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About the article
Published Online: 2013-05-27
Published in Print: 2013-06-01
In equation (2), we take the fraction to be
A digraph is weakly connected if replacing its arcs with undirected edges yields a connected graph.
Recall that for a matrix with non-negative entries, there exists a positive, real eigenvalue (called the Perron-Frobenius eigenvalue) such that any other eigenvalue is smaller in magnitude. The Perron-Frobenius eigenvalue is simple and the corresponding eigenvector (called the Perron-Frobenius eigenvector) has non-negative entries. See, for example, Horn and Johnson (1991).
The matrices W and S are irreducible if the corresponding directed graph is strongly connected. The matrix W is irreducible if there is no partition of the teams V=V1ߎV2 such that no team in V1 has beat a team in V2.