Attribute grammars are useful for combinatorics
Theoretical Computer Science - Selected papers of the 2nd workshop on algebraic and computer-theoretic aspects of formal power series
Attribute grammars and automatic complexity analysis
Advances in Applied Mathematics - Special issue on: Formal power series and algebraic combinatorics in memory of Rodica Simion, 1955-2000
Using Automatic Clustering to Produce High-Level System Organizations of Source Code
IWPC '98 Proceedings of the 6th International Workshop on Program Comprehension
Graph-based hierarchical conceptual clustering
The Journal of Machine Learning Research
Cluster Validity Indices for Graph Partitioning
IV '04 Proceedings of the Information Visualisation, Eighth International Conference
Mining Graph Data
Graph mining: Laws, generators, and algorithms
ACM Computing Surveys (CSUR)
Engineering graph clustering: Models and experimental evaluation
Journal of Experimental Algorithmics (JEA)
A Quality Measure for Multi-Level Community Structure
SYNASC '06 Proceedings of the Eighth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
Scalable graph clustering using stochastic flows: applications to community discovery
Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining
The community-search problem and how to plan a successful cocktail party
Proceedings of the 16th ACM SIGKDD international conference on Knowledge discovery and data mining
Post-processing hierarchical community structures: Quality improvements and multi-scale view
Theoretical Computer Science
Multiscale visualization of small world networks
INFOVIS'03 Proceedings of the Ninth annual IEEE conference on Information visualization
Computer Science Review
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"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 $$q$$q to keep track of the depth of clusters at which edges occur. Our multilevel measure relies on a recursive definition involving variable $$q$$q 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.