I have a graph representing scores of various players of a game against each other in match-ups of 2 players. It is a game where players are in "attacking" and "defending" positions. The game is such that some players are good at "attacking" others, but may be bad at "defending" against the same opponent.
Every player has played against every other player in both attacking and defending position. So we have a fully connected directed graph.
I convert it into a Markov Chain by dividing each entry of each row by the sum of the entries in the row (still not clear if this is necessary, or if there's a simpler way to do it).
I then compute the left eigenvector of said matrix for eigenvalue 1 and normalize. I am told this is to be interpreted as a "stationary distribution" of the Markov Chain.
My question is this: Can I interpret the resulting vector as a list of "the best players", represented by different nodes? Eg lower/higher means more/less skilled?