I have a certain finite (but huge and without an apparent pattern, so that only numerical studies seem feasible) graph $G = (V,E)$, and a function $f: V \rightarrow \mathbb{R}$. On each edge $e = (u,v)$ with endpoints $u, v \in V$, we assign an orientation so that we go from $u$ to $v$ if $f(u) > f(v)$. Define the ``potential'' of $e$ to be $f(u) - f(v)$.
Now pick a random (uniformly if you will) point in $V$, and continue walking down along the edges of the greatest potential until one reaches a local minimum, i.e. a vertex without an edge starting from it. I am intending a graph-theory analogue of the gradient descent method on a function $F: \mathbb{R}^n \rightarrow \mathbb{R}$ (see https://en.wikipedia.org/wiki/Gradient_descent).
My computer experiments suggest that some local minima are reached way more often than others. I wish to predict how attractive a given local minimum is by studying $f$ around that minimum. For instance, in the case of gradient descent, one can look at the Hessian (of course it's wrong in general, but in case our function $F$ is computationally cumbersome even for a computer, wouldn't it be a reasonable first guess). So my question is, in other words, does there exist an analogue of this Hessian in the graph-theory context?
Or is there a better way of thinking about this? Instances where a similar problem has been discussed? I feel some people should think about the size of basins of attractions on graphs, but my literature search hasn't been so fruitful.