CS 599 : Structure and Dynamics of Networked

So far, we have mostly talked about communities in the sense of discovering one, or a few, densely linked subgraphs. We departed from this interpretation at the end of last lecture, when we defined the notion of the modularity of a clustering. There, we are interested in the division of a graph into disjoint partitions (or clusters) of nodes, and the quality of this clustering. Clustering of data, mostly in metric spaces, is one of the most well-studied problems in CS, mostly due to its applications to machine learning and classification. Here, we look at the relatively new concept of correlation clustering [1]. In correlation clustering, each edge of the graph is annotated with a label of ‘+’ or ‘-’, expressing that the two endpoints were observed to be similar or dissimilar, respectively. The goal is then to find a clustering that puts many ’+’ edges inside clusters, and ’-’ edges between clusters. However, these goals may be conflicting, as can be seen for a triangle with two edges labeled ’+’ and one labeled ’-’. Notice that the number of clusters is not pre-specified in this problem. This notion of clustering can be useful when we can identify if a link constitutes an endorsement, or the opposite. For instance, in many competitive scenarios (for instance, politics or sports), pages will link to other pages with the explicit goal of deriding the content. This can be frequently identified from anchor text and similar clues. In this sense, correlation clustering may help us in identifying communities with aligned interests, which compete with other communities. More formally, given the graph G = (V,E) on n vertices, we write +(i, j) if the edge between i and j is labeled ‘+’ and similarly for −(i, j). The optimization problem can now be expressed in two ways: