Stochastic block models (SBMs) have been playing an important role in modeling clusters or community structures of network data. But, it is incapable of handling several complex features ubiquitously exhibited in real-world networks, one of which is the power-law degree characteristic. To this end, we propose a new variant of SBM, termed power-law degree SBM (PLD-SBM), by introducing degree decay variables to explicitly encode the varying degree distribution over all nodes. With an exponential prior, it is proved that PLD-SBM approximately preserves the scale-free feature in real networks. In addition, from the inference of variational E-Step, PLD-SBM is indeed to correct the bias inherited in SBM with the introduced degree decay factors. Furthermore, experiments conducted on both synthetic networks and two real-world datasets including Adolescent Health Data and the political blogs network verify the effectiveness of the proposed model in terms of cluster prediction accuracies.
Understanding a complex network's structure holds the key to understanding its function. The physics community has contributed a multitude of methods and analyses to this cross-disciplinary endeavor. Structural features exist on both the microscopic level, resulting from differences between single node properties, and the mesoscopic level resulting from properties shared by groups of nodes. Disentangling the determinants of network structure on these different scales has remained a major, and so far unsolved, challenge. Here we show how multiscale generative probabilistic exponential random graph models combined with efficient, distributive message-passing inference techniques can be used to achieve this separation of scales, leading to improved detection accuracy of latent classes as demonstrated on benchmark problems. It sheds new light on the statistical significance of motif-distributions in neural networks and improves the link-prediction accuracy as exemplified for gene-disease associations in the highly consequential Online Mendelian Inheritance in Man database. Traditional image representations are not suited to conventional classification methods such as the linear discriminant analysis (LDA) because of the undersample problem (USP): the dimensionality of the feature space is much higher than the number of training samples. Motivated by the successes of the two-dimensional LDA (2DLDA) for face recognition, we develop a general tensor discriminant analysis (GTDA) as a preprocessing step for LDA. The benefits of GTDA, compared with existing preprocessing methods such as the principal components analysis (PCA) and 2DLDA, include the following: 1) the USP is reduced in subsequent classification by, for example, LDA, 2) the discriminative information in the training tensors is preserved, and 3) GTDA provides stable recognition rates because the alternating projection optimization algorithm to obtain a solution of GTDA converges, whereas that of 2DLDA does not. We use human gait recognition to validate the proposed GTDA. The averaged gait images are utilized for gait representation. Given the popularity of Gabor-function-based image decompositions for image understanding and object recognition, we develop three different Gabor-function-based image representations: 1) GaborD is the sum of Gabor filter responses over directions, 2) GaborS is the sum of Gabor filter responses over scales, and 3) GaborSD is the sum of Gabor filter responses over scales and directions. The GaborD, GaborS, and GaborSD representations are applied to the problem of recognizing people from their averaged gait images. A large number of experiments were carried out to evaluate the effectiveness (recognition rate) of gait recognition based on first obtaining a Gabor, GaborD, GaborS, or GaborSD image representation, then using GDTA to extract features and, finally, using LDA for classification. The proposed methods achieved good performance for gait recognition based on image sequences from the University of South Florida (USF) HumanID Database. Experimental comparisons are made with nine state-of-the-art classification methods in gait recognition. We extend spectral methods to random graphs with skewed degree distributions through a degree based normalization closely connected to the normalized Laplacian. The normalization is based on intuition drawn from perturbation theory of random matrices, and has the effect of boosting the expectation of the random adjacency matrix without increasing the variances of its entries, leading to better perturbation bounds. The primary implication of this result lies in the realm of spectral analysis of random graphs with skewed degree distributions, such as the ubiquitous "power law graphs". Mihail and Papadimitriou (2002) argued that for randomly generated graphs satisfying a power law degree distribution, spectral analysis of the adjacency matrix simply produces the neighborhoods of the high degree nodes as its eigenvectors, and thus miss any embedded structure. We present a generalization of their model, incorporating latent structure, and prove that after applying our transformation, spectral analysis succeeds in recovering the latent structure with high probability. In this paper, we examine a spectral clustering algorithm for similarity graphs drawn from a simple random graph model, where nodes are allowed to have varying degrees, and we provide theoretical bounds on its performance. The random graph model we study is the Extended Planted Partition (EPP) model, a variant of the classical planted partition model. The standard approach to spectral clustering of graphs is to compute the bottom k singular vectors or eigenvectors of a suitable graph Laplacian, project the nodes of the graph onto these vectors, and then use an iterative clustering algorithm on the projected nodes. However a challenge with applying this approach to graphs generated from the EPP model is that unnormalized Laplacians do not work, and normalized Laplacians do not concentrate well when the graph has a number of low degree nodes. We resolve this issue by introducing the notion of a degree-corrected graph Laplacian. For graphs with many low degree nodes, degree correction has a regularizing eect on the Laplacian. Our spectral clustering algorithm projects the nodes in the graph onto the bottom k right singular vectors of the degree-corrected random-walk Laplacian, and clusters the nodes in this subspace. We show guarantees on the performance of this algorithm, demonstrating that it outputs the correct partition under a wide range of parameter values. Unlike some previous work, our algorithm does not require access to any generative parameters of the model.