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.
@cite_33: ion X = E [X ] + (X − E [X ]) = “signal” + “noise” Challenge. Quantity of interest (labels) is contained in E [X ] I observed with only one snapshot I covered by centered-Bernoulli noise I O(n) parameters, O(n) measurements Solution. Using ‘sparsity’ of E [X ] Jiashun Jin Fast Network Community Detection by SCORE
@cite_6: In this paper, we study the linking patterns and discussion topics of political bloggers. Our aim is to measure the degree of interaction between liberal and conservative blogs, and to uncover any differences in the structure of the two communities. Specifically, we analyze the posts of 40 "A-list" blogs over the period of two months preceding the U.S. Presidential Election of 2004, to study how often they referred to one another and to quantify the overlap in the topics they discussed, both within the liberal and conservative communities, and also across communities. We also study a single day snapshot of over 1,000 political blogs. This snapshot captures blogrolls (the list of links to other blogs frequently found in sidebars), and presents a more static picture of a broader blogosphere. Most significantly, we find differences in the behavior of liberal and conservative blogs, with conservative blogs linking to each other more frequently and in a denser pattern.
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 .