The stochastic block model (SBM) is a popular framework for studying community detection in networks. This model is limited by the assumption that all nodes in the same community are statistically equivalent and have equal expected degrees. The degree-corrected stochastic block model (DCSBM) is a natural extension of SBM that allows for degree heterogeneity within communities. This paper proposes a convexified modularity maximization approach for estimating the hidden communities under DCSBM. Our approach is based on a convex programming relaxation of the classical (generalized) modularity maximization formulation, followed by a novel doubly-weighted @math -norm @math -median procedure. We establish non-asymptotic theoretical guarantees for both approximate clustering and perfect clustering. Our approximate clustering results are insensitive to the minimum degree, and hold even in sparse regime with bounded average degrees. In the special case of SBM, these theoretical results match the best-known performance guarantees of computationally feasible algorithms. Numerically, we provide an efficient implementation of our algorithm, which is applied to both synthetic and real-world networks. Experiment results show that our method enjoys competitive performance compared to the state of the art in the literature.
It is observed in @cite_33 that spectral clustering based directly the adjacency matrix (or their normalized version) often result in inconsistent clustering in real data, such as the political blogs dataset @cite_6 , a popular benchmark for testing community detection approaches. To address this issue, a new spectral clustering algorithm called SCORE is proposed in @cite_33 . Specifically, the second to @math -th leading eigenvectors are divided by the first leading eigenvector elementwisely, and spectral clustering is applied to the resulting ratio matrix. In their theoretical results, an implicit assumption is that the number of communities @math is bounded by a constant, as implied by the condition (2.14) in @cite_33 . In comparison, our convexified modularity maximization approach works for growing @math both theoretically and empirically. As illustrated in sec:synthetic , our method exhibited better performance on both the synthetic and real datasets considered there, especially when @math .
@cite_33 @cite_6
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We analyze the performance of spectral clustering for community extraction in stochastic block models. We show that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as @math , with @math the number of nodes. This result applies to some popular polynomial time spectral clustering algorithms and is further extended to degree corrected stochastic block models using a spherical @math -median spectral clustering method. A key component of our analysis is a combinatorial bound on the spectrum of binary random matrices, which is sharper than the conventional matrix Bernstein inequality and may be of independent interest. We introduce a novel spectral clustering algorithm that allows us to automatically determine the number of clusters in a dataset. The algorithm is based on a theoretical analysis of the spectral properties of block diagonal affinity matrices; in contrast to established methods, we do not normalise the rows of the matrix of eigenvectors, and argue that the non-normalised data contains key information that allows the automatic determination of the number of clusters present. We present several examples of datasets successfully clustered by our algorithm, both artificial and real, obtaining good results even without employing refined feature extraction techniques Based on spectral graph theory, spectral clustering is an optimal graph partition problem. It has been proven that the spectral clustering is equivalent to nonnegative matrix factorization (NMF) under certain conditions. Based on the equivalence, some spectral clustering methods are proposed, but the global discriminative information of the dataset is neglected. In this paper, based on the equivalence between spectral clustering and NMF, we simultaneously maximize the between-class scatter matrix and minimize the within-class scatter matrix to enhance the discriminating power. We integrate the geometrical structure and discriminative structure in a joint framework. With a global discriminative regularization term added into the nonnegative matrix factorization framework, we propose two novel spectral clustering methods, named global discriminative-based nonnegative and spectral clustering (GDBNSC-Ncut and GDBNSC-Rcut) These new spectral clustering algorithms can preserve both the global geometrical structure and global discriminative structure. The intrinsic geometrical information of the dataset is detected, and clustering quality is improved with enhanced discriminating power. In addition, the proposed algorithms also have very good abilities of handling out-of-sample data. Experimental results on real word data demonstrate that the proposed algorithms outperform some state-of-the-art methods with good clustering qualities. We maximize the between-class scatter matrix.Meanwhile, we minimize the within-class scatter matrix.We integrate the geometrical structure and discriminative structure in a framework.A global discriminative regularization term added into the objective functions.The new algorithms can preserve the global geometrical and discriminative structure. Spectral clustering is a popular and successful approach for partitioning the nodes of a graph into clusters for which the ratio of outside connections compared to the volume (sum of degrees) is small. In order to partition into k clusters, one first computes an approximation of the bottom k eigenvectors of the (normalized) Laplacian of G, uses it to embed the vertices of G into k-dimensional Euclidean space R^k, and then partitions the resulting points via a k-means clustering algorithm. It is an important task for theory to explain the success of spectral clustering. (COLT, 2015) made an important step in this direction. They showed that spectral clustering provably works if the gap between the (k+1)-th and the k-th eigenvalue of the normalized Laplacian is sufficiently large. They proved a structural and an algorithmic result. The algorithmic result needs a considerably stronger gap assumption and does not analyze the standard spectral clustering paradigm; it replaces spectral embedding by heat kernel embedding and k-means clustering by locality sensitive hashing. We extend their work in two directions. Structurally, we improve the quality guarantee for spectral clustering by a factor of k and simultaneously weaken the gap assumption. Algorithmically, we show that the standard paradigm for spectral clustering works. Moreover, it even works with the same gap assumption as required for the structural result.