Spectral clustering is a popular and versatile clustering method based on a relaxation of the normalised graph cut objective. Despite its popularity, however, there is no single agreed upon method for tuning the important scaling parameter, nor for determining automatically the number of clusters to extract. Popular heuristics exist, but corresponding theoretical results are scarce. In this paper we investigate the asymptotic value of the normalised cut for an increasing sample assumed to arise from an underlying probability distribution, and based on this result provide recommendations for improving spectral clustering methodology. A corresponding algorithm is proposed with strong empirical performance.

Partitioning a graph into groups of vertices such that those within each group are more densely connected than vertices assigned to different groups, known as graph clustering, is often used to gain insight into the organisation of large scale networks and for visualisation purposes. Whereas a large number of dedicated techniques have been recently proposed for static graphs, the design of on-line graph clustering methods tailored for evolving networks is a challenging problem, and much less documented in the literature. Motivated by the broad variety of applications concerned, ranging from the study of biological networks to the analysis of networks of scientific references through the exploration of communications networks such as the World Wide Web, it is the main purpose of this paper to introduce a novel, computationally efficient, approach to graph clustering in the evolutionary context. Namely, the method promoted in this article can be viewed as an incremental eigenvalue solution for the spectral clustering method described by Ng. et al. (2001). The incremental eigenvalue solution is a general technique for finding the approximate eigenvectors of a symmetric matrix given a change. As well as outlining the approach in detail, we present a theoretical bound on the quality of the approximate eigenvectors using perturbation theory. We then derive a novel spectral clustering algorithm called Incremental Approximate Spectral Clustering (IASC). The IASC algorithm is simple to implement and its efficacy is demonstrated on both synthetic and real datasets modelling the evolution of a HIV epidemic, a citation network and the purchase history graph of an e-commerce website.

Another important question is the absoluteness of the constraints. Lately several web-based tagging systems such as Technorati, Even if we use this approach to turn tags into constraints, Flickr or Delicious have become very popular. In this a fair amount of them are bound to be inaccurate paper we will exploit the information created by the community (i.e., linking documents which should not be in the same in Delicious: a social bookmarking service where cluster) until a high value of the parameter t, due to the polysemy the users can save the URLs of their favourite webpages of the terms used as tags or to differences in the criteria offering also the possibility of associating tags to them. of the taggers. Consequently, we have used soft positive On the other hand the clustering methods are a very important constraints, meaning that the documents affected by one of data mining tool in order to exploit the knowledge them are likely to be in the same cluster, without forcing the present in data collections. In the last years a new family of clustering algorithm to actually put them so.