An axiomatic study of objective functions for graph clustering

We investigate axioms that intuitively ought to be satisfied by graph clustering objective functions. Two tailored for graph clustering objectives are introduced, and the four axioms introduced in previous work on distance based clustering are reformulated and generalized for the graph setting. We show that modularity, a standard objective for graph clustering, does not satisfy all these axioms. This leads us to consider adaptive scale modularity, a variant of modularity, that does satisfy the axioms. Adaptive scale modularity has two parameters, which give greater control over the clustering. Standard graph clustering objectives, such as normalized cut and unnormalized cut, are obtained as special cases of adaptive scale modularity. We furthermore show that adaptive scale modularity does not have a resolution limit. In general, the results of our investigation indicate that the considered axioms cover existing ‘good’ objective functions for graph clustering, and can be used to derive an interesting new family of objectives.