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In this paper, we develop semi-external and external memory algorithms for graph partitioning and clustering problems. Graph partitioning and clustering are key tools for processing and analyzing large complex networks. We address both problems in the (semi-)external model by adapting the size-constrained label propagation technique. Our (semi-)external size-constrained label propagation algorithm can be used to compute graph clusterings and is a prerequisite for the (semi-)external graph partitioning algorithm. The algorithm is then used for both the coarsening and the refinement phase of a multilevel algorithm to compute graph partitions. Our algorithm is able to partition and cluster huge complex networks with billions of edges on cheap commodity machines. Experiments demonstrate that the semi-external graph partitioning algorithm is scalable and can compute high quality partitions in time that is comparable to the running time of an efficient internal memory implementation. A parallelization of the algorithm in the semi-external model further reduces running time.

Many data types arising from data mining applications can be modeled as bipartite graphs, examples include terms and documents in a text corpus, customers and purchasing items in market basket analysis and reviewers and movies in a movie recommender system. In this paper, we propose a new data clustering method based on partitioning the underlying bipartite graph. The partition is constructed by minimizing a normalized sum of edge weights between unmatched pairs of vertices of the bipartite graph. We show that an approximate solution to the minimization problem can be obtained by computing a partial singular value decomposition (SVD) of the associated edge weight matrix of the bipartite graph. We point out the connection of our clustering algorithm to correspondence analysis used in multivariate analysis. We also briefly discuss the issue of assigning data objects to multiple clusters. In the experimental results, we apply our clustering algorithm to the problem of document clustering to illustrate its effectiveness and efficiency.

Spectral clustering approaches have led to well-accepted algorithms for finding accurate clusters in a given dataset. However, their application to large-scale datasets has been hindered by computational complexity of eigenvalue decompositions. Several algorithms have been proposed in the recent past to accelerate spectral clustering, however they compromise on the accuracy of the spectral clustering to achieve faster speed. In this paper, we propose a novel spectral clustering algorithm based on a mixing process on a graph. Unlike the existing spectral clustering algorithms, our algorithm does not require computing eigenvectors. Specifically, it finds the equivalent of a linear combination of eigenvectors of the normalized similarity matrix weighted with corresponding eigenvalues. This linear combination is then used to partition the dataset into meaningful clusters. Simulations on real datasets show that partitioning datasets based on such linear combinations of eigenvectors achieves better accuracy than standard spectral clustering methods as the number of clusters increase. Our algorithm can easily be implemented in a distributed setting.

Spectral clustering approaches have led to well-accepted algorithms for finding accurate clusters in a given dataset. However, their application to large-scale datasets has been hindered by computational complexity of eigenvalue decompositions. Several algorithms have been proposed in the recent past to accelerate spectral clustering, however they compromise on the accuracy of the spectral clustering to achieve faster speed. In this paper, we propose a novel spectral clustering algorithm based on a mixing process on a graph. Unlike the existing spectral clustering algorithms, our algorithm does not require computing eigenvectors. Specifically, it finds the equivalent of a linear combination of eigenvectors of the normalized similarity matrix weighted with corresponding eigenvalues. This linear combination is then used to partition the dataset into meaningful clusters. Simulations on real datasets show that partitioning datasets based on such linear combinations of eigenvectors achieves better accuracy than standard spectral clustering methods as the number of clusters increase. Our algorithm can easily be implemented in a distributed setting.

Complex design problems are characterized by a multitude of competing requirements. System designers frequently find the scope of the problem beyond their conceptual abilities, and attempt to cope with this difficulty by decomposing the original design problem into smaller, more manageable subproblems. In the SDM research effort, a systematic approach has been proposed for the decomposition of the set of functional requirements of a design problem into subsets (called subproblems) to form a design structure that will exhibit key characteristics of good design: strong coupling among requirements within subproblems and weak coupling between subproblems. This report documents the implementation of an efficient graph partitioning technique based on a high-density clustering model. The new method identifies the 'high-density regions' in the graph, which are sets of functional requirements exhibiting a relatively high degree of interdependency, and associates them with the different subsets of the design problem. The new technique, as currently implemented, is applied to several problems from the design literature. The results indicate that the proposed approach gives solutions that are conceptually and intuitively appealing, and that these partitions are consistent with the currently accepted decomposition. Although direct comparison with computational requirements of other partitioning procedures is difficult due to different machine implementations, the empirical evidence suggest that the new method is useful for decomposing design problems too large for the procedures currently in use. (Author)

We study the design of local algorithms for massive graphs. A local algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster--a subset of vertices whose internal connections are significantly richer than its external connections--near a given vertex. The running time of our algorithm, when it finds a non-empty local cluster, is nearly linear in the size of the cluster it outputs. Our clustering algorithm could be a useful primitive for handling massive graphs, such as social networks and web-graphs. As an application of this clustering algorithm, we present a partitioning algorithm that finds an approximate sparsest cut with nearly optimal balance. Our algorithm takes time nearly linear in the number edges of the graph. Using the partitioning algorithm of this paper, we have designed a nearly-linear time algorithm for constructing spectral sparsifiers of graphs, which we in turn use in a nearly-linear time algorithm for solving linear systems in symmetric, diagonally-dominant matrices. The linear system solver also leads to a nearly linear-time algorithm for approximating the second-smallest eigenvalue and corresponding eigenvector of the Laplacian matrix of a graph. These other results are presented in two companion papers.