A fast kernel-based multilevel algorithm for graph clustering

Abstract

Graph clustering (also called graph partitioning)-clustering the nodes of a graph-is an important problem in diverse data mining applications. Traditional approaches involve optimization of graph clustering objectives such as normalized cut or ratio association; spectral methods are widely used for these objectives, but they require eigenvector computation which can be slow. Recently, graph clustering with a general cut objective has been shown to be mathematically equivalent to an appropriate weighted kernel k-means objective function. In this paper, we exploit this equivalence to develop a very fast multilevel algorithm for graph clustering. Multilevel approaches involve coarsening, initial partitioning and refinement phases, all of which may be specialized to different graph clustering objectives. Unlike existing multilevel clustering approaches, such as METIS, our algorithm does not constrain the cluster sizes to be nearly equal. Our approach gives a theoretical guarantee that the refinement step decreases the graph cut objective under consideration. Experiments show that we achieve better final objective function values as compared to a state-of-the-art spectral clustering algorithm: on a series of benchmark test graphs with up to thirty thousand nodes and one million edges, our algorithm achieves lower normalized cut values in 67% of our experiments and higher ratio association values in 100% of our experiments. Furthermore, on large graphs, our algorithm is significantly faster than spectral methods. Finally, our algorithm requires far less memory than spectral methods; we cluster a 1.2 million node movie network into 5000 clusters, which due to memory requirements cannot be done directly with spectral methods.

Graph summarization with bounded error

“…• Graclus (GRAC) [8]: This is a graph clustering algorithm that divides the nodes of a given weighted graph into clusters such that the sum of weights of the inter-cluster edges is minimized. We ran it on a new derived graph having the same set of nodes as G, while edges exist between any two nodes with a non-zero cost reduction and the weight on the edge equal to the cost reduction s(·).…”

“…In contrast, we use information-theoretic metrics to group nodes such that the graph representation is as compact as possible. Another problem with many of the widely-used clustering algorithms, such as METIS [17], Graclus [8], kmeans and spectral clustering [24], is that they require the user to specify the number of partitions beforehand, which is typically hard to estimate and not required in our setting.…”

“…Thus, it is amenable to visualization and other graph analysis techniques (e.g., finding communities, customer segments); specifically, it provides insight into the highlevel structure of the graph, and the dominant relationships among the various node clusters. Unlike clustering algorithms [17,8] that group nodes based on their similarity or distances, our summary is computed using information-theoretic principles.…”

Applying Electromagnetic Field Theory Concepts to Clustering with Constraints

“…Cannot-link constraints (like charges) increase the weights of affected edge. The situation is illustrated in Figure 1, where must-link constraints are {e (2,3), e (6,7), e (14,15)} and the cannot-link constraints are {e(9,10)}.…”

Kernel-based clustering and low rank approximation

“…These algorithms use the eigenvectors of the affinity matrix that encodes the local data structure to make a global decision of clustering. They demonstrate promising performance in the domains of image segmentation [95], document clustering [35], data mining [112], dimensionality reduction [16], and semi-supervised learning [15] [116].…”