If I understand the algorithm correctly, (which I'm not 100% sure that I do) the 'paths' can be interpreted as the result of Dijkstra's algorithm on a digraph. So for instance, given a graph with adjacency lists:
I think it might be helpful to read the paper. See "A New Monotonic and Clone-Independent Single-Winner Election Method" under Condorcet method.
It's actually Floyd's algorithm (all-points shortest paths rather than one-source, which is Dijkstra's) and in reverse, sorta: as you're trying to find the all-pairs greatest path. The edges are already weighted: the weight of (A, B) is how many votes ranked A higher than B. And (as I understand it) if A beat C (and C beat B) harder than A beat B directly, then A beats B with the greater strength.