Under mild conditions of the undirected graph (basically that it is not bipartite and that it is connected) one can use the Perron-Frobenius theorem. So, to remove the ambiguity you are concerned with consider that you want an x s.t. all the entries in x are positive. Then PF guarantees that the largest eigenvalue of A has multiplicity one and its associated eigenspace is one-dimensional. PF also guarantees that the eigenvector associated with the largest eigenvalue has positive entries and is the only eigenvector with positive entries. So, you want the largest eigenvalue (which is positive) and the unique positive valued eigenvector associated with it. Finally, to obtain this eigenvector/value pair one can use the power method which this author recommends.
Thanks for the explanation! My only concern now is the validity of the meaning of the principal eigenvalue: "λ determines how much influence people share with each other through their connections. If λ is small then the CEO has a lot of influence, if it is large then he has little." It seems to depend on λ>1 or λ<1. Also, has this method been applied in practice, if you know?
I don't understand the interpretation of the principal eigenvalue either. Perhaps there is a more suitable interpretation in the directed case, but I'm not sure of that either.
I think this method is generally known as eigenvector centrality, that is to say, the entries in the vector x are generally known as eigenvector centralities. I think this method is quite popular, but I do not know who uses it or how often.
