Bipartite four-cycle density ($t_4(G)={\mathbb{E}}_{x,x^{\prime },y,y^{\prime }}G(x,y)G(x,y^{\prime })G(x^{\prime },y)G(x^{\prime },y^{\prime })$)

This is the part-preserving homomorphism density of a four-cycle in a bipartite graph with independently uniform labels $x,x^{\prime }$ in the first part and $y,y^{\prime }$ in the second. Repeated labels are included. It equals the fourth power of the four-cycle norm of the adjacency indicator function. It differs at finite sizes from a count requiring four distinct vertices.