Dirac synchronization is rhythmic and explosive

Calmon, Lucille; Restrepo, Juan G.; Torres, Joaquín J.; Bianconi, Ginestra

doi:10.48550/arxiv.2107.05107

Cited by 5 publications

(8 citation statements)

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“…The K-block Hodge-Laplacian L K is related to the Dirac operator in differential geometry (i.e., Dirac operator is a square root of L K ) (Lloyd, Garnerone, and Zanardi 2016). As such, L K has multiple implications for analysis of synchronization dynamics and coupling of various topological signals on graphs, with applications in physics and quantum information processing (Calmon et al 2021).…”

Section: Block Simplicial Complex Neural Networkmentioning

confidence: 99%

BScNets: Block Simplicial Complex Neural Networks

Chen

Gel

Poor

2022

AAAI

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Simplicial neural networks (SNNs) have recently emerged as a new direction in graph learning which expands the idea of convolutional architectures from node space to simplicial complexes on graphs. Instead of predominantly assessing pairwise relations among nodes as in the current practice, simplicial complexes allow us to describe higher-order interactions and multi-node graph structures. By building upon connection between the convolution operation and the new block Hodge-Laplacian, we propose the first SNN for link prediction. Our new Block Simplicial Complex Neural Networks (BScNets) model generalizes existing graph convolutional network (GCN) frameworks by systematically incorporating salient interactions among multiple higher-order graph structures of different dimensions. We discuss theoretical foundations behind BScNets and illustrate its utility for link prediction on eight real-world and synthetic datasets. Our experiments indicate that BScNets outperforms the state-of-the-art models by a significant margin while maintaining low computation costs. Finally, we show utility of BScNets as a new promising alternative for tracking spread of infectious diseases such as COVID-19 and measuring the effectiveness of the healthcare risk mitigation strategies.

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Section: Block Simplicial Complex Neural Networkmentioning

confidence: 99%

BScNets: Block Simplicial Complex Neural Networks

Chen

Gel

Poor

2022

AAAI

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“…The Hodge decomposition has played an important role in several analyzes and applications. For instance Hodge decomposition is central for defining higher-order synchronization of k-chains and of coupled chains of different dimension [21][22][23][24]. Moreover, the space of 1-chains on simplicial complexes have been studied extensively [18] as a natural way of modeling 'flows'.…”

Section: Hodge Decompositionmentioning

confidence: 99%

“…Exploiting the relationship with topology, recent works have investigated higher-order dynamics [21][22][23][24] and data analyses using Hodge theory [25], as well as persistent homology [26]. Additionally, applied topology studies the underlying properties such as the Betti numbers (the number of high-dimensional holes) of simplicial complexes applied to real data.…”

Section: Introductionmentioning

confidence: 99%

Spectral detection of simplicial communities via Hodge Laplacians

Krishnagopal

Bianconi

2021

Phys. Rev. E

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While the study of graphs has been very popular, simplicial complexes are relatively new in the network science community. Despite being are a source of rich information, graphs are limited to pairwise interactions. However, several real world networks such as social networks, neuronal networks etc. involve simultaneous interactions between more than two nodes. Simplicial complexes provide a powerful mathematical way to model such interactions. Now, the spectrum of the graph Laplacian is known to be indicative of community structure, with nonzero eigenvectors encoding the identity of communities. Here, we propose that the spectrum of the Hodge Laplacian, a higherorder Laplacian applied to simplicial complexes, encodes simplicial communities. We formulate an algorithm to extract simplicial communities (of arbitrary dimension). We apply this algorithm on simplicial complex benchmarks and on real data including social networks and language-networks, where higher-order relationships are intrinsic. Additionally, datasets for simplicial complexes are scarce. Hence, we introduce a method of optimally generating a simplicial complex from its network backbone through estimating the true higher-order relationships when its community structure is known. We do so by using the adjusted mutual information to identify the configuration that best matches the expected data partition. Lastly, we demonstrate an example of persistent simplicial communities inspired by the field of persistence homology.

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“…Moreover edge signals might also represent a number of climate data such as currents in the ocean and velocity of wind that can be projected on a suitable triangulation of the Earth surface [35,36]. Topological signals can undergo higher-order simplicial synchronization [37][38][39][40][41][42][43], and higher-order diffusion [39,44,45]. Moreover datasets of topological signals can be treated with topological signal processing [33,36,46] and with topological machine learning tools [47][48][49][50].…”

Section: Introductionmentioning

confidence: 99%

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

Giambagli¹,

Calmon²,

Muolo³

et al. 2022

Preprint

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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 the brain and the climate, dynamical variables are not only defined on nodes but also on links, triangles and higher-dimensional simplices, 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 of a given dimension to simplices 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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Dirac synchronization is rhythmic and explosive

Cited by 5 publications

References 0 publications

BScNets: Block Simplicial Complex Neural Networks

BScNets: Block Simplicial Complex Neural Networks

Spectral detection of simplicial communities via Hodge Laplacians

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

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