 Bipartite four-cycle density

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.