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5 pages, 164 KiB

Open AccessEditorial

Signal and Information Processing in Networks

by Minyu Feng, Liang-Jian Deng and Feng Chen

Entropy 2023, 25(12), 1643; https://doi.org/10.3390/e25121643 - 11 Dec 2023

Viewed by 1335

Abstract

Networks are omnipresent in the realm of science, serving as a central focus in our modern world [...] Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

Research

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19 pages, 1278 KiB

Open AccessArticle

Deep-Learning-Based Classification of Cyclic-Alternating-Pattern Sleep Phases

by Yoav Kahana, Aviad Aberdam, Alon Amar and Israel Cohen

Entropy 2023, 25(10), 1395; https://doi.org/10.3390/e25101395 - 28 Sep 2023

Viewed by 1346

Abstract

Determining the cyclic-alternating-pattern (CAP) phases in sleep using electroencephalography (EEG) signals is crucial for assessing sleep quality. However, most current methods for CAP classification primarily rely on classical machine learning techniques, with limited implementation of deep-learning-based tools. Furthermore, these methods often require manual [...] Read more.

Determining the cyclic-alternating-pattern (CAP) phases in sleep using electroencephalography (EEG) signals is crucial for assessing sleep quality. However, most current methods for CAP classification primarily rely on classical machine learning techniques, with limited implementation of deep-learning-based tools. Furthermore, these methods often require manual feature extraction. Herein, we propose a fully automatic deep-learning-based algorithm that leverages convolutional neural network architectures to classify the EEG signals via their time-frequency representations. Through our investigation, we explored using time-frequency analysis techniques and found that Wigner-based representations outperform the commonly used short-time Fourier transform for CAP classification. Additionally, our algorithm incorporates contextual information of the EEG signals and employs data augmentation techniques specifically designed to preserve the time-frequency structure. The model is developed using EEG signals of healthy subjects from the publicly available CAP sleep database (CAPSLPDB) on Physionet. An experimental study demonstrates that our algorithm surpasses existing machine-learning-based methods, achieving an accuracy of 77.5% on a balanced test set and 81.8% when evaluated on an unbalanced test set. Notably, the proposed algorithm exhibits efficiency and scalability, making it suitable for on-device implementation to enhance CAP identification procedures. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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14 pages, 5689 KiB

Open AccessArticle

Network Analysis of Depression Using Magnetoencephalogram Based on Polynomial Kernel Granger Causality

by Yijia Ma, Jing Qian, Qizhang Gu, Wanyi Yi, Wei Yan, Jianxuan Yuan and Jun Wang

Entropy 2023, 25(9), 1330; https://doi.org/10.3390/e25091330 - 13 Sep 2023

Cited by 1 | Viewed by 1456

Abstract

Depression is a psychiatric disorder characterized by anxiety, pessimism, and suicidal tendencies, which has serious impact on human’s life. In this paper, we use Granger causality index based on polynomial kernel as network node connectivity coefficient to construct brain networks from the magnetoencephalogram [...] Read more.

Depression is a psychiatric disorder characterized by anxiety, pessimism, and suicidal tendencies, which has serious impact on human’s life. In this paper, we use Granger causality index based on polynomial kernel as network node connectivity coefficient to construct brain networks from the magnetoencephalogram (MEG) of 5 depressed patients and 11 healthy individuals under positive, neutral, and negative emotional stimuli, respectively. We found that depressed patients had more information exchange between the frontal and occipital regions compared to healthy individuals and less causal connections in the parietal and central regions. We further analyzed the topological properties of the network revealed and found that depressed patients had higher average degrees under negative stimuli (p = 0.008) and lower average clustering coefficients than healthy individuals (p = 0.034). When comparing the average degree and average clustering coefficient of the same sample under different emotional stimuli, we found that depressed patients had a higher average degree and average clustering coefficient under negative stimuli than neutral and positive stimuli. We also found that the characteristic path lengths of patients under negative and neutral stimuli significantly deviated from small-world network. Our results suggest that the analysis of polynomial kernel Granger causality brain networks can effectively characterize the pathology of depression. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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33 pages, 13769 KiB

Open AccessArticle

Prediction of Contact Fatigue Performance Degradation Trends Based on Multi-Domain Features and Temporal Convolutional Networks

by Yu Liu, Yuanbo Liu and Yan Yang

Entropy 2023, 25(9), 1316; https://doi.org/10.3390/e25091316 - 9 Sep 2023

Cited by 2 | Viewed by 1243

Abstract

Contact fatigue is one of the most common failure forms of typical basic components such as bearings and gears. Accurate prediction of contact fatigue performance degradation trends in components is conducive to the scientific formulation of maintenance strategies and health management of equipment, [...] Read more.

Contact fatigue is one of the most common failure forms of typical basic components such as bearings and gears. Accurate prediction of contact fatigue performance degradation trends in components is conducive to the scientific formulation of maintenance strategies and health management of equipment, which is of great significance for industrial production. In this paper, to realize the performance degradation trend prediction accurately, a prediction method based on multi-domain features and temporal convolutional networks (TCNs) was proposed. Firstly, a multi-domain and high-dimensional feature set of vibration signals was constructed, and performance degradation indexes with good sensitivity and strong trends were initially screened using comprehensive evaluation indexes. Secondly, the kernel principal component analysis (KPCA) method was used to eliminate redundant information among multi-domain features and construct health indexes (HIs) based on a convolutional autoencoder (CAE) network. Then, the performance degradation trend prediction model based on TCN was constructed, and the degradation trend prediction for the monitored object was realized using direct multi-step prediction. On this basis, the effectiveness of the proposed method was verified using a bearing common-use data set, and it was successfully applied to performance degradation trend prediction for rolling contact fatigue specimens. The results show that using KPCA can reduce the feature set from 14 dimensions to 4 dimensions and retain 98.33% of the information in the original preferred feature set. The method of constructing the HI based on CAE is effective, and change processes versus time of the constructed HI can truly reflect the degradation process of rolling contact fatigue specimen performance; this method has obvious advantages over the two commonly used methods for constructing HIs including auto-encoding (AE) networks and gaussian mixture models (GMMs). The model based on TCN can accurately predict the performance degradation of rolling contact fatigue specimens. Compared with prediction models based on long short-term memory (LSTM) networks and gating recurrent units (GRUs), the model based on TCN has better performance and higher prediction accuracy. The RMS error and average absolute error for a prediction step of 3 are 0.0146 and 0.0105, respectively. Overall, the proposed method has universal significance and can be applied to predict the performance degradation trend of other mechanical equipment/parts. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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20 pages, 5422 KiB

Open AccessArticle

Projective Synchronization of Delayed Uncertain Coupled Memristive Neural Networks and Their Application

by Zhen Han, Naipeng Chen, Xiaofeng Wei, Manman Yuan and Huijia Li

Entropy 2023, 25(8), 1241; https://doi.org/10.3390/e25081241 - 21 Aug 2023

Cited by 2 | Viewed by 1276

Abstract

In this article, the authors analyzed the nonlinear effects of projective synchronization between coupled memristive neural networks (MNNs) and their applications. Since the complete signal transmission is difficult under parameter mismatch and different projective factors, the delays, which are time-varying, and uncertainties have [...] Read more.

In this article, the authors analyzed the nonlinear effects of projective synchronization between coupled memristive neural networks (MNNs) and their applications. Since the complete signal transmission is difficult under parameter mismatch and different projective factors, the delays, which are time-varying, and uncertainties have been taken to realize the projective synchronization of MNNs with multi-links under the nonlinear control method. Through the extended comparison principle and a new approach to dealing with the mismatched parameters, sufficient criteria have been determined under different types of projective factors and the framework of the Lyapunov–Krasovskii functional (LKF) for projective convergence of the coupled MNNs. Instead of the classical treatment for secure communication, the concept of error of synchronization between the drive and response systems has been applied to solve the signal encryption/decryption problem. Finally, the simulations in numerical form have been demonstrated graphically to confirm the adaptability of the theoretical results. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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17 pages, 701 KiB

Open AccessArticle

Density-Based Entropy Centrality for Community Detection in Complex Networks

by Krista Rizman Žalik and Mitja Žalik

Entropy 2023, 25(8), 1196; https://doi.org/10.3390/e25081196 - 11 Aug 2023

Viewed by 1779

Abstract

One of the most important problems in complex networks is the location of nodes that are essential or play a main role in the network. Nodes with main local roles are the centers of real communities. Communities are sets of nodes of complex [...] Read more.

One of the most important problems in complex networks is the location of nodes that are essential or play a main role in the network. Nodes with main local roles are the centers of real communities. Communities are sets of nodes of complex networks and are densely connected internally. Choosing the right nodes as seeds of the communities is crucial in determining real communities. We propose a new centrality measure named density-based entropy centrality for the local identification of the most important nodes. It measures the entropy of the sum of the sizes of the maximal cliques to which each node and its neighbor nodes belong. The proposed centrality is a local measure for explaining the local influence of each node, which provides an efficient way to locally identify the most important nodes and for community detection because communities are local structures. It can be computed independently for individual vertices, for large networks, and for not well-specified networks. The use of the proposed density-based entropy centrality for community seed selection and community detection outperforms other centrality measures. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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16 pages, 2385 KiB

Open AccessArticle

Leveraging Deep Learning for IoT Transceiver Identification

by Jiayao Gao, Hongfei Fan, Yumei Zhao and Yang Shi

Entropy 2023, 25(8), 1191; https://doi.org/10.3390/e25081191 - 10 Aug 2023

Cited by 2 | Viewed by 1303

Abstract

With the increasing demand for Internet of Things (IoT) network applications, the lack of adequate identification and authentication has become a significant security concern. Radio frequency fingerprinting techniques, which utilize regular radio traffic as the identification source, were then proposed to provide a [...] Read more.

With the increasing demand for Internet of Things (IoT) network applications, the lack of adequate identification and authentication has become a significant security concern. Radio frequency fingerprinting techniques, which utilize regular radio traffic as the identification source, were then proposed to provide a more secured identification approach compared to traditional security methods. Such solutions take hardware-level characteristics as device fingerprints to mitigate the risk of pre-shared key leakage and lower computational complexity. However, the existing studies suffer from problems such as location dependence. In this study, we have proposed a novel scheme for further exploiting the spectrogram and the carrier frequency offset (CFO) as identification sources. A convolutional neural network (CNN) is chosen as the classifier. The scheme addressed the location-dependence problem in the existing identification schemes. Experimental evaluations with data collected in the real world have indicated that the proposed approach can achieve 80% accuracy even if the training and testing data are collected on different days and at different locations, which is 13% higher than state-of-the-art approaches. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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13 pages, 660 KiB

Open AccessArticle

Robustness of Sparsely Distributed Representations to Adversarial Attacks in Deep Neural Networks

by Nida Sardar, Sundas Khan, Arend Hintze and Priyanka Mehra

Entropy 2023, 25(6), 933; https://doi.org/10.3390/e25060933 - 13 Jun 2023

Cited by 2 | Viewed by 1997

Abstract

Deep learning models have achieved an impressive performance in a variety of tasks, but they often suffer from overfitting and are vulnerable to adversarial attacks. Previous research has shown that dropout regularization is an effective technique that can improve model generalization and robustness. [...] Read more.

Deep learning models have achieved an impressive performance in a variety of tasks, but they often suffer from overfitting and are vulnerable to adversarial attacks. Previous research has shown that dropout regularization is an effective technique that can improve model generalization and robustness. In this study, we investigate the impact of dropout regularization on the ability of neural networks to withstand adversarial attacks, as well as the degree of “functional smearing” between individual neurons in the network. Functional smearing in this context describes the phenomenon that a neuron or hidden state is involved in multiple functions at the same time. Our findings confirm that dropout regularization can enhance a network’s resistance to adversarial attacks, and this effect is only observable within a specific range of dropout probabilities. Furthermore, our study reveals that dropout regularization significantly increases the distribution of functional smearing across a wide range of dropout rates. However, it is the fraction of networks with lower levels of functional smearing that exhibit greater resilience against adversarial attacks. This suggests that, even though dropout improves robustness to fooling, one should instead try to decrease functional smearing. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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12 pages, 480 KiB

Open AccessArticle

TTANAD: Test-Time Augmentation for Network Anomaly Detection

by Seffi Cohen, Niv Goldshlager, Bracha Shapira and Lior Rokach

Entropy 2023, 25(5), 820; https://doi.org/10.3390/e25050820 - 19 May 2023

Viewed by 1943

Abstract

Machine learning-based Network Intrusion Detection Systems (NIDS) are designed to protect networks by identifying anomalous behaviors or improper uses. In recent years, advanced attacks, such as those mimicking legitimate traffic, have been developed to avoid alerting such systems. Previous works mainly focused on [...] Read more.

Machine learning-based Network Intrusion Detection Systems (NIDS) are designed to protect networks by identifying anomalous behaviors or improper uses. In recent years, advanced attacks, such as those mimicking legitimate traffic, have been developed to avoid alerting such systems. Previous works mainly focused on improving the anomaly detector itself, whereas in this paper, we introduce a novel method, Test-Time Augmentation for Network Anomaly Detection (TTANAD), which utilizes test-time augmentation to enhance anomaly detection from the data side. TTANAD leverages the temporal characteristics of traffic data and produces temporal test-time augmentations on the monitored traffic data. This method aims to create additional points of view when examining network traffic during inference, making it suitable for a variety of anomaly detector algorithms. Our experimental results demonstrate that TTANAD outperforms the baseline in all benchmark datasets and with all examined anomaly detection algorithms, according to the Area Under the Receiver Operating Characteristic (AUC) metric. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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17 pages, 14035 KiB

Open AccessArticle

Few Shot Class Incremental Learning via Efficient Prototype Replay and Calibration

by Wei Zhang and Xiaodong Gu

Entropy 2023, 25(5), 776; https://doi.org/10.3390/e25050776 - 10 May 2023

Cited by 4 | Viewed by 2633

Abstract

Few shot class incremental learning (FSCIL) is an extremely challenging but valuable problem in real-world applications. When faced with novel few shot tasks in each incremental stage, it should take into account both catastrophic forgetting of old knowledge and overfitting of new categories [...] Read more.

Few shot class incremental learning (FSCIL) is an extremely challenging but valuable problem in real-world applications. When faced with novel few shot tasks in each incremental stage, it should take into account both catastrophic forgetting of old knowledge and overfitting of new categories with limited training data. In this paper, we propose an efficient prototype replay and calibration (EPRC) method with three stages to improve classification performance. We first perform effective pre-training with rotation and mix-up augmentations in order to obtain a strong backbone. Then a series of pseudo few shot tasks are sampled to perform meta-training, which enhances the generalization ability of both the feature extractor and projection layer and then helps mitigate the over-fitting problem of few shot learning. Furthermore, an even nonlinear transformation function is incorporated into the similarity computation to implicitly calibrate the generated prototypes of different categories and alleviate correlations among them. Finally, we replay the stored prototypes to relieve catastrophic forgetting and rectify prototypes to be more discriminative in the incremental-training stage via an explicit regularization within the loss function. The experimental results on CIFAR-100 and miniImageNet demonstrate that our EPRC significantly boosts the classification performance compared with existing mainstream FSCIL methods. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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18 pages, 5615 KiB

Open AccessArticle

Robust Superpixel Segmentation for Hyperspectral-Image Restoration

by Ya-Ru Fan

Entropy 2023, 25(2), 260; https://doi.org/10.3390/e25020260 - 31 Jan 2023

Cited by 1 | Viewed by 1900

Abstract

Hyperspectral-image (HSI) restoration plays an essential role in remote sensing image processing. Recently, superpixel segmentation-based the low-rank regularized methods for HSI restoration have shown outstanding performance. However, most of them simply segment the HSI according to its first principal component, which is suboptimal. [...] Read more.

Hyperspectral-image (HSI) restoration plays an essential role in remote sensing image processing. Recently, superpixel segmentation-based the low-rank regularized methods for HSI restoration have shown outstanding performance. However, most of them simply segment the HSI according to its first principal component, which is suboptimal. In this paper, integrating the superpixel segmentation with principal component analysis, we propose a robust superpixel segmentation strategy to better divide the HSI, which can further enhance the low-rank attribute of the HSI. To better employ the low-rank attribute, the weighted nuclear norm by three types of weighting is proposed to efficiently remove the mixed noise in degraded HSI. Experiments conducted on simulated and real HSI data verify the performance of the proposed method for HSI restoration. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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17 pages, 3019 KiB

Open AccessArticle

Robustness of Multi-Project Knowledge Collaboration Network in Open Source Community

by Xiaodong Zhang, Shaojuan Lei, Jiazheng Sun and Weijie Kou

Entropy 2023, 25(1), 108; https://doi.org/10.3390/e25010108 - 4 Jan 2023

Cited by 2 | Viewed by 1575

Abstract

Multi-project parallelism is an important feature of open source communities (OSCs), and multi-project collaboration among users is a favorable condition for an OSC’s development. This paper studies the robustness of this type of community. Based on the characteristics of knowledge collaboration behavior and [...] Read more.

Multi-project parallelism is an important feature of open source communities (OSCs), and multi-project collaboration among users is a favorable condition for an OSC’s development. This paper studies the robustness of this type of community. Based on the characteristics of knowledge collaboration behavior and the large amount of semantic content generated from user collaboration in open source projects, we construct a directed, weighted, semantic-based multi-project knowledge collaboration network. Using analysis of the KCN’s structure and user attributes, nodes are divided into knowledge collaboration nodes and knowledge dissemination nodes that participate in either multi- or single-project collaboration. From the perspectives of user churn and behavior degradation, two types of failure modes are constructed: node failure and edge failure. Based on empirical data from the Local Motors open source vehicle design community, we then carry out a dynamic robustness analysis experiment. Our results show that the robustness of our constructed network varies for different failure modes and different node types: the network has (1) a high robustness to random failure and a low robustness to deliberate failure, (2) a high robustness to edge failure and a low robustness to node failure, and (3) a high robustness to the failure of single-project nodes (or their edges) and a low robustness to the failure of multi-project nodes (or their edges). These findings can be used to provide a more comprehensive and targeted management reference, promoting the efficient development of OSCs. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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20 pages, 651 KiB

Open AccessArticle

Distributed Support Vector Ordinal Regression over Networks

by Huan Liu, Jiankai Tu and Chunguang Li

Entropy 2022, 24(11), 1567; https://doi.org/10.3390/e24111567 - 31 Oct 2022

Cited by 2 | Viewed by 1806

Abstract

Ordinal regression methods are widely used to predict the ordered labels of data, among which support vector ordinal regression (SVOR) methods are popular because of their good generalization. In many realistic circumstances, data are collected by a distributed network. In order to protect [...] Read more.

Ordinal regression methods are widely used to predict the ordered labels of data, among which support vector ordinal regression (SVOR) methods are popular because of their good generalization. In many realistic circumstances, data are collected by a distributed network. In order to protect privacy or due to some practical constraints, data cannot be transmitted to a center for processing. However, as far as we know, existing SVOR methods are all centralized. In the above situations, centralized methods are inapplicable, and distributed methods are more suitable choices. In this paper, we propose a distributed SVOR (dSVOR) algorithm. First, we formulate a constrained optimization problem for SVOR in distributed circumstances. Since there are some difficulties in solving the problem with classical methods, we used the random approximation method and the hinge loss function to transform the problem into a convex optimization problem with constraints. Then, we propose subgradient-based algorithm dSVOR to solve it. To illustrate the effectiveness, we theoretically analyze the consensus and convergence of the proposed method, and conduct experiments on both synthetic data and a real-world example. The experimental results show that the proposed dSVOR could achieve close performance to that of the corresponding centralized method, which needs all the data to be collected together. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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14 pages, 946 KiB

Open AccessArticle

Decentralized Primal-Dual Proximal Operator Algorithm for Constrained Nonsmooth Composite Optimization Problems over Networks

by Liping Feng, Liang Ran, Guoyang Meng, Jialong Tang, Wentao Ding and Huaqing Li

Entropy 2022, 24(9), 1278; https://doi.org/10.3390/e24091278 - 11 Sep 2022

Viewed by 1687

Abstract

In this paper, we focus on the nonsmooth composite optimization problems over networks, which consist of a smooth term and a nonsmooth term. Both equality constraints and box constraints for the decision variables are also considered. Based on the multi-agent networks, the objective [...] Read more.

In this paper, we focus on the nonsmooth composite optimization problems over networks, which consist of a smooth term and a nonsmooth term. Both equality constraints and box constraints for the decision variables are also considered. Based on the multi-agent networks, the objective problems are split into a series of agents on which the problems can be solved in a decentralized manner. By establishing the Lagrange function of the problems, the first-order optimal condition is obtained in the primal-dual domain. Then, we propose a decentralized algorithm with the proximal operators. The proposed algorithm has uncoordinated stepsizes with respect to agents or edges, where no global parameters are involved. By constructing the compact form of the algorithm with operators, we complete the convergence analysis with the fixed-point theory. With the constrained quadratic programming problem, simulations verify the effectiveness of the proposed algorithm. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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16 pages, 3302 KiB

Open AccessArticle

Research on Throughput-Guaranteed MAC Scheduling Policies in Wireless Networks

by Fan Zhang, Baozhu Li and Gangqiang Yang

Entropy 2022, 24(9), 1246; https://doi.org/10.3390/e24091246 - 4 Sep 2022

Cited by 1 | Viewed by 1543

Abstract

In wireless networks, MAC scheduling methods can be divided into two types according to the implementation model: centralized and distributed scheduling. By reasonably designing MAC scheduling policies, both centralized and distributed schedulers can ensure a reliable throughput capacity region, i.e., realizing throughput-guaranteed. However, [...] Read more.

In wireless networks, MAC scheduling methods can be divided into two types according to the implementation model: centralized and distributed scheduling. By reasonably designing MAC scheduling policies, both centralized and distributed schedulers can ensure a reliable throughput capacity region, i.e., realizing throughput-guaranteed. However, it can be found that some existing throughput-guaranteed scheduling schemes cannot further ensure bounded end-to-end average delay, and the reason for this phenomenon has not been deeply analyzed. In practical communication networks, throughput and delay are equally important. Based on this idea, the existing MAC scheduling strategies are investigated systematically in this paper from two aspects of throughput and delay, and their performances are evaluated and compared through both theoretical analysis and simulation experiments. The work of this paper provides a theoretical basis for the improvement of MAC scheduling technology in the next-generation wireless networks. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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41 pages, 2684 KiB

Open AccessArticle

A Useful Criterion on Studying Consistent Estimation in Community Detection

by Huan Qing

Entropy 2022, 24(8), 1098; https://doi.org/10.3390/e24081098 - 9 Aug 2022

Cited by 4 | Viewed by 1329

Abstract

In network analysis, developing a unified theoretical framework that can compare methods under different models is an interesting problem. This paper proposes a partial solution to this problem. We summarize the idea of using a separation condition for a standard network and sharp [...] Read more.

In network analysis, developing a unified theoretical framework that can compare methods under different models is an interesting problem. This paper proposes a partial solution to this problem. We summarize the idea of using a separation condition for a standard network and sharp threshold of the Erdös–Rényi random graph to study consistent estimation, and compare theoretical error rates and requirements on the network sparsity of spectral methods under models that can degenerate to a stochastic block model as a four-step criterion SCSTC. Using SCSTC, we find some inconsistent phenomena on separation condition and sharp threshold in community detection. In particular, we find that the original theoretical results of the SPACL algorithm introduced to estimate network memberships under the mixed membership stochastic blockmodel are sub-optimal. To find the formation mechanism of inconsistencies, we re-establish the theoretical convergence rate of this algorithm by applying recent techniques on row-wise eigenvector deviation. The results are further extended to the degree-corrected mixed membership model. By comparison, our results enjoy smaller error rates, lesser dependence on the number of communities, weaker requirements on network sparsity, and so forth. The separation condition and sharp threshold obtained from our theoretical results match the classical results, so the usefulness of this criterion on studying consistent estimation is guaranteed. Numerical results for computer-generated networks support our finding that spectral methods considered in this paper achieve the threshold of separation condition. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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