${{{\mathcal W}}}$ define: $${d}_{{\hbox{\hskip 1pt\vrule width3pt height2pt depth1pt \hskip1pt}}}(f,g)\equiv \sup_{S,T\subseteq [0,1]}\Big| \int_{S\times
  T}[f(x,y)-g(x,y)]\, dxdy\Big|.$$ Then on $\tilde {{\mathcal W}}$ define the cut metric by:

$${\tilde d}_{{\hbox{\hskip 1pt\vrule width3pt height2pt depth1pt \hskip1pt}}}(\tilde f,\tilde g)\equiv \inf_{\sigma,\sigma'\in \Sigma}
{d}_{{\hbox{\hskip 1pt\vrule width3pt height2pt depth1pt \hskip1pt}}}(f_{\sigma},g_{\sigma'}).$$

We will use the fact, which follows easily from Lemma 4.1 in [@LS1], that the cut metric is equivalent to the metric $$\delta_{_{\hbox{hom}}}(\tilde f,\tilde g)\equiv \sum_{j\ge 1}
\frac{1}{2^j}|t(H_j,f)-t(H_j,g)|,$$ where $\{H_j\}$ is a countable set of simple graphs, one from each graph-equivalence class. Also note that if each vertex of a finite graph is split into the same number of ‘twins’, each connected to the same vertices, the result stays in the same equivalence class, so for a convergent sequence $\tilde g^{G_j}$ one may assume $|V(G_j)|\to \infty$.

The value of this graphon formalism here is that one can use large deviations on graphs with independent edges, proven in [@CV], to give an optimization formula for $s_{e,t}$, which allows us to analyze $s_{e,t}$ near the graphons of minimal triangle density, the lower boundary of $R$ in Figure 1. We next use the large deviations rate function for graphs with independent edges to give a variational characterization for the entropy density. (There 