Assessing the quality of multilevel graph clustering

"Lifting up" a non-hierarchical approach to handle hierarchical clustering by iteratively applying the approach to hierarchically cluster a graph is a popular strategy. However, these lifted iterative strategies cannot reasonably guide the overall nesting process precisely because they fail to evaluate the very hierarchical character of the clustering they produce. In this study, we develop a criterion that can evaluate the quality of the subgraph hierarchy. The multilevel criterion we present and discuss in this paper generalizes a measure designed for a one-level (flat) graph clustering to take nesting of the clusters into account. We borrow ideas from standard techniques in algebraic combinatorics and exploit a variable to keep track of the depth of clusters at which edges occur. Our multilevel measure relies on a recursive definition involving variable outputting a one-variable polynomial. This paper examines archetypal examples as proofs-of-concept; these simple cases are useful in understanding how the multilevel measure actually works. We also apply this multilevel modularity to real world networks to demonstrate how it can be used to compare hierarchical clusterings of graphs.