Graphs determined by their (signless) Laplacian spectra

Liu, Muhuo; Liu, Bolian; Wei, Fuyi

doi:10.13001/1081-3810.1428

Cited by 20 publications

(9 citation statements)

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“…Spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original graph on all real vector inputs. This is equivalent to saying that the Laplacian of the sparsifier is a good preconditioner for the Laplacian of the original [32].…”

Section: Discussionmentioning

confidence: 99%

Quantifying loss of information in network-based dimensionality reduction techniques

2015

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To cope with the complexity of large networks, a number of dimensionality reduction techniques for graphs have been developed. However, the extent to which information is lost or preserved when these techniques are employed has not yet been clear. Here we develop a framework, based on algorithmic information theory, to quantify the extent to which information is preserved when network motif analysis, graph spectra and spectral sparsification methods are applied to over twenty different biological and artificial networks. We find that the spectral sparsification is highly sensitive to high number of edge deletion, leading to significant inconsistencies, and that graph spectral methods are the most irregular, capturing algebraic information in a condensed fashion but largely losing most of the information content of the original networks. However, the approach shows that network motif analysis excels at preserving the relative algorithmic information content of a network, hence validating and generalizing the remarkable fact that despite their inherent combinatorial possibilities, local regularities preserve information to such an extent that essential properties are fully recoverable across different networks to determine their family group to which they belong to (eg genetic vs social network). Our algorithmic information methodology thus provides a rigorous framework enabling a fundamental assessment and comparison between different data dimensionality reduction methods thereby facilitating the identification and evaluation of the capabilities of old and new methods.

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Section: Discussionmentioning

confidence: 99%

Quantifying loss of information in network-based dimensionality reduction techniques

2015

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“…In [21], it was proved that S(n, c, k) and its complement graph are determined by their Laplacian spectra for k = 0 and c 0. In this section, we shall show that S(n, c, k) and its complement graph are also determined by their Laplacian spectra for k 1 and c 0.…”

Section: S(n C K) Is Determined By Its Laplacian Spectrummentioning

confidence: 99%

“…Thus, it seems rather interesting to consider the problem: Which graphs are determined by their signless Laplacian spectra? Recently, the lollipop graph was proved to be determined by its signless Laplacian spectrum [31], and the bundle graph S(n, c, k) and its complement graph were shown to be determined by their signless Laplacian and Laplacian spectra [21], respectively, for k = 0 and c 0.…”

Section: Introductionmentioning

confidence: 99%

Some graphs determined by their (signless) Laplacian spectra

Liu

Shan

Das

2014

Linear Algebra and its Applications

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“…In [21], S(n, c, k) was proved to be determined by its signless Laplacian spectrum for c 0 and k = 0. In this section, we shall show that S(n, c, k) is also determined by its signless Laplacian spectrum for c 1 and k 1.…”

Section: S(n C K) Is Determined By Its Signless Laplacian Spectrummentioning

confidence: 99%