Pub Date : 2022-12-01DOI: 10.48550/arXiv.2212.00445
Thomas Jahn, T. Ullrich, Felix Voigtländer
Using techniques developed recently in the field of compressed sensing we prove new upper bounds for general (nonlinear) sampling numbers of (quasi-)Banach smoothness spaces in $L^2$. In particular, we show that in relevant cases such as mixed and isotropic weighted Wiener classes or Sobolev spaces with mixed smoothness, sampling numbers in $L^2$ can be upper bounded by best $n$-term trigonometric widths in $L^infty$. We describe a recovery procedure from $m$ function values based on $ell^1$-minimization (basis pursuit denoising). With this method, a significant gain in the rate of convergence compared to recently developed linear recovery methods is achieved. In this deterministic worst-case setting we see an additional speed-up of $m^{-1/2}$ (up to log factors) compared to linear methods in case of weighted Wiener spaces. For their quasi-Banach counterparts even arbitrary polynomial speed-up is possible. Surprisingly, our approach allows to recover mixed smoothness Sobolev functions belonging to $S^r_pW(mathbb{T}^d)$ on the $d$-torus with a logarithmically better rate of convergence than any linear method can achieve when $1
利用压缩感知领域最新发展的技术，我们证明了$L^2$中(拟-)Banach平滑空间的一般(非线性)采样数的新上界。特别地，我们证明了在相关的情况下，如混合和各向同性加权Wiener类或具有混合平滑性的Sobolev空间中，$L^2$中的采样数可以被$L^ inty $中的最佳$n$项三角宽度的上界。我们描述了基于$ well ^1$最小化(基追求去噪)的$m$函数值的恢复过程。与最近开发的线性恢复方法相比，这种方法的收敛速度有了显著的提高。在这种确定的最坏情况设置中，我们看到与加权Wiener空间的线性方法相比，$m^{-1/2}$(高达对数因子)的额外加速。对于它们的拟巴拿赫对应物，甚至任意多项式加速是可能的。令人惊讶的是，我们的方法允许在$d$-环面上恢复属于$S^r_pW(mathbb{T}^d)$的混合平滑Sobolev函数，其收敛速度比任何线性方法在$1时都要高
{"title":"Sampling numbers of smoothness classes via 𝓁1-minimization","authors":"Thomas Jahn, T. Ullrich, Felix Voigtländer","doi":"10.48550/arXiv.2212.00445","DOIUrl":"https://doi.org/10.48550/arXiv.2212.00445","url":null,"abstract":"Using techniques developed recently in the field of compressed sensing we prove new upper bounds for general (nonlinear) sampling numbers of (quasi-)Banach smoothness spaces in $L^2$. In particular, we show that in relevant cases such as mixed and isotropic weighted Wiener classes or Sobolev spaces with mixed smoothness, sampling numbers in $L^2$ can be upper bounded by best $n$-term trigonometric widths in $L^infty$. We describe a recovery procedure from $m$ function values based on $ell^1$-minimization (basis pursuit denoising). With this method, a significant gain in the rate of convergence compared to recently developed linear recovery methods is achieved. In this deterministic worst-case setting we see an additional speed-up of $m^{-1/2}$ (up to log factors) compared to linear methods in case of weighted Wiener spaces. For their quasi-Banach counterparts even arbitrary polynomial speed-up is possible. Surprisingly, our approach allows to recover mixed smoothness Sobolev functions belonging to $S^r_pW(mathbb{T}^d)$ on the $d$-torus with a logarithmically better rate of convergence than any linear method can achieve when $1","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80386334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-04DOI: 10.48550/arXiv.2211.02346
Chris Jones, K. Wiesner
Mean field theory models of percolation on networks provide analytic estimates of network robustness under node or edge removal. We introduce a new mean field theory model based on generating functions that includes information about the tree-likeness of each node's local neighbourhood. We show that our new model outperforms all other generating function models in prediction accuracy when testing their estimates on a wide range of real-world network data. We compare the new model's performance against the recently introduced message passing models and provide evidence that the standard version is also outperformed, while the `loopy' version is only outperformed on a targeted attack strategy. As we show, however, the computational complexity of our model implementation is much lower than that of message passing algorithms. We provide evidence that all discussed models are poor in predicting networks with highly modular structure with dispersed modules, which are also characterised by high mixing times, identifying this as a general limitation of percolation prediction models.
{"title":"Improving mean-field network percolation models with neighbourhood information and their limitations on highly modular, highly dispersed networks","authors":"Chris Jones, K. Wiesner","doi":"10.48550/arXiv.2211.02346","DOIUrl":"https://doi.org/10.48550/arXiv.2211.02346","url":null,"abstract":"Mean field theory models of percolation on networks provide analytic estimates of network robustness under node or edge removal. We introduce a new mean field theory model based on generating functions that includes information about the tree-likeness of each node's local neighbourhood. We show that our new model outperforms all other generating function models in prediction accuracy when testing their estimates on a wide range of real-world network data. We compare the new model's performance against the recently introduced message passing models and provide evidence that the standard version is also outperformed, while the `loopy' version is only outperformed on a targeted attack strategy. As we show, however, the computational complexity of our model implementation is much lower than that of message passing algorithms. We provide evidence that all discussed models are poor in predicting networks with highly modular structure with dispersed modules, which are also characterised by high mixing times, identifying this as a general limitation of percolation prediction models.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75813974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-26DOI: 10.48550/arXiv.2210.15009
Bogumil Kami'nski, P. Prałat, F. Théberge
The Artificial Benchmark for Community Detection (ABCD) graph is a recently introduced random graph model with community structure and power-law distribution for both degrees and community sizes. The model generates graphs with similar properties as the well-known Lancichinetti, Fortunato, Radicchi (LFR) one, and its main parameter ξ can be tuned to mimic its counterpart in the LFR model, the mixing parameter μ. In this article, we introduce hypergraph counterpart of the ABCD model, h–ABCD, which also produces random hypergraph with distributions of ground-truth community sizes and degrees following power-law. As in the original ABCD, the new model h–ABCD can produce hypergraphs with various levels of noise. More importantly, the model is flexible and can mimic any desired level of homogeneity of hyperedges that fall into one community. As a result, it can be used as a suitable, synthetic playground for analyzing and tuning hypergraph community detection algorithms. [Received on 22 October 2022; editorial decision on 18 July 2023; accepted on 19 July 2023]
ABCD (Artificial Benchmark for Community Detection)图是近年来提出的一种随机图模型，它具有社团结构和社团大小的幂律分布。该模型生成的图与著名的Lancichinetti, Fortunato, Radicchi (LFR)模型具有相似的性质，并且其主要参数ξ可以被调整以模拟LFR模型中的对应参数，即混合参数μ。在本文中，我们引入了ABCD模型的对应超图h-ABCD, h-ABCD也产生了基于真值社区大小和度服从幂律分布的随机超图。与原来的ABCD一样，新模型h-ABCD可以产生具有不同程度噪声的超图。更重要的是，该模型是灵活的，可以模拟属于一个社区的任何期望级别的超边缘同质性。因此，它可以作为一个合适的综合平台，用于分析和调优超图社区检测算法。[2022年10月22日收到;2023年7月18日的编辑决定;于2023年7月19日接受]
{"title":"Hypergraph Artificial Benchmark for Community Detection (h-ABCD)","authors":"Bogumil Kami'nski, P. Prałat, F. Théberge","doi":"10.48550/arXiv.2210.15009","DOIUrl":"https://doi.org/10.48550/arXiv.2210.15009","url":null,"abstract":"\u0000 The Artificial Benchmark for Community Detection (ABCD) graph is a recently introduced random graph model with community structure and power-law distribution for both degrees and community sizes. The model generates graphs with similar properties as the well-known Lancichinetti, Fortunato, Radicchi (LFR) one, and its main parameter ξ can be tuned to mimic its counterpart in the LFR model, the mixing parameter μ. In this article, we introduce hypergraph counterpart of the ABCD model, h–ABCD, which also produces random hypergraph with distributions of ground-truth community sizes and degrees following power-law. As in the original ABCD, the new model h–ABCD can produce hypergraphs with various levels of noise. More importantly, the model is flexible and can mimic any desired level of homogeneity of hyperedges that fall into one community. As a result, it can be used as a suitable, synthetic playground for analyzing and tuning hypergraph community detection algorithms.\u0000 [Received on 22 October 2022; editorial decision on 18 July 2023; accepted on 19 July 2023]","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80601680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose closed-form analytical expressions to determine the minimum number of driver nodes that is needed to control a specific class of networks. We consider swarm signaling networks with regular out-degree distribution where a fraction $p$ of the links is unavailable. We further apply our method to networks with bi-modal out-degree distributions. Our approximations are validated through intensive simulations. Results show that our approximations have high accuracy when compared with simulation results for both types of out-degree distribution.
{"title":"Controllability of a Class of Swarm Signaling Networks","authors":"Peng Sun, R. Kooij, Roland Bouffanais","doi":"10.1093/comnet/cnac054","DOIUrl":"https://doi.org/10.1093/comnet/cnac054","url":null,"abstract":"In this paper, we propose closed-form analytical expressions to determine the minimum number of driver nodes that is needed to control a specific class of networks. We consider swarm signaling networks with regular out-degree distribution where a fraction $p$ of the links is unavailable. We further apply our method to networks with bi-modal out-degree distributions. Our approximations are validated through intensive simulations. Results show that our approximations have high accuracy when compared with simulation results for both types of out-degree distribution.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80293718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Signed networks and balance theory provide a natural setting for real-world scenarios that show polarization dynamics, positive/negative relationships and political partisanship. For example, they have been proven effective in studying the increasing polarization of the votes in the two chambers of the U.S. Congress from World War II on Andris, Lee, Hamilton, Martino, Gunning & Selden (2015, PLoS ONE, 10, 1–14) and Aref & Neal (2020, Sci. Rep., 10, 1–10). To provide further insights into this particular case study, we propose the application of a pipeline to analyze and visualize a signed graphs configuration based on the exploitation of the corresponding Laplacian matrix spectral properties. The overall methodology is comparable with others based on the frustration index, but it has at least two main advantages: first, it requires a much lower computational cost and second, it allows for a quantitative and visual assessment of how arbitrarily small subgraphs (even single nodes) contribute to the overall balance (or unbalance) of the network. The proposed pipeline allows the exploration of polarization dynamics shown by the U.S. Congress from 1945 to 2020 at different resolution scales. In fact, we are able to spot and point out the influence of some (groups of) congressmen in the overall balance, as well as to observe and explore polarizations evolution of both chambers across the years.
{"title":"Analyzing and visualizing polarization and balance with signed networks: the U.S. Congress case study","authors":"A. Capozzi, Alfonso Semeraro, G. Ruffo","doi":"10.1093/comnet/cnad027","DOIUrl":"https://doi.org/10.1093/comnet/cnad027","url":null,"abstract":"\u0000 Signed networks and balance theory provide a natural setting for real-world scenarios that show polarization dynamics, positive/negative relationships and political partisanship. For example, they have been proven effective in studying the increasing polarization of the votes in the two chambers of the U.S. Congress from World War II on Andris, Lee, Hamilton, Martino, Gunning & Selden (2015, PLoS ONE, 10, 1–14) and Aref & Neal (2020, Sci. Rep., 10, 1–10). To provide further insights into this particular case study, we propose the application of a pipeline to analyze and visualize a signed graphs configuration based on the exploitation of the corresponding Laplacian matrix spectral properties. The overall methodology is comparable with others based on the frustration index, but it has at least two main advantages: first, it requires a much lower computational cost and second, it allows for a quantitative and visual assessment of how arbitrarily small subgraphs (even single nodes) contribute to the overall balance (or unbalance) of the network. The proposed pipeline allows the exploration of polarization dynamics shown by the U.S. Congress from 1945 to 2020 at different resolution scales. In fact, we are able to spot and point out the influence of some (groups of) congressmen in the overall balance, as well as to observe and explore polarizations evolution of both chambers across the years.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81449196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-29DOI: 10.48550/arXiv.2208.13610
P. Kritzer
Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-$1$ lattice rule to approximate an $s$-dimensional integral is fully specified by its emph{generating vector} $boldsymbol{z} in mathbb{Z}^s$ and its number of points~$N$. While there are many results on the existence of ``good'' rank-$1$ lattice rules, there are no explicit constructions of good generating vectors for dimensions $s ge 3$. This is why one usually resorts to computer search algorithms. In a recent paper by Ebert et al. in the Journal of Complexity, we showed a component-by-component digit-by-digit (CBC-DBD) construction for good generating vectors of rank-1 lattice rules for integration of functions in weighted Korobov classes. However, the result in that paper was limited to product weights. In the present paper, we shall generalize this result to arbitrary positive weights, thereby answering an open question posed in the paper of Ebert et al. We also include a short section on how the algorithm can be implemented in the case of POD weights, by which we see that the CBC-DBD construction is competitive with the classical CBC construction.
格规则是研究最突出的拟蒙特卡罗方法来近似多元积分。近似$s$维积分的秩- $1$点阵规则由其emph{生成向量}$boldsymbol{z} in mathbb{Z}^s$及其点数$N$完全指定。虽然有许多关于“好”秩- $1$格规则存在的结果，但没有关于维度$s ge 3$的好生成向量的明确构造。这就是为什么人们通常求助于计算机搜索算法。在Ebert等人最近发表在《复杂性杂志》上的一篇论文中，我们展示了一种组件-组件-数字-数字(CBC-DBD)构造，用于加权Korobov类中函数积分的秩-1格规则的良好生成向量。然而，该论文的结果仅限于产品权重。在本文中，我们将这个结果推广到任意正权，从而回答了Ebert等人的论文中提出的一个开放性问题。我们还包括一个关于如何在POD权重的情况下实现算法的简短部分，通过该部分我们可以看到CBC- dbd结构与经典CBC结构相竞争。
{"title":"A note on the CBC-DBD construction of lattice rules with general positive weights","authors":"P. Kritzer","doi":"10.48550/arXiv.2208.13610","DOIUrl":"https://doi.org/10.48550/arXiv.2208.13610","url":null,"abstract":"Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-$1$ lattice rule to approximate an $s$-dimensional integral is fully specified by its emph{generating vector} $boldsymbol{z} in mathbb{Z}^s$ and its number of points~$N$. While there are many results on the existence of ``good'' rank-$1$ lattice rules, there are no explicit constructions of good generating vectors for dimensions $s ge 3$. This is why one usually resorts to computer search algorithms. In a recent paper by Ebert et al. in the Journal of Complexity, we showed a component-by-component digit-by-digit (CBC-DBD) construction for good generating vectors of rank-1 lattice rules for integration of functions in weighted Korobov classes. However, the result in that paper was limited to product weights. In the present paper, we shall generalize this result to arbitrary positive weights, thereby answering an open question posed in the paper of Ebert et al. We also include a short section on how the algorithm can be implemented in the case of POD weights, by which we see that the CBC-DBD construction is competitive with the classical CBC construction.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75738677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Haji Gul, F. Al-Obeidat, Adnan Amin, Muhammad Mohsin Tahir, Kaizhu Huang
Link prediction in a complex network is a difficult and challenging issue to address. Link prediction tries to better predict relationships, interactions and friendships based on historical knowledge of the complex network graph. Many link prediction techniques exist, including the common neighbour, Adamic-Adar, Katz and Jaccard coefficient, which use node information, local and global routes, and previous knowledge of a complex network to predict the links. These methods are extensively used in various applications because of their interpretability and convenience of use, irrespective of the fact that the majority of these methods were designed for a specific field. This study offers a unique link prediction approach based on the matrix-forest metric and vertex local structural information in a real-world complex network. We empirically examined the proposed link prediction method over 13 real-world network datasets obtained from various sources. Extensive experiments were performed that demonstrated the superior efficacy of the proposed link prediction method compared to other methods and outperformed the existing state-of-the-art in terms of prediction accuracy.
{"title":"Efficient link prediction model for real-world complex networks using matrix-forest metric with local similarity features","authors":"Haji Gul, F. Al-Obeidat, Adnan Amin, Muhammad Mohsin Tahir, Kaizhu Huang","doi":"10.1093/comnet/cnac039","DOIUrl":"https://doi.org/10.1093/comnet/cnac039","url":null,"abstract":"\u0000 Link prediction in a complex network is a difficult and challenging issue to address. Link prediction tries to better predict relationships, interactions and friendships based on historical knowledge of the complex network graph. Many link prediction techniques exist, including the common neighbour, Adamic-Adar, Katz and Jaccard coefficient, which use node information, local and global routes, and previous knowledge of a complex network to predict the links. These methods are extensively used in various applications because of their interpretability and convenience of use, irrespective of the fact that the majority of these methods were designed for a specific field. This study offers a unique link prediction approach based on the matrix-forest metric and vertex local structural information in a real-world complex network. We empirically examined the proposed link prediction method over 13 real-world network datasets obtained from various sources. Extensive experiments were performed that demonstrated the superior efficacy of the proposed link prediction method compared to other methods and outperformed the existing state-of-the-art in terms of prediction accuracy.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85414906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Estimating the number of eigenvalues $mu_{[a,b]}$ of a network’s adjacency matrix in a given interval $[a,b]$ is essential in several fields. The straightforward approach consists of calculating all the eigenvalues in $O(n^3)$ (where $n$ is the number of nodes in the network) and then counting the ones that belong to the interval $[a,b]$. Another approach is to use Sylvester’s law of inertia, which also requires $O(n^3)$. Although both methods provide the exact number of eigenvalues in $[a,b]$, their application for large networks is computationally infeasible. Sometimes, an approximation of $mu_{[a,b]}$ is enough. In this case, Chebyshev’s method approximates $mu_{[a,b]}$ in $O(|E|)$ (where $|E|$ is the number of edges). This study presents two alternatives to compute $mu_{[a,b]}$ for locally tree-like networks: edge- and degree-based algorithms. The former presented a better accuracy than Chebyshev’s method. It runs in $O(d|E|)$, where $d$ is the number of iterations. The latter presented slightly lower accuracy but ran linearly ($O(n)$).
{"title":"Efficient eigenvalue counts for tree-like networks","authors":"Grover E. C. Guzman, P. Stadler, André Fujita","doi":"10.1093/comnet/cnac040","DOIUrl":"https://doi.org/10.1093/comnet/cnac040","url":null,"abstract":"\u0000 Estimating the number of eigenvalues $mu_{[a,b]}$ of a network’s adjacency matrix in a given interval $[a,b]$ is essential in several fields. The straightforward approach consists of calculating all the eigenvalues in $O(n^3)$ (where $n$ is the number of nodes in the network) and then counting the ones that belong to the interval $[a,b]$. Another approach is to use Sylvester’s law of inertia, which also requires $O(n^3)$. Although both methods provide the exact number of eigenvalues in $[a,b]$, their application for large networks is computationally infeasible. Sometimes, an approximation of $mu_{[a,b]}$ is enough. In this case, Chebyshev’s method approximates $mu_{[a,b]}$ in $O(|E|)$ (where $|E|$ is the number of edges). This study presents two alternatives to compute $mu_{[a,b]}$ for locally tree-like networks: edge- and degree-based algorithms. The former presented a better accuracy than Chebyshev’s method. It runs in $O(d|E|)$, where $d$ is the number of iterations. The latter presented slightly lower accuracy but ran linearly ($O(n)$).","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84515635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The stochastic block model (SBM) is a popular model for community detecting problems. Many community detecting approaches have been proposed, and most of them assume that the number of communities is given previously. However, in practice, the number of communities is often unknown. Plenty of approaches were proposed to estimate the number of communities, but most of them were computationally intensive. Moreover, when outliers exist, there are no approaches to consistently estimate the number of communities. In this article, we propose a fast method based on the eigenvalues of the regularized and normalized adjacency matrix to estimate the number of communities under the SBM with outliers. We show that our method can consistently estimate the number of communities when outliers exist. Moreover, we extend our method to the degree-corrected SBM. We show that our approach is comparable to the other existing approaches in simulations. We also illustrate our approach on four real-world networks.
{"title":"Estimating the number of communities in the stochastic block model with outliers","authors":"Jingsong Xiao, Fei Ye, Weidong Ma, Ying Yang","doi":"10.1093/comnet/cnac042","DOIUrl":"https://doi.org/10.1093/comnet/cnac042","url":null,"abstract":"\u0000 The stochastic block model (SBM) is a popular model for community detecting problems. Many community detecting approaches have been proposed, and most of them assume that the number of communities is given previously. However, in practice, the number of communities is often unknown. Plenty of approaches were proposed to estimate the number of communities, but most of them were computationally intensive. Moreover, when outliers exist, there are no approaches to consistently estimate the number of communities. In this article, we propose a fast method based on the eigenvalues of the regularized and normalized adjacency matrix to estimate the number of communities under the SBM with outliers. We show that our method can consistently estimate the number of communities when outliers exist. Moreover, we extend our method to the degree-corrected SBM. We show that our approach is comparable to the other existing approaches in simulations. We also illustrate our approach on four real-world networks.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84026306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-19DOI: 10.48550/arXiv.2208.09108
D. Dung
We investigate the approximation of weighted integrals over $mathbb{R}^d$ for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to $n$ integration nodes for functions from these spaces. In the one-dimensional case $(d=1)$, we obtain the right convergence rate of optimal quadratures. For $d ge 2$, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain $mathbb{R}^d$.
{"title":"Numerical weighted integration of functions having mixed smoothness","authors":"D. Dung","doi":"10.48550/arXiv.2208.09108","DOIUrl":"https://doi.org/10.48550/arXiv.2208.09108","url":null,"abstract":"We investigate the approximation of weighted integrals over $mathbb{R}^d$ for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to $n$ integration nodes for functions from these spaces. In the one-dimensional case $(d=1)$, we obtain the right convergence rate of optimal quadratures. For $d ge 2$, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain $mathbb{R}^d$.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79655330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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