Nonlinear Joint Spectral Radius
We introduce a nonlinear extension of the joint spectral radius (JSR) for switched discrete-time dynamical systems. The classical JSR characterizes the exponential growth rate of linear switched systems, but its extension to nonlinear systems has remained an open problem. Our approach builds on the framework of …
A Low-Rank Perspective on Oversmoothing in Graph Neural Networks
AAAI Conference on Artificial Intelligence (AAAI), Workshop on Connecting Low-Rank Representations in AI, 2025
Oversmoothing is a fundamental challenge in graph neural networks (GNNs): as the number of layers increases, node embeddings become increasingly similar, and model performance drops sharply. Traditionally, oversmoothing has been quantified using metrics that measure the similarity of neighbouring node features, such as …
Subhomogeneous Deep Equilibrium Models
International Conference on Machine Learning (ICML), 2024
Implicit-depth neural networks have grown as powerful alternatives to traditional networks in various applications in recent years. However, these models often lack guarantees of existence and uniqueness, raising stability, performance, and reproducibility issues. In this paper, we present a new analysis of the …
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 …
Nonlinear Higher-Order Label Spreading
The Web Conference (WWW), pp. 2402–2413, 2021
Label spreading is a general technique for semi-supervised learning with point cloud or network data, which can be interpreted as a diffusion of labels on a graph. While there are many variants of label spreading, nearly all of them are linear models, where the incoming information to a node is a weighted sum of …
Ergodicity coefficients for higher-order stochastic processes
SIAM Journal on Mathematics of Data Science, vol. 2, pp. 740–769, 2020
The use of higher-order stochastic processes such as nonlinear Markov chains or vertex-reinforced random walks is significantly growing in recent years as they are much better at modeling high dimensional data and nonlinear dynamics in numerous application settings. In many cases of practical interest, these processes …
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, …
Nonlocal PageRank
ESAIM: Mathematical Modelling and Numerical Analysis, vol. 55, pp. 77–97, 2021
In this work we introduce and study a nonlocal version of the PageRank. In our approach, the random walker explores the graph using longer excursions than just moving between neighboring nodes. As a result, the corresponding ranking of the nodes, which takes into account a long-range interaction between them, does not …
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 …
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 …
Multi-Dimensional, Multilayer, Nonlinear and Dynamic HITS
SIAM International Conference on Data Mining (SDM), pp. 369–377, 2019
We introduce a ranking model for temporal multi-dimensional weighted and directed networks based on the Perron eigenvector of a multi-homogeneous order-preserving map. The model extends to the temporal multilayer setting the HITS algorithm and defines five centrality vectors: two for the nodes, two for the layers, and …
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 …
Node and layer eigenvector centralities for multiplex networks
SIAM Journal on Applied Mathematics, vol. 78, pp. 853–876, 2018
Eigenvector-based centrality measures are among the most popular centrality measures in network science. The underlying idea is intuitive and the mathematical description is extremely simple in the framework of standard, mono-layer networks. Moreover, several efficient computational tools are available for their …
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 …
On complex power nonnegative matrices
Linear Algebra and its Applications, vol. 471, pp. 449–468, 2015
Power nonnegative matrices are defined as complex matrices having at least one nonnegative integer power. We exploit the possibility of deriving a Perron–Frobenius-like theory for these matrices, obtaining three main results and drawing several consequences. We study, in particular, the relationships with the set of …