Fast Computation of Eigenvector Centralities for Multilayer Networks with Nonnegative Tensor Train
Lobachevskii Journal of Mathematics, vol. 46, pp. 3804–3814, 2025
In this article, we introduce eigenvector centralities for higher-order multilayer networks and suggest to use nonnegative tensor train decomposition for fast computations of dominant eigenvectors of multi-homogeneous maps via power iterates. The analysis of the approximation error for using nonnegative tensor train …
A nonlinear spectral core-periphery detection method for multiplex networks
Proceedings of the Royal Society A, 2024
Core-periphery detection aims to separate the nodes of a complex network into two subsets: a core that is densely connected to the entire network and a periphery that is densely connected to the core but sparsely connected internally. The definition of core-periphery structure in multiplex networks that record …
Nonlinear Perron-Frobenius theorems for nonnegative tensors
SIAM Review (SIGEST paper), vol. 65, pp. 495–536, 2023
We present a unifying Perron–Frobenius theory for nonlinear spectral problems defined in terms of nonnegative tensors. By using the concept of tensor shape partition, our results include, as a special case, a wide variety of particular tensor spectral problems considered in the literature and can be applied to a broad …
Core-periphery detection in hypergraphs
SIAM Journal on Mathematics of Data Science, vol. 5, 2023
Core-periphery detection is a key task in exploratory network analysis where one aims to find a core, a set of nodes well-connected internally and with the periphery, and a periphery, a set of nodes connected only (or mostly) with the core. In this work we propose a model of core-periphery for higher-order networks …
Node and Edge Eigenvector Centrality for Hypergraphs
Communications Physics, vol. 4:201, 2021
Network scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency matrix. Recent work indicates that there are further benefits from accounting …
The Global Convergence of the Nonlinear Power Method for Mixed-Subordinate Matrix Norms
Journal of Scientific Computing, vol. 88, pp. 21, 2021
We analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $\ell^p$ matrix norms. In particular, …
A framework for second order eigenvector centralities and clustering coefficients
Proceedings of the Royal Society A, vol. 476, pp. 20190724, 2020
We propose and analyse a general tensor-based framework for incorporating second order features into network measures. This approach allows us to combine traditional pairwise links with information that records whether triples of nodes are involved in wedges or triangles. Our treatment covers classical spectral methods …
Shifted and extrapolated power methods for tensor $\ell^p$-eigenpairs
Electronic Transactions on Numerical Analysis (ETNA), vol. 53, pp. 1–27, 2020
This work is concerned with the computation of $\ell^p$-eigenvalues and eigenvectors of square tensors with $d$ modes. In the first part we propose two possible shifted variants of the popular (higher-order) power method for the computation of $\ell^p$-eigenpairs proving the convergence of both the schemes to the …
A fast and robust kernel optimization method for core–periphery detection in directed and weighted graphs
Applied Network Science, vol. 4:75, 2019
Many graph mining tasks can be viewed as classification problems on high dimensional data. Within this class we consider the issue of discovering core-periphery structure, which has wide applications in the economic and social sciences. In contrast to many current approaches, we allow for weighted and directed edges …
Extrapolation methods for fixed-point multilinear PageRank computations
Numerical Linear Algebra with Applications, vol. 27, pp. e2280, 2020
Nonnegative tensors arise very naturally in many applications that involve large and complex data flows. Due to the relatively small requirement in terms of memory storage and number of operations per step, the (shifted) higher-order power method is one of the most commonly used technique for the computation of …
A nonlinear spectral method for core-periphery detection in networks
SIAM Journal on Mathematics of Data Science, vol. 1, pp. 269–292, 2019
We derive and analyse a new iterative algorithm for detecting network core–periphery structure. Using techniques in nonlinear Perron-Frobenius theory, we prove global convergence to the unique solution of a relaxed version of a natural discrete optimization problem. On sparse networks, the cost of each iteration scales …
The contractivity of cone-preserving multilinear mappings
Nonlinearity, vol. 32, pp. 4713, 2019
With the notion of mode-$j$ Birkhoff contraction ratio, we prove a multilinear version of the Birkhoff-Hopf and the Perron-Fronenius theorems, which provide conditions on the existence and uniqueness of a solution to a large family of systems of nonlinear equations of the type $$f_i(x_1,…,x_ν)= λ_i x_i, ,$$ being $x_i$ …
A unifying Perron-Frobenius theorem for nonnegative tensors via multihomogeneous maps
SIAM Journal on Matrix Analysis and Applications, vol. 40, pp. 1206–1231, 2019
We introduce the concept of shape partition of a tensor and formulate a general tensor eigenvalue problem that includes all previously studied eigenvalue problems as special cases. We formulate irreducibility and symmetry properties of a nonnegative tensor $T$ in terms of the associated shape partition. We recast the …
The Perron-Frobenius theorem for multihomogeneous mappings
SIAM Journal on Matrix Analysis and Applications, vol. 40, pp. 1179–1205, 2019
The Perron-Frobenius theory for nonnegative matrices has been generalized to order-preserving homogeneous mappings on a cone and more recently to nonnegative multilinear forms. We unify both approaches by introducing the concept of order-preserving multi-homogeneous mappings, their associated nonlinear spectral …