We propose two spectral algorithms for partitioning nodes in directed graphs respectively with a cyclic and an acyclic pattern of connection between groups of nodes. Our methods are based on the computation of extremal eigenvalues of the transition matrix associated to the directed graph. The two algorithms outperform state-of-the art methods for directed graph clustering on synthetic datasets, including methods based on blockmodels, bibliometric symmetrization and random walks. Our algorithms have the same space complexity as classical spectral clustering algorithms for undirected graphs and their time complexity is also linear in the number of edges in the graph. One of our methods is applied to a trophic network based on predator-prey relationships. It successfully extracts common categories of preys and predators encountered in food chains. The same method is also applied to highlight the hierarchical structure of a worldwide network of Autonomous Systems depicting business agreements between Internet Service Providers.
Bibliometric symmetrization refers to a symmetrization of the adjacency matrix @math of @math . The symmetrized matrix @math is defined as the adjacency matrix of an undirected graph @math for a certain choice of weighing parameter @math . This symmetric adjacency matrix is a linear combination of the co-coupling matrix @math and the co-citation matrix @math . Then clustering methods for undirected graphs are applied to @math such as a spectral clustering algorithm. This method is efficient to detect co-citation networks @cite_18 .
@cite_18
2002850151 2256361316 1504886279 2764230287
A discrete uncoupling process for finite spaces is introduced, called the Markov Cluster Process or the MCL process. The process is the engine for the graph clustering algorithm called the MCL algorithm. The MCL process takes a stochastic matrix as input, and then alternates expansion and inflation, each step defining a stochastic matrix in terms of the previous one. Expansion corresponds with taking the @math th power of a stochastic matrix, where @math . Inflation corresponds with a parametrized operator @math , @math , that maps the set of (column) stochastic matrices onto itself. The image @math is obtained by raising each entry in @math to the @math th power and rescaling each column to have sum 1 again. In practice the process converges very fast towards a limit that is invariant under both matrix multiplication and inflation, with quadratic convergence around the limit points. The heuristic behind the process is its expected behavior for (Markov) graphs possessing cluster structure. The process is typically applied to the matrix of random walks on a given graph @math , and the connected components of (the graph associated with) the process limit generically allow a clustering interpretation of @math . The limit is in general extremely sparse and iterands are sparse in a weighted sense, implying that the MCL algorithm is very fast and highly scalable. Several mathematical properties of the MCL process are established. Most notably, the process (and algorithm) iterands posses structural properties generalizing the mapping from process limits onto clusterings. The inflation operator @math maps the class of matrices that are diagonally similar to a symmetric matrix onto itself. The phrase diagonally positive semi-definite (dpsd) is used for matrices that are diagonally similar to a positive semi-definite matrix. For @math and for @math a stochastic dpsd matrix, the image @math is again dpsd. Determinantal inequalities satisfied by a dpsd matrix @math imply a natural ordering among the diagonal elements of @math , generalizing the mapping of process limits onto clusterings. The spectrum of @math is of the form @math , where @math is the number of endclasses of the ordering associated with @math , and @math is the dimension of @math . This attests to the uncoupling effect of the inflation operator. Based on spectral graph theory, spectral clustering is an optimal graph partition problem. It has been proven that the spectral clustering is equivalent to nonnegative matrix factorization (NMF) under certain conditions. Based on the equivalence, some spectral clustering methods are proposed, but the global discriminative information of the dataset is neglected. In this paper, based on the equivalence between spectral clustering and NMF, we simultaneously maximize the between-class scatter matrix and minimize the within-class scatter matrix to enhance the discriminating power. We integrate the geometrical structure and discriminative structure in a joint framework. With a global discriminative regularization term added into the nonnegative matrix factorization framework, we propose two novel spectral clustering methods, named global discriminative-based nonnegative and spectral clustering (GDBNSC-Ncut and GDBNSC-Rcut) These new spectral clustering algorithms can preserve both the global geometrical structure and global discriminative structure. The intrinsic geometrical information of the dataset is detected, and clustering quality is improved with enhanced discriminating power. In addition, the proposed algorithms also have very good abilities of handling out-of-sample data. Experimental results on real word data demonstrate that the proposed algorithms outperform some state-of-the-art methods with good clustering qualities. We maximize the between-class scatter matrix.Meanwhile, we minimize the within-class scatter matrix.We integrate the geometrical structure and discriminative structure in a framework.A global discriminative regularization term added into the objective functions.The new algorithms can preserve the global geometrical and discriminative structure. Nonnegative matrix factorization (NMF) provides a lower rank approximation of a nonnegative matrix, and has been successfully used as a clustering method. In this paper, we offer some conceptual understanding for the capabilities and shortcomings of NMF as a clustering method. Then, we propose Symmetric NMF (SymNMF) as a general framework for graph clustering, which inherits the advantages of NMF by enforcing nonnegativity on the clustering assignment matrix. Unlike NMF, however, SymNMF is based on a similarity measure between data points, and factorizes a symmetric matrix containing pairwise similarity values (not necessarily nonnegative). We compare SymNMF with the widely-used spectral clustering methods, and give an intuitive explanation of why SymNMF captures the cluster structure embedded in the graph representation more naturally. In addition, we develop a Newton-like algorithm that exploits second-order information efficiently, so as to show the feasibility of SymNMF as a practical framework for graph clustering. Our experiments on artificial graph data, text data, and image data demonstrate the substantially enhanced clustering quality of SymNMF over spectral clustering and NMF. Therefore, SymNMF is able to achieve better clustering results on both linear and nonlinear manifolds, and serves as a potential basis for many extensions We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs @math converging locally to a Galton--Watson tree @math (GWT), we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function @math of @math . In the first part, we show that the adjacency operator associated with @math is always self-adjoint; we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we establish a sufficient condition on @math for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of @math . Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory and statistical physics.