Nonlinear Spectral Graph Theory
SIAM Review, 2026
Nonlinear spectral graph theory is an extension of the traditional (linear) spectral graph theory and studies relationships between spectral properties of nonlinear operators defined on a graph and topological properties of the graph itself. Many of these relationships get tighter when going from the linear to the …
Efficient Sparsification of Simplicial Complexes via Local Densities of States
SIAM Journal on Mathematics of Data Science, 2026
Simplicial complexes (SCs), a generalization of graph models for relational data that account for higher-order relations between data items, have become a popular abstraction for analyzing complex data using tools from topological data analysis or topological signal processing. However, the analysis of many real-world …
The graph $\infty$-Laplacian eigenvalue problem
SIAM Journal on Mathematical Analysis, 2026
We analyze various formulations of the $\infty$-Laplacian eigenvalue problem on graphs, comparing their properties and highlighting their respective advantages and limitations. First, we investigate the graph $\infty$-eigenpairs arising as limits of $p$-Laplacian eigenpairs, extending key results from the continuous …
Cholesky-like Preconditioner for Hodge Laplacians via Heavy Collapsible Subcomplex
SIAM Journal on Matrix Analysis and Applications, 2024
Techniques based on $k$-th order Hodge Laplacian operators $L_k$ are widely used to describe the topology as well as the governing dynamics of high-order systems modeled as simplicial complexes. In all of them, it is required to solve a number of least square problems with $L_k$ as coefficient matrix, for example in …
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 …
Quantifying the structural stability of simplicial homology
Journal of Scientific Computing, vol. 97, 2023
The homology groups of a simplicial complex reveal fundamental properties of the topology of the data or the system and the notion of topological stability naturally poses an important yet not fully investigated question. In the current work, we study the stability in terms of the smallest perturbation sufficient to …
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 …
Nodal domain count for the generalized graph $p$-Laplacian
Applied and Computational Harmonic Analysis, vol. 64, pp. 1–32, 2023
Inspired by the linear Schrödinger operator, we consider a generalized $p$-Laplacian operator on discrete graphs and present new results that characterize several spectral properties of this operator with particular attention to the nodal domain count of its eigenfunctions. Just like the one-dimensional continuous …
The self-consistent field iteration for p-spectral clustering
The self-consistent field (SCF) iteration, combined with its variants, is one of the most widely used algorithms in quantum chemistry. We propose a procedure to adapt the SCF iteration for the p-Laplacian eigenproblem, which is an important problem in the field of unsupervised learning. We formulate the p-Laplacian …
Nonlinear Feature Diffusion on Hypergraphs
International Conference on Machine Learning (ICML), 2022
Hypergraphs are a common model for multiway relationships in data, and hypergraph semi-supervised learning is the problem of assigning labels to all nodes in a hypergraph, given labels on just a few nodes. Diffusions and label spreading are classical techniques for semi-supervised learning in the graph setting, and …
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 …
Generalized matrix means for semisupervised learning with multilayer graphs
Advances in Neural Information Processing Systems (NeurIPS), 2019
We study the task of semi-supervised learning on multilayer graphs by taking into account both labeled and unlabeled observations together with the information encoded by each individual graph layer. We propose a regularizer based on the generalized matrix mean, which is a one-parameter family of matrix means that …
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 …
Spectral Clustering of Signed Graphs via Matrix Power Means
International Conference on Machine Learning (ICML), pp. 4526–4536, 2019
Signed graphs encode positive (attractive) and negative (repulsive) relations between nodes. We extend spectral clustering to signed graphs via the one-parameter family of Signed Power Mean Laplacians, defined as the matrix power mean of normalized standard and signless Laplacians of positive and negative edges. We …
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 Power Mean Laplacian for Multilayer Graph Clustering
International Conference on Artificial Intelligence and Statistics (AISTATS), vol. 84, pp. 1828–1838, 2018
Multilayer graphs encode different kind of interactions between the same set of entities. When one wants to cluster such a multilayer graph, the natural question arises how one should merge the information form different layers. We introduce in this paper a one-parameter family of matrix power means for merging the …
A modularity based spectral method for simultaneous community and anti-community detection
Linear Algebra and its Applications, vol. 542, pp. 605–623, 2018
In a graph or complex network, communities and anti-communities are node sets whose modularity attains extremely large values, positive and negative, respectively. We consider the simultaneous detection of communities and anti-communities, by looking at spectral methods based on various matrix-based definitions of the …
Community detection in networks via nonlinear modularity eigenvectors
SIAM Journal on Applied Mathematics, vol. 78, pp. 2393–2419, 2018
Revealing a community structure in a network or dataset is a central problem arising in many scientific areas. The modularity function $Q$ is an established measure quantifying the quality of a community, being identified as a set of nodes having high modularity. In our terminology, a set of nodes with positive …
A nodal domain theorem and a higher-order Cheeger inequality for the graph p-Laplacian
EMS Journal of Spectral Theory, vol. 8, pp. 883–908, 2018
We consider the nonlinear graph $p$-Laplacian and its set of eigenvalues and associated eigenfunctions of this operator defined by a variational principle. We prove a nodal domain theorem for the graph $p$-Laplacian for any $p\geq 1$. While for $p>1$ the bounds on the number of weak and strong nodal domains are the …
Localization of dominant eigenpairs and planted communities by means of Frobenius inner products
Czechoslovak Mathematical Journal, vol. 66, pp. 881–893, 2016
We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix $A$. The result exploits the Frobenius inner product between $A$ and a given rank-one landmark matrix $X$. Different choices for $X$ may be used, depending on the problem under investigation. In particular, we show …
Clustering Signed Networks with the Geometric Mean of Laplacians
Advances in Neural Information Processing Systems (NeurIPS), vol. 29, 2016
Signed networks allow to model positive and negative relationships. We analyze existing extensions of spectral clustering to signed networks. It turns out that existing approaches do not recover the ground truth clustering in several situations where either the positive or the negative network structures contain no …
An algebraic analysis of the graph modularity
SIAM Journal on Matrix Analysis and Applications, vol. 35, pp. 997–1018, 2014
One of the most relevant tasks in network analysis is the detection of community structures, or clustering. Most popular techniques for community detection are based on the maximization of a quality function called modularity, which in turn is based upon particular quadratic forms associated to a real symmetric …