Partitioning Networks by Eigenvectors

A survey of published methods for partitioning sparse arrays is presented. These include early attempts to describe the partitioning properties of eigenvectors of the adjacency matrix. More direct methods of partitioning are developed by introducing the Laplacian of the adjacency matrix via the directed (signed) edge-vertex incidence matrix. It is shown that the Laplacian solves the minimization of total length of connections between adjacent nodes, which induces clustering of connected nodes by partitioning the underlying graph. Another matrix derived from the adjacency matrix is also introduced via the unsigned edge-vertex matrix. This (the Normal) matrix is not symmetric, and it also is shown to solve the minimization of total length in its own non-Euclidean metric. In this case partitions are induced by clustering the connected nodes. The Normal matrix is closely related to Correspondence Analysis. The problem of colouring graphs (which is in a sense dual to the problem of clustering induced by minimizing total distances) is also considered. A heuristic method to produce approximate correct colourings using sign patterns of eigenvectors with large negative eigenvalues is described. The formulation used to solve the minimization problem may also be used to solve the maximization problem, which leads to approximate colourings from both the Laplacian and the Normal matrix. Introduction There has been a recent renewal of interest in spectral methods for partitioning large sparse arrays, based on results that first appeared in the literature over twenty years ago. Hagen (1992) speculates that this is because growth in problem complexity has exposed scaling weaknesses in iterative methods such as Kernighan and Lin (1970), Kirkpatrick, et al., (1983), where problems now include finding efficient methods for distributing large adaptive finite mesh problems to arrays of parallel processors. The methods currently being developed exploit global properties of the underlying graph representation of the sparse array and so should be of interest to Social Network researchers. We present a brief survey of some early attempts to understand