Suppose I have an exponentially large graph $G$ ($|G|=2^n$) supplied with an efficient (of size $poly(n)$) randomized circuit $C_G$ implementing the random walk on $G$ - that is, $C_G$ takes a vertex index $i$ and outputs a random neighbor of $i$.
Has this type of graph specification been studied and is it more powerful that the standard succinct representation, where $G$ is given as an efficient circuit that given $i,j$ outputs whether $(i,j)$ is an edge in $G$? I could imagine that being able to perform a random walk could help e.g. in detecting triangles in a dense graph (e.g. by choosing a random starting vertex and performing a random walk of length $3$; on the other hand, deciding triangle-freeness in the usual succinct model in NP-hard)