Principal eigenvector of the signless Laplacian matrix

Cardoso, Kauê

doi:10.1007/s40314-021-01440-7

Cited by 2 publications

(3 citation statements)

References 23 publications

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“…The degree of a vertex in a hypergraph is equal to the number of hyperedges that contain that vertex, and a k -regular hypergraph is one in which all vertices have degree equal to k . The clique multigraph corresponding to a hypergraph is the multigraph (another extension of simple graphs where two vertices can have multiple simple edges between them) with the same vertices as the hypergraph, and as many edges between two vertices as the number of times those vertices co-occur in a hyperedge of the hypergraph (i. e., if two vertices are both in two separate hyperedges of the hypergraph, then they will have two edges between them in the clique multigraph) [65]. The ( i, j ) th element of the adjacency matrix of a multigraph is equal to the number of edges that connect the i th and j th vertices.…”

Section: Proofs and Additional Theoretical Resultsmentioning

confidence: 99%

Section: Proofs and Additional Theoretical Resultsmentioning

confidence: 99%

“…Then the eigenvalues of FF T will simply be given by the sum of the eigenvalues of the F [j] (F [j] ) T . We are further assisted by the following result regarding the outer product of incidence matrices of regular hypergraphs, due to [65]. Lemma 8 tells us if we can determine the spectrum of the adjacency matrices A [j] of the clique multigraphs C [j] , then it is straightforward to calculate the spectrum of F [j] (F [j] ) T .…”

Section: Lemma 7 Ffmentioning

confidence: 99%

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On the sparsity of fitness functions and implications for learning

Brookes

Aghazadeh

2021

Preprint

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Fitness functions map biological sequences to a scalar property of interest. Accurate estimation of these functions yields biological insight and sets the foundation for model-based sequence design. However, the amount of fitness data available to learn these functions is typically small relative to the large combinatorial space of sequences; characterizing how much data is needed for accurate estimation remains an open problem. There is a growing body of evidence demonstrating that empirical fitness functions display substantial sparsity when represented in terms of epistatic interactions. Moreover, the theory of Compressed Sensing provides scaling laws for the number of samples required to exactly recover a sparse function. Motivated by these results, we study the sparsity of fitness functions sampled from a generalization of the NK model, a widely-used random field model of fitness functions. In particular, we present theoretical results that allow us to test the effect of the Generalized NK (GNK) model's interpretable parameters---sequence length, alphabet size, and assumed interactions between sequence positions---on the sparsity of fitness functions sampled from the model and, consequently, the number of measurements required to exactly recover these functions. Further, we show that GNK fitness functions with parameters set according to protein structural contacts can be used to accurately approximate the number of samples required to estimate two empirical protein fitness functions, and are able to identify important higher-order epistatic interactions in these functions using only structural information.

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Section: Proofs and Additional Theoretical Resultsmentioning

confidence: 99%

Section: Proofs and Additional Theoretical Resultsmentioning

confidence: 99%

Section: Lemma 7 Ffmentioning

confidence: 99%

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On the sparsity of fitness functions and implications for learning

Brookes

Aghazadeh

2021

Preprint

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The signless Laplacian matrix of hypergraphs

Cardoso

Trevisan

2022

Special Matrices

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In this article, we define signless Laplacian matrix of a hypergraph and obtain structural properties from its eigenvalues. We generalize several known results for graphs, relating the spectrum of this matrix to structural parameters of the hypergraph such as the maximum degree, diameter, and the chromatic number. In addition, we characterize the complete signless Laplacian spectrum for the class of power hypergraphs from the spectrum of its base hypergraph.

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Principal eigenvector of the signless Laplacian matrix

Cited by 2 publications

References 23 publications

On the sparsity of fitness functions and implications for learning

On the sparsity of fitness functions and implications for learning

The signless Laplacian matrix of hypergraphs

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