Weighted simplicial complexes and their representation power of higher-order network data and topology

Baccini, Federica; Geraci, Filippo; Bianconi, Ginestra

doi:10.1103/physreve.106.034319

Cited by 39 publications

(28 citation statements)

References 75 publications

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“…In particular, we identify the geometric and topological properties of both non-homology and homology eigenvectors for molecular structures. We generalize these results to weighted simplicial complexes on top of which the weighted Dirac operator [63] is carefully defined. In particular, here we analyse the influence of weighting schemes on the spectral properties of molecular structures.…”

Section: Introductionmentioning

confidence: 79%

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Persistent Dirac for molecular representation

Wee¹,

Bianconi²,

Xia³

2023

Preprint

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Molecular representations are of fundamental importance for the modeling and analysis of molecular systems. Representation models and in general approaches based on topological data analysis (TDA) have demonstrated great success in various steps of drug design and materials discovery.Here we develop a mathematically rigorous computational framework for molecular representation based on the persistent Dirac operator. The properties of the spectrum of the discrete weighted and unweighted Dirac matrices are systemically discussed and used to demonstrate the geometric and topological properties of both non-homology and homology eigenvectors of real molecular structures. This allows us to asses the influence of weighting schemes on the information encoded in the Dirac eigenspectrum. A series of physical persistent attributes, which characterize the spectrum of the Dirac matrices across a filtration, are proposed and used as efficient molecular fingerprints. Finally, our persistent Dirac-based model is used for clustering molecular configurations from nine types of organic-inorganic halide perovskites. We found that our model can cluster the structures very well, demonstrating the representation and featurization power of the current approach.

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Section: Introductionmentioning

confidence: 79%

“…Discrete Dirac models a. Weighted Dirac matrix Recently, weighted Dirac matrices have been proposed based on a weighted simplicial complex [63]. For a d-dimensional weighted simplicial complex K, let us define the n p × n p metric matrix G p (0 ≤ p ≤ d) to be a diagonal matrix with positive entries.…”

Section: Persistent Diracmentioning

confidence: 99%

Persistent Dirac for molecular representation

Wee¹,

Bianconi²,

Xia³

2023

Preprint

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“…The Dirac operator D is a linear operator that acts on the topological spinor [47,54,55]. In the canonical basis of topological spinors of a d = 2 dimensional simplicial complex, the Dirac operator D has the block structure…”

Section: B the Dirac Operatormentioning

confidence: 99%

“…We remark that it is further possible to define a (weighted) normalized Dirac operators [52,55]. In this paper we focus exclusively on the unnormalized Dirac operator.…”

Section: Eigenvalues and Eigenvectors Of The Dirac Operatormentioning

confidence: 99%

Dirac signal processing of higher-order topological signals

Calmon¹,

Bianconi²

2023

Preprint

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We consider topological signals corresponding to variables supported on nodes, links and triangles of higher-order networks and simplicial complexes. So far such signals are typically processed independently of each other, and algorithms that can enforce a consistent processing of topological signals across different levels are largely lacking. Here we propose Dirac signal processing, an adaptive, unsupervised signal processing algorithm that learns to jointly filter topological signals supported on nodes, links and (filled) triangles of simplicial complexes in a consistent way. The proposed Dirac signal processing algorithm is rooted in algebraic topology and formulated in terms of the discrete Dirac operator which can be interpreted as "square root" of a higher-order (Hodge) Laplacian matrix acting on nodes, links and triangles of simplicial complexes. We test our algorithms on noisy synthetic data and noisy data of drifters in the ocean and find that the algorithm can learn to efficiently reconstruct the true signals outperforming algorithms based exclusively on the Hodge Laplacian.

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“…Interestingly, both the Hodge-Laplacians and the Dirac operator can be extended to treat weighted simplicial complexes (see for instance [85]).…”

Section: Appendix A: Basics Properties Of Algebraic Topologymentioning

confidence: 99%

Diffusion-driven instability of topological signals coupled by the Dirac operator

Giambagli

Calmon²,

Muolo³

et al. 2022

Phys. Rev. E

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The study of reaction-diffusion systems on networks is of paramount relevance for the understanding of nonlinear processes in systems where the topology is intrinsically discrete, such as the brain. Until now reactiondiffusion systems have been studied only when species are defined on the nodes of a network. However, in a number of real systems including, e.g., the brain and the climate, dynamical variables are not only defined on nodes but also on links, faces and higher-dimensional cells of simplicial or cell complexes, leading to topological signals. In this work we study reaction-diffusion processes of topological signals coupled through the Dirac operator. The Dirac operator allows topological signals of different dimension to interact or cross-diffuse as it projects the topological signals defined on simplices or cells of a given dimension to simplices or cells of one dimension up or one dimension down. By focusing on the framework involving nodes and links we establish the conditions for the emergence of Turing patterns and we show that the latter are never localized only on nodes or only on links of the network. Moreover when the topological signals display Turing pattern their projection does as well. We validate the theory hereby developed on a benchmark network model and on square lattices with periodic boundary conditions.

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Weighted simplicial complexes and their representation power of higher-order network data and topology

Cited by 39 publications

References 75 publications

Persistent Dirac for molecular representation

Persistent Dirac for molecular representation

Dirac signal processing of higher-order topological signals

Diffusion-driven instability of topological signals coupled by the Dirac operator

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