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A Perron-Frobenius theory for block matrices associated to a multiplex network. (English) Zbl 1352.90018

Summary: The uniqueness of the Perron vector of a nonnegative block matrix associated to a multiplex network is discussed. The conclusions come from the relationships between the irreducibility of some nonnegative block matrix associated to a multiplex network and the irreducibility of the corresponding matrices to each layer as well as the irreducibility of the adjacency matrix of the projection network. In addition the computation of that Perron vector in terms of the Perron vectors of the blocks is also addressed. Finally we present the precise relations that allow to express the Perron eigenvector of the multiplex network in terms of the Perron eigenvectors of its layers.

MSC:

90B10 Deterministic network models in operations research
15A18 Eigenvalues, singular values, and eigenvectors
15B34 Boolean and Hadamard matrices
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