The nonlinear Perron-Frobenius theory addresses existence, uniqueness and maximality of positive eigenpairs for order-preserving homogeneous functions.
This is an important and relatively recent generalization of the famous result for nonnegative matrices. In this
talk I present a further generalization of this theory to ''multi-dimensional'' order-preserving and homogeneous maps, which we briefly call multi-homogeneous maps. The results presented are then used to discuss some nonlinear matrix and tensor eigenvalue problems and a new eigenvector-based centrality measure for nodes and layers of multi-layer networks.