Community discovery is a prominent issue in com-plex network analysis. Symmetric non-negative matrix factorization (SNMF) is frequently adopted to tackle this issue. The use of a single feature matrix can depict network symmetry, but it limits its ability to learn node representations. To break this limitation, we present a novel Relaxed Symmetric NMF (RSN) approach to boost an SNMF-based community detector. It works by 1) expanding the representational space and its degrees of freedom with multiple feature factors; 2) integrating the well-designed equality-constraints to make the model well-aware of the network’s intrinsic symmetry; 3) employing graph regularization to pre-serve the local geometric invariance of the network structure; and 4) separating constraints from decision variables for efficient optimization via the principle of alternating-direction-method of multi-pliers. RSN’s effectiveness is verified through empirical studies on six real social networks, show-casing superior precision in community discovery over existing models and baselines.

Community detection sheds light on various graph mining tasks such as social recommendation, which is becoming a long-standing issue in the realm of complex network analysis. A non-negative matrix factorization (NMF) is frequently used to tackle this task. However, it builds based on the principle of linear representation and has difficulty in capturing non-linear features from irregularly non-Euclidean data. To address this issue, this study boosts an NMF-based community detector by combining with a graph convolution module, and a novel Graph Convolution and Graph-Laplacian bi-regularized, Symmetric non-negative matrix factorization (GCGS) model is proposed relying on two main ideas: a) taking a graph convolution network (GCN) as a non-linear constraint module on the feature matrix to ensure its non-linearity; and b) adopting graph regularization to preserve the local geometric features of the network topology. A non-negative and multiplicative update (NMU) algorithm is then derived to solve the unified objective function. Extensive experimental results on six real networks indicate that GCGS achieves higher precision in community detection than its peers.

An Undirected Weighted Network (UWN) is frequently encountered in a big-data-related application concerning the complex interactions among numerous nodes. A Symmetric High-Dimensional and Incomplete (SHDI) matrix can smoothly illustrate such a UWN, which contains rich knowledge like node interaction behaviors and local complexes. To extract desired knowledge from an SHDI matrix, an analysis model should carefully consider its topology for describing a UWN's intrinsic symmetry precisely. Representation learning to a UWN borrows the success of a pyramid of symmetry-aware models like a Symmetric Nonnegative Matrix Factorization (SNMF) model whose objective function utilizes a sole Latent Factor (LF) matrix for representing SHDI's symmetry precisely. However, they suffer from the following drawbacks: 1) their computational complexity is high; and 2) their modeling strategy narrows their representation features, making them suffer from low learning ability. Aiming at addressing the above critical issues, this paper proposes a Multi-constrained Symmetric Nonnegative Latent-factor-analysis (MSNL) model with two-fold ideas: 1) introducing multi-constraints composed of multiple LF matrices, i.e., inequality and equality ones into a data-density-oriented objective function for precisely representing the intrinsic symmetry of an SHDI matrix with broadened feature space; and 2) implementing an alternating direction method of multipliers (ADMM)-incorporated learning scheme for efficiently solving such a multi-constrained model. Empirical studies on three SHDI matrices from a real bioinformatics or industrial application demonstrate that the proposed MSNL model achieves higher representation accuracy than state-of-the-art models do, as well as promising computational efficiency.

Large-scale undirected weighted networks are frequently encountered in big-data-related applications concerning interactions among a large unique set of entities. Such a network can be described by a Symmetric, High-Dimensional, and Incomplete (SHDI) matrix whose symmetry and incompleteness should be addressed with care. However, existing models fail in either correctly representing its symmetry or efficiently handling its incomplete data. For addressing this critical issue, this study proposes an Alternating-Direction-Method of Multipliers (ADMM)-based Symmetric Non-negative Latent Factor Analysis (ASNL) model. It adopts fourfold ideas: 1) implementing the data density-oriented modeling for efficiently representing an SHDI matrix's incomplete and imbalanced data; 2) separating the non-negative constraints from the decision parameters to avoid truncations during the training process; 3) incorporating the ADMM principle into its learning scheme for fast model convergence; and 4) parallelizing the training process with load balance considerations for high efficiency. Empirical studies on four SHDI matrices demonstrate that ASNL significantly outperforms several state-of-the-art models in both prediction accuracy for missing data of an SHDI and computational efficiency. It is a promising model for handling large-scale undirected networks raised in real applications.

Symmetric high-dimensional and sparse (SHiDS) networks are frequently found in various industrial applications. A symmetric non-negative latent factor (SNLF) model can acquire essential features from them precisely, yet it suffers from slow convergence. To address this issue, this article integrates a generalized momentum method into a symmetric, single latent factor-dependent, non-negative and multiplicative update (S2LF-NMU) algorithm, thereby achieving amomentum-incorporated,symmetric,single-latent-factor-dependentnon-negative-multiplicative-update (MS2N) algorithm. Based on an MS2N algorithm, momentum-incorporated symmetric non-negative latent factor (MSNLF) models are proposed for an SHiDS network, which ensures fast convergence as well as high representative learning ability. Empirical studies on four SHiDS networks from industrial applications demonstrate that compared with state-of-the-art models, the proposed MSNLF models have significantly higher computational efficiency and representative learning ability.

Big-data related applications frequently concern how to analyze large-scale undirected weighted network effectively. Such a network can be quantized into a Symmetric, High-Dimensional and Sparse (SHiDS) matrix, owing to its sparsity and symmetry. For considering these characteristics of an SHiDS matrix with care, a symmetric non-negative latent factor (SNLF) model is proposed. However, representation learning ability of an SNLF model is limited owing to its commonly-adopted learning objective, i.e., Euclidean distance. For addressing this issue, this study proposes a generalized symmetric nonnegative latent factor analysis (GSNL) model. Its main idea is two-fold: a) adopting$\alpha-\beta$-divergence to generalize SNLF's learning objective for achieving accurate representation ability of an SHiDS matrix; and b) utilizing a self-adaptive scheme on all involved hyperparameters brought by the resultant model for strong practicability. Empirical studies on four SHiDS matrices demonstrate that a GSNL model outperforms its peers regarding accuracy gain and computational efficiency.

A large-scale undirected weighted network (LUWN) is described by a symmetric high-dimensional and sparse (SHiDS) matrix. To correctly represent its symmetry, existing models adopt a strong symmetry assumption, i.e., reducing the solution space to adopt a unique latent factor matrix to describe an SHiDS matrix’s symmetry. Yet it may impair a resultant model’s representativeness to its numerical features. Aiming at addressing this issue, this work proposes a Relaxed Symmetric Non-negative latent factor analysis (RSN) model that adopts three-fold ideas: a) Introducing a triple equation constraint into its learning objective for relaxing the strong symmetry assumption, thereby greatly improving its representativeness to the numerical features of an SHiDS matrix; b) Adopting the framework of Alternating Direction Method of Multipliers to fast realize its learning objective subject to multiple constraints; and c) utilizing a data density-oriented principle during its modeling and optimization, thereby precisely representing an SHiDS matrix’s imbalanced data. Empirical studies on four industrial SHiDS matrices describing real LUWNs demonstrate that RSN outperforms state-of-the-art models in representing an SHiDS matrix precisely, as well as achieves highly competitive computational efficiency. Hence, this work greatly advances the area of LUWN analysis.