according to the paper https://arxiv.org/abs/math/0107140, we have known that the diameter of the component graph of the USF on d-dimensional transitive graph is $[n-1]/4$, I wonder if the diameter of the component graph of the USF on the amenable transitive graph with superpolynoimal growth is infinite. Actually, I think the answer is yes for every transitive graph with superpolynomial growth. A concrete example is that I think the component graph of WUSF on 3-regular tree has infinite diameter. But 3-regular tree is not amenable.