  This paper studies the $p$-biharmonic equation on graphs, which arises in
point cloud processing and can be interpreted as a natural extension of the
graph $p$-Laplacian from the perspective of hypergraph. The asymptotic behavior
of the solution is investigated when the random geometric graph is considered
and the number of data points goes to infinity. We show that the continuum
limit is an appropriately weighted $p$-biharmonic equation with homogeneous
Neumann boundary conditions. The result relies on the uniform $L^p$ estimates
for solutions and gradients of nonlocal and graph Poisson equations. The
$L^\infty$ estimates of solutions are also obtained as a byproduct.
