Homological scaffold via minimal homology bases

Guerra, Marco; Gregorio, Alessandro De; Fugacci, Ulderico; Petri, Giovanni; Vaccarino, Francesco

doi:10.1038/s41598-021-84486-1

“…For instance, a very elastic coffee cup could be continuously contracted into the shape of a donut, as they share the same toroidal topology. For the sake of convenience, a homological cycle is often represented as the minimal representative cycle ( Guerra, De Gregorio, Fugacci, Petri, & Vaccarino, 2020 ; Petri et al, 2014 ).…”

Section: Methodsmentioning

confidence: 99%

Simplicial and topological descriptions of human brain dynamics

Billings

¹

,

Saggar

²

,

Hlinka

³

et al. 2021

Network Neuroscience

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Whereas brain imaging tools like functional Magnetic Resonance Imaging (fMRI) afford measurements of whole-brain activity, it remains unclear how best to interpret patterns found amid the data’s apparent self-organization. To clarify how patterns of brain activity support brain function, one might identify metric spaces that optimally distinguish brain states across experimentally defined conditions. Therefore, the present study considers the relative capacities of several metric spaces to disambiguate experimentally defined brain states. One fundamental metric space interprets fMRI data topographically, i.e, as the vector of amplitudes of a multivariate signal, changing with time. Another perspective compares the brain’s Functional Connectivity, i.e., the similarity matrix computed between signals from different brain regions. More recently, metric spaces that consider the data’s topology have become available. Such methods treat data as a sample drawn from an abstract geometric object. To recover the structure of that object, topological data analysis detects features that are invariant under continuous deformations (such as coordinate rotation and nodal misalignment). Moreover, the methods explicitly consider features that persist across multiple geometric scales. While, certainly, there are strengths and weaknesses of each brain dynamics metric space, we find that those that track topological features optimally distinguish experimentally defined brain states.

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“…Interestingly, for cases with holes, the harmonic representatives themselves are in the kernel of L u k , however, we are still able to find simplicial communities. There exist several algorithms that find sparse eigenvectors which typically separate the support when degeneracy arises [66]. In this work, we use LApack functions built into the sklearn package in python.…”

Section: Append(w) Visualize Communities End For VII Higher-order Sim...mentioning

confidence: 99%

Spectral detection of simplicial communities via Hodge Laplacians

Krishnagopal

¹

,

Bianconi

²

2021

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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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“…There exist several algorithms that find sparse eigenvectors which typically separate the support when degeneracy arises [66]. In this work, we use LApack functions built into the sklearn package in python.…”

Section: A 2-down Communitiesmentioning

confidence: 99%

Spectral Detection of Simplicial Communities via Hodge Laplacians

Krishnagopal,

Bianconi

2021

Preprint

0

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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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Homological scaffold via minimal homology bases

Cited by 9 publications

References 91 publications

Simplicial and topological descriptions of human brain dynamics

Simplicial and topological descriptions of human brain dynamics

Spectral detection of simplicial communities via Hodge Laplacians

Spectral Detection of Simplicial Communities via Hodge Laplacians

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