Spectral preorder and perturbations of discrete weighted graphs

In this article, we introduce a geometric and a spectral preorder relation on the class of weighted graphs with a magnetic potential. The first preorder is expressed through the existence of a graph homomorphism respecting the magnetic potential and fulfilling certain inequalities for the weights. The second preorder refers to the spectrum of the associated Laplacian of the magnetic weighted graph. These relations give a quantitative control of the effect of elementary and composite perturbations of the graph (deleting edges, contracting vertices, etc.) on the spectrum of the corresponding Laplacians, generalising interlacing of eigenvalues. We give several applications of the preorders: we show how to classify graphs according to these preorders and we prove the stability of certain eigenvalues in graphs with a maximal d-clique. Moreover, we show the monotonicity of the eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic Cheeger constants with respect to the geometric preorder. Finally, we prove a refined procedure to detect spectral gaps in the spectrum of an infinite covering graph.Comment: 26 pages; 8 figure

Covering graphs, magnetic spectral gaps and applications to polymers and nanoribbons

In this article, we analyze the spectrum of discrete magnetic Laplacians (DML) on an infinite covering graph $\widetilde{G} \rightarrow G=\widetilde{G} /\Gamma$ with (Abelian) lattice group $\Gamma$ and periodic magnetic potential $\widetilde{\beta}$. We give sufficient conditions for the existence of spectral gaps in the spectrum of the DML and study how these depend on $\widetilde{\beta}$. The magnetic potential may be interpreted as a control parameter for the spectral bands and gaps. We apply these results to describe the spectral band/gap structure of polymers (polyacetylene) and of nanoribbons in the presence of a constant magnetic field.Comment: 17 pages; 6 figure

Dispersion entropy: A Measure of Irregularity for Graph Signals

We introduce a novel method, called Dispersion Entropy for Graph Signals, $DE_G$, as a powerful tool for analysing the irregularity of signals defined on graphs. We demonstrate the effectiveness of $DE_G$ in detecting changes in the dynamics of signals defined on synthetic and real-world graphs, by defining mixed processing on random geometric graphs or those exhibiting with small-world properties. Remarkably, $DE_G$ generalises the classical dispersion entropy for univariate time series, enabling its application in diverse domains such as image processing, time series analysis, and network analysis, as well as in establishing theoretical relationships (i.e., graph centrality measures, spectrum). Our results indicate that $DE_G$ effectively captures the irregularity of graph signals across various network configurations, successfully differentiating between distinct levels of randomness and connectivity. Consequently, $DE_G$ provides a comprehensive framework for entropy analysis of various data types, enabling new applications of dispersion entropy not previously feasible, and revealing relationships between graph signals and its graph topology.Comment: 9 pages, 10 figures, 1 tabl

Multivariate permutation entropy, a Cartesian graph product approach

Entropy metrics are nonlinear measures to quantify the complexity of time series. Among them, permutation entropy is a common metric due to its robustness and fast computation. Multivariate entropy metrics techniques are needed to analyse data consisting of more than one time series. To this end, we present a multivariate permutation entropy, $MPE_G$, using a graph-based approach. Given a multivariate signal, the algorithm $MPE_G$ involves two main steps: 1) we construct an underlying graph G as the Cartesian product of two graphs G1 and G2, where G1 preserves temporal information of each times series together with G2 that models the relations between different channels, and 2) we consider the multivariate signal as samples defined on the regular graph G and apply the recently introduced permutation entropy for graphs. Our graph-based approach gives the flexibility to consider diverse types of cross channel relationships and signals, and it overcomes with the limitations of current multivariate permutation entropy.Comment: 5 pages, 4 figures, 2 table

A noise-robust Multivariate Multiscale Permutation Entropy for two-phase flow characterisation

Using a graph-based approach, we propose a multiscale permutation entropy to explore the complexity of multivariate time series over multiple time scales. This multivariate multiscale permutation entropy (MPEG) incorporates the interaction between channels by constructing an underlying graph for each coarse-grained time series and then applying the recent permutation entropy for graph signals. Given the challenge posed by noise in real-world data analysis, we investigate the robustness to noise of MPEG using synthetic time series and demonstrating better performance than similar multivariate entropy metrics. Two-phase flow data is an important industrial process characterised by complex, dynamic behaviour. MPEG characterises the flow behaviour transition of two-phase flow by incorporating information from different scales. The experimental results show that MPEG is sensitive to the dynamic of flow patterns, allowing us to distinguish between different flow patterns

Permutation Entropy for Graph Signals

Entropy metrics (for example, permutation entropy) are nonlinear measures of irregularity in time series (one-dimensional data). Some of these entropy metrics can be generalised to data on periodic structures such as a grid or lattice pattern (two-dimensional data) using its symmetry, thus enabling their application to images. However, these metrics have not been developed for signals sampled on irregular domains, defined by a graph. Here, we define for the first time an entropy metric to analyse signals measured over irregular graphs by generalising permutation entropy, a well-established nonlinear metric based on the comparison of neighbouring values within patterns in a time series. Our algorithm is based on comparing signal values on neighbouring nodes, using the adjacency matrix. We show that this generalisation preserves the properties of classical permutation for time series and the recent permutation entropy for images, and it can be applied to any graph structure with synthetic and real signals. We expect the present work to enable the extension of other nonlinear dynamic approaches to graph signals.Comment: 11 pares, 12 figures, 2 table

A geometric construction of isospectral magnetic graphs

We present a geometrical construction of families of finite isospectral graphs labelled by different partitions of a natural number r of given length s (the number of summands). Isospectrality here refers to the discrete magnetic Laplacian with normalised weights (including standard weights). The construction begins with an arbitrary finite graph GG with normalised weight and magnetic potential as a building block from which we construct, in a first step, a family of so-called frame graphs (FFa)a∈N . A frame graph FFa is constructed contracting a copies of G along a subset of vertices V0 . In a second step, for any partition A=(a1,…,as) of length s of a natural number r (i.e., r=a1+⋯+as ) we construct a new graph FFA contracting now the frames FFa1,…,FFas selected by A along a proper subset of vertices V1⊂V0 . All the graphs obtained by different s-partitions of r≥4 (for any choice of V0 and V1 ) are isospectral and non-isomorphic. In particular, we obtain increasing finite families of graphs which are isospectral for given r and s for different types of magnetic Laplacians including the standard Laplacian, the signless standard Laplacian, certain kinds of signed Laplacians and, also, for the (unbounded) Kirchhoff Laplacian of the underlying equilateral metric graph. The spectrum of the isospectral graphs is determined by the spectrum of the Laplacian of the building block G and the spectrum for the Laplacian with Dirichlet conditions on the set of vertices V0 and V1 with multiplicities determined by the numbers r and s of the partition.JSFC was supported by the Leverhulme Trust via a Research Project Grant (RPG-2020-158). FLl was supported by the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0554) and from the Spanish National Research Council, through the Ayuda extraordinaria a Centros de Excelencia Severo Ochoa (20205CEX001) and by the Madrid Government under the Agreement with UC3M in the line of Research Funds for Beatriz Galindo Fellowships (C&QIG-BG-CM-UC3M), and in the context of the V PRICIT.Publicad