Spectral Theory of Unsigned and Signed Graphs. Applications to Graph Clustering: a Survey

This is a survey of the method of graph cuts and its applications to graph clustering of weighted unsigned and signed graphs. I provide a fairly thorough treatment of the method of normalized graph cuts, a deeply original method due to Shi and Malik, including complete proofs. I also cover briefly the method of ratio cuts, and show how it can be viewed as a special case of normalized cuts. I include the necessary background on graphs and graph Laplacians. I then explain in detail how the eigenvectors of the graph Laplacian can be used to draw a graph. This is an attractive application of graph Laplacians. The main thrust of this paper is the method of normalized cuts. I give a detailed account for K = 2 clusters, and also for K > 2 clusters, based on the work of Yu and Shi. I also show how both graph drawing and normalized cut K-clustering can be easily generalized to handle signed graphs, which are weighted graphs in which the weight matrix W may have negative coefficients. Intuitively, negative coefficients indicate distance or dissimilarity. The solution is to replace the degree matrix D by the matrix D in which absolute values of the weights are used, and to replace the Laplacian L = D −W by the signed Laplacian L = D −W . The signed Laplacian L is always positive semidefinite, and it may be positive definite (for unbalanced graphs, see Chapter 5). As far as I know, the generalization of K-way normalized clustering to signed graphs is new. Finally, I show how the method of ratio cuts, in which a cut is normalized by the size of the cluster rather than its volume, is just a special case of normalized cuts. All that needs to be done is to replace the normalized Laplacian Lsym by the unormalized Laplacian L. This is also true for signed graphs (where we replace Lsym by L). Three points that do not appear to have been clearly articulated before are elaborated: 1. The solutions of the main optimization problem should be viewed as tuples in the K-fold cartesian product of projective space RPN−1. 2. When K > 2, the solutions of the relaxed problem should be viewed as elements of the Grassmannian G(K,N). 3. Two possible Riemannian distances are available to compare the closeness of solutions: (a) The distance on (RPN−1)K . (b) The distance on the Grassmannian. I also clarify what should be the necessary and sufficient conditions for a matrix to represent a partition of the vertices of a graph to be clustered.