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Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously…
Models of network diffusion typically rely on the Laplacian matrix, capturing interactions via direct connections. Beyond direct interactions, information in many systems can also flow via indirect pathways, where influence typically…
Motivated by the relationship between the eigenvalue spectrum of the Laplacian matrix of a network and the behavior of dynamical processes evolving in it, we propose a distributed iterative algorithm in which a group of $n$ autonomous…
The use of Laplacian eigenfunctions is ubiquitous in a wide range of computer graphics and geometry processing applications. In particular, Laplacian eigenbases allow generalizing the classical Fourier analysis to manifolds. A key drawback…
This paper deals with spectral graph theory issues related to questions of monotonicity and comparison of eigenvalues. We consider finite directed graphs with non symmetric edge weights and we introduce a special self-adjoint operator as…
It is reported that dynamical systems over digraphs have superior performance in terms of system damping and tolerance to time delays if the underlying graph Laplacian has a purely real spectrum. This paper investigates the topological…
In this paper we present a decentralized algorithm to estimate the eigenvalues of the Laplacian matrix that encodes the network topology of a multi-agent system. We consider network topologies modeled by undirected graphs. The basic idea is…
This paper investigates spectral properties of the deformed Laplacian matrix, which merges the Laplacian and signless Laplacian matrices of a graph through a one-parameter family of matrices. We present general results on the eigenvalues of…
Different from most existing distributed localization approaches in static networks where the agents in a network are static, this paper addresses the distributed localization problem in dynamic networks where the positions of the agents…
An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of…
A network's community structure commonly impacts its functions. For instance, networks seeking synchronisation will see this process follow the topology's hierarchical and community structuring. Herein, the interplay of network adjacency…
Determining the effect of structural perturbations on the eigenvalue spectra of networks is an important problem because the spectra characterize not only their topological structures, but also their dynamical behavior, such as…
Networks with a prescribed power-law scaling in the spectrum of the graph Laplacian can be generated by evolutionary optimization. The Laplacian spectrum encodes the dynamical behavior of many important processes. Here, the networks are…
Learning a graph with a specific structure is essential for interpretability and identification of the relationships among data. It is well known that structured graph learning from observed samples is an NP-hard combinatorial problem. In…
The standard notion of the Laplacian of a graph is generalized to the setting of a graph with the extra structure of a ``transmission`` system. A transmission system is a mathematical representation of a means of transmitting…
This is a tutorial and survey paper for nonlinear dimensionality and feature extraction methods which are based on the Laplacian of graph of data. We first introduce adjacency matrix, definition of Laplacian matrix, and the interpretation…
Several graph data mining, signal processing, and machine learning downstream tasks rely on information related to the eigenvectors of the associated adjacency or Laplacian matrix. Classical eigendecomposition methods are powerful when the…
We study the Laplacian of the undirected De Bruijn graph over an alphabet $A$ of order $k$. While the eigenvalues of this Laplacian were found in 1998 by Delorme and Tillich [1], an explicit description of its eigenvectors has remained…
It is an increasingly common practice in several natural and social sciences to rely on network visualisations both as heuristic tools to get a first overview of relational datasets and as a way to offer an illustration of network analysis…
The first part of this paper deals with electrical networks and symplectic reductions. We consider two operations on electrical networks (the "trace map" and the "gluing map") and show that they correspond to symplectic reductions. We also…