A weighted directed graph on a set \(V\) consists of a directed graph on \(V\), with loops permitted, together with a real weight \(w(x,y)\) for every ordered pair \(x,y\in V\); no positivity or support condition is imposed.

For \(n\in \mathbb N\) and \(\rho \in \mathbb {R}\), the \(\rho \)-stable cube is the complete directed graph, including loops, on \(\{ 0,1\} ^n\), with weight \(w(x,y)=2^{-n}\prod _{i=1}^n\frac{1+\rho (-1)^{x_i+y_i}}{2}\). For \(-1\le \rho \le 1\), these weights are the joint probabilities of a uniform correlated pair.

For \(n\in \mathbb N\) and \(\delta \in \mathbb {R}\), the \(\delta \)-noisy cube is the \((1-2\delta )\)-stable cube: the complete directed graph on \(\{ 0,1\} ^n\) with weights \(2^{-n}\prod _{i=1}^n\frac{1+(1-2\delta )(-1)^{x_i+y_i}}{2}\).