Networks have been a general tool for representing, analyzing, and modeling relational data arising in several domains. One of the most important aspect of network analysis is community detection or network clustering. Until recently, the major focus have been on discovering community structure in single (i.e., monoplex) networks. However, with the advent of relational data with multiple modalities, multiplex networks, i.e., networks composed of multiple layers representing different aspects of relations, have emerged. Consequently, community detection in multiplex network, i.e., detecting clusters of nodes shared by all layers, has become a new challenge. In this paper, we propose Network Fusion for Composite Community Extraction (NF-CCE), a new class of algorithms, based on four different non-negative matrix factorization models, capable of extracting composite communities in multiplex networks. Each algorithm works in two steps: first, it finds a non-negative, low-dimensional feature representation of each network layer; then, it fuses the feature representation of layers into a common non-negative, low-dimensional feature representation via collective factorization. The composite clusters are extracted from the common feature representation. We demonstrate the superior performance of our algorithms over the state-of-the-art methods on various types of multiplex networks, including biological, social, economic, citation, phone communication, and brain multiplex networks.
Multi-view graph clustering aims to enhance clustering performance by integrating heterogeneous information collected in different domains. Each domain provides a different view of the data instances. Leveraging cross-domain information has been demonstrated an effective way to achieve better clustering results. Despite the previous success, existing multi-view graph clustering methods usually assume that different views are available for the same set of instances. Thus instances in different domains can be treated as having strict one-to-one relationship. In many real-life applications, however, data instances in one domain may correspond to multiple instances in another domain. Moreover, relationships between instances in different domains may be associated with weights based on prior (partial) knowledge. In this paper, we propose a flexible and robust framework, CGC (Co-regularized Graph Clustering), based on non-negative matrix factorization (NMF), to tackle these challenges. CGC has several advantages over the existing methods. First, it supports many-to-many cross-domain instance relationship. Second, it incorporates weight on cross-domain relationship. Third, it allows partial cross-domain mapping so that graphs in different domains may have different sizes. Finally, it provides users with the extent to which the cross-domain instance relationship violates the in-domain clustering structure, and thus enables users to re-evaluate the consistency of the relationship. Extensive experimental results on UCI benchmark data sets, newsgroup data sets and biological interaction networks demonstrate the effectiveness of our approach. In graph-based learning models, entities are often represented as vertices in an undirected graph with weighted edges describing the relationships between entities. In many real-world applications, however, entities are often associated with relations of different types and or from different sources, which can be well captured by multiple undirected graphs over the same set of vertices. How to exploit such multiple sources of information to make better inferences on entities remains an interesting open problem. In this paper, we focus on the problem of clustering the vertices based on multiple graphs in both unsupervised and semi-supervised settings. As one of our contributions, we propose Linked Matrix Factorization (LMF) as a novel way of fusing information from multiple graph sources. In LMF, each graph is approximated by matrix factorization with a graph-specific factor and a factor common to all graphs, where the common factor provides features for all vertices. Experiments on SIAM journal data show that (1) we can improve the clustering accuracy through fusing multiple sources of information with several models, and (2) LMF yields superior or competitive results compared to other graph-based clustering methods. Observational data usually comes with a multimodal nature, which means that it can be naturally represented by a multi-layer graph whose layers share the same set of vertices (objects) with different edges (pairwise relationships). In this paper, we address the problem of combining different layers of the multi-layer graph for an improved clustering of the vertices compared to using layers independently. We propose two novel methods, which are based on a joint matrix factorization and a graph regularization framework respectively, to efficiently combine the spectrum of the multiple graph layers, namely the eigenvectors of the graph Laplacian matrices. In each case, the resulting combination, which we call a “joint spectrum” of multiple layers, is used for clustering the vertices. We evaluate our approaches by experiments with several real world social network datasets. Results demonstrate the superior or competitive performance of the proposed methods compared to state-of-the-art techniques and common baseline methods, such as co-regularization and summation of information from individual graphs. Given a co-authorship collaboration network, how well can we cluster the participating authors into communities? If we also consider their citation network, based on the same individuals, is it possible to do a better job? In general, given a network with multiple types (or views) of edges (e.g., collaboration, citation, friendship), can community detection and graph clustering benefit? In this work, we propose Multi-CLUS and GraphFuse, two multi-graph clustering techniques powered by Minimum Description Length and Tensor analysis, respectively. We conduct experiments both on real and synthetic networks, evaluating the performance of our approaches. Our results demonstrate higher clustering accuracy than state-of-the-art baselines that do not exploit the multi-view nature of the network data. Finally, we address the fundamental question posed in the title, and provide a comprehensive answer, based on our systematic analysis. Simple structure and other common principles of factor rotation do not in general provide strong grounds for attributing explanatory significance to the factors which they select. In contrast, it is shown that an extension of Cattell's principle of rotation to Proportional Profiles (PP) offers a basis for determining explanatory factors for three-way or higher order multi-mode data. Conceptual models are developed for two basic patterns of multi-mode data variation, systemand object-variation, and PP analysis is found to apply in the system-variation case. Although PP was originally formulated as a principle of rotation to be used with classic two-way factor analysis, it is shown to embody a latent three-mode factor model, which is here made explicit and generalized frown two to N "parallel occasions". As originally formulated, PP rotation was restricted to orthogonal factors. The generalized PP model is demonstrated to give unique "correct" solutions with oblique, non-simple structure, and even non-linear factor structures. A series of tests, conducted with synthetic data of known factor composition, demonstrate the capabilities of linear and non-linear versions of the model, provide data on the minimal necessary conditions of uniqueness, and reveal the properties of the analysis procedures when these minimal conditions are not fulfilled. In addition, a mathematical proof is presented for the uniqueness of the solution given certain conditions on the data. Three-mode PP factor analysis is applied to a three-way set of real data consisting of the fundamental and first three formant frequencies of 11 persons saying 8 vowels. A unique solution is extracted, consisting of three factors which are highly meaningful and consistent with prior knowledge and theory concerning vowel quality. The relationships between the three-mode PP model and Tucker's multi-modal model, McDonald's non-linear model and Carroll and Chang's multi-dimensional scaling model are explored.