Using Maximum Entry-Wise Deviation to Test the Goodness of Fit for Stochastic Block Models

Abstract

The stochastic block model is widely used for detecting community structures in network data. How to test the goodness-of-fit of the model is one of the fundamental problems and has gained growing interests in recent years. In this paper, we propose a novel goodness-of-fit test based on the maximum entry of the centered and re-scaled adjacency matrix for the stochastic block model. One noticeable advantage of the proposed test is that the number of communities can be allowed to grow linearly with the number of nodes ignoring a logarithmic factor. We prove that the null distribution of the test statistic converges in distribution to a Gumbel distribution, and we show that both the number of communities and the membership vector can be tested via the proposed method. Further, we show that the proposed test has asymptotic power guarantee against a class of alternatives. We also demonstrate that the proposed method can be extended to the degree-corrected stochastic block model. Both simulation studies and real-world data examples indicate that the proposed method works well.KEY WORDS: Community detection; Degree-corrected stochastic block model; Goodnessof-fit test; Network data; Stochastic block model.

Stochastic Block Model Reveals Maps of In Applied Mathematics Studies Using VOS Viewer

“…VOS Viewer, also called VV, is software used to visualize bibliometric maps (Barra & Zotti, 2017;Hu et al, 2020;Mundt & Mundt, 2020) or data sets containing bibliographic fields such as title, author, journal, and others. In the world of research, VV is used for bibliometric analysis, mapping topics for the latest research, finding the most widely used references in certain fields, and others.…”

Selective inference for latent block models

“…Besides estimating the block structure from a given observed data matrix based on an LBM, it is also important to test the validity of a model (i.e., the number of blocks) or an estimation result. Until now, several tests [3,14,22,35,37] have been proposed for determining the number of blocks in block models, such as a stochastic block model (SBM), which is a model for a square symmetric matrix (e.g., an adjacency matrix of the network structure). Among these studies, only [35]'s test can be applied to the LBM setting; however, its target is different from ours in that it is limited to the number of blocks, not to the cluster memberships.…”

“…In regard to an SBM, several studies have proposed a statistical test for a given set of community memberships of an observed matrix [8,14,16]. In [8], based on the numbers of edges within and across the clusters, two tests were proposed for an SBM; one of these tests included a goodness-of-fit test of community memberships.…”

“…Although this study's objective is similar to ours, its problem setting is quite different from ours in various aspects, such as the setting of the alternative hypothesis and the assumptions in the network structure (e.g., there are two equal-sized communities in a given network and more intra-community edges than inter-community ones). Another study [14] proposed an asymptotic test on both the number of communities and the community memberships of an SBM, whose validity is guaranteed with the infinite matrix size. This study is different from ours in that our proposed test is validated with a finite size matrix.…”

A Goodness-of-fit Test on the Number of Biclusters in a Relational Data Matrix

“…Statistical test on the number of biclusters K. Although there have been many studies for testing whether an observed matrix A contains any large average submatrix or not [6,7,10,32,35], few statistical test methods have been proposed for the number of biclusters K in matrix A. Recently, statistical tests on K has been proposed in [3,28,30,55] with the constraint that the underlying bicluster structure should be represented by a regular grid (as shown in Figure 1 (b-2)) 1 . However, if the latent bicluster structure does not satisfy the regular grid constraint (as shown in Figure 1 (b-1)), such a test needs larger hypothetical number of biclusters K 0 (i.e., finer bicluster structure) to accept the null hypothesis K = K 0 than necessary, which would not be desirable in the perspective of interpreting the accepted bicluster structure.…”