Revisiting Power-law Distributions in Spectra of Real World Networks

Eikmeier, Nicole; Gleich, David F.

doi:10.1145/3097983.3098128

Cited by 24 publications

(15 citation statements)

References 46 publications

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“…The bulk of the spectrum has triangular shape. This is very different from the classical semi-circle law by Wigner which holds for Erdős-Rényi graph (e.g., [1], [12], [13], [22]). The edge eigenvalues seem to follow a power law.…”

Section: Introductioncontrasting

confidence: 63%

Spectrum of Complex Networks

Montealegre¹,

Vu²

2019

Internet Mathematics

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The study of complex networks has been one of the most active fields in science in recent decades. Spectral properties of networks (or graphs that represent them) are of fundamental importance. Researchers have been investigating these properties for many years, and, based on numerical data, have raised a number of questions about the distribution of the eigenvalues and eigenvectors.In this paper, we give the solution to some of these questions. In particular, we determine the limiting distribution of (the bulk of) the spectrum as the size of the network grows to infinity and show that the leading eigenvectors are strongly localized.We focus on the preferential attachment graph, which is the most popular mathematical model for growing complex networks. Our analysis is, on the other hand, general and can be applied to other models.

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Section: Introductioncontrasting

confidence: 63%

Spectrum of Complex Networks

Montealegre¹,

Vu²

2019

Internet Mathematics

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“…The quantum algorithm is sublinear, while the dequantized one is not. Based on the distribution of eigenvalues of the Laplacian for real-world graphs (Eikmeier and Gleich 2017) and for kernel matrices (Wathen and Zhu 2015), which decay fast, we can expect this is the most realistic case. These observations agree with a recent work Arrazola et al (2019) that studied the performance of quantuminspired algorithms in practice and concluded that their performance degrades significantly when the rank and condition number of the input matrix are increased, and high performance requires very low rank and condition number.…”

Section: Complexity Quantum Semi-supervised Ls-svmmentioning

confidence: 99%

Quantum semi-supervised kernel learning

Saeedi

Panahi

Arodz

2021

Quantum Mach. Intell.

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Quantum machine learning methods have the potential to facilitate learning using extremely large datasets. While the availability of data for training machine learning models is steadily increasing, oftentimes it is much easier to collect feature vectors to obtain the corresponding labels. One of the approaches for addressing this issue is to use semi-supervised learning, which leverages not only the labeled samples, but also unlabeled feature vectors. Here, we present a quantum machine learning algorithm for training semi-supervised kernel support vector machines. The algorithm uses recent advances in quantum sample-based Hamiltonian simulation to extend the existing quantum LS-SVM algorithm to handle the semisupervised term in the loss. Through a theoretical study of the algorithm's computational complexity, we show that it maintains the same speedup as the fully-supervised quantum LS-SVM.

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“…By studying the structure of datasets, we find evidence that controversial issues follow the power-law distribution [5], a small number of classes gather at the top of the distribution and take up the great majority of the whole controversial issues. Since such social problems are mostly composed of complex networks that follow the power-law distribution [6], [7], it is not surprising that the data structure of controversial issues has similar properties. The power-law distribution of controversial issues indicates that a few classes of them are common, while most classes are rare.…”

Section: Introductionmentioning

confidence: 99%

Few-Shot Learning for Chinese Legal Controversial Issues Classification

Yin

Tian

et al. 2020

IEEE Access

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Chinese courts organize debates surrounding controversial issues along with the gradual formation of the new procedural system. With the progress of China's judicial reform, more than 80 million judgement documents have been made public online. Similar controversial issues identified in and among the massive public judgment documents are of significant value for judges in their trial work. Hence, homogeneous controversial issues classification becomes the basis for similar cases retrieval. However, controversial issues follow the power-law distribution, not all of them are within the labels provided by manual annotation and their categories cannot be exhausted. In order to generalize those unfamiliar categories without necessitating extensive retraining, we propose a controversial issues classification algorithm based on few-shot learning. Two few-shot learning algorithms are proposed for our controversial issues problem, Relation Network and Induction Network, respectively. With only a handful of given instances, both of them have shown excellent results on the two datasets, which proves their effectiveness in adapting to accommodating new categories not seen in training. The proposed method provides trial assistance for judges, promotes the dissemination of experience and improves fairness of adjudication. INDEX TERMS Controversial issues, few-shot learning, text classification, power-law, BERT.

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Revisiting Power-law Distributions in Spectra of Real World Networks

Cited by 24 publications

References 46 publications

Spectrum of Complex Networks

Spectrum of Complex Networks

Quantum semi-supervised kernel learning

Few-Shot Learning for Chinese Legal Controversial Issues Classification

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