ graphon $w$ is the limit object of a sequence of edge density-normalized simple graphs $\{G_n/\|G_n\|_1\}$ under the cut distance $\delta_{\Box}$. A graphon $w$ is Robinson if it satisfies the Robinson property: if $x\leq y\leq z$, then $w(x,z)\leq \min\{w(x,y),w(y,z)\}$, and it is Robinsonian if $\delta_{\Box}(w,u)=0$ for some Robinson $u$. In previous work, the author and collaborators introduced a graphon parameter $\Lambda$ that recognizes the Robinson property, where $\Lambda(w) = 0$ precisely when $w$ is Robinson. Using functional analytic arguments, we show here that for $p > 5$, the Robinsonian $L^p$ graphons $w$ are precisely those that are the cut distance limit object of graphs $G_n$ such that $\Lambda(G_n/\|G_n\|_1) \to 0$.