Post-Processing Partitions to Identify Domains of Modularity Optimization

Abstract

We introduce the Convex Hull of Admissible Modularity Partitions (CHAMP) algorithm to prune and prioritize different network community structures identified across multiple runs of possibly various computational heuristics. Given a set of partitions, CHAMP identifies the domain of modularity optimization for each partition—i.e., the parameter-space domain where it has the largest modularity relative to the input set—discarding partitions with empty domains to obtain the subset of partitions that are “admissible” candidate community structures that remain potentially optimal over indicated parameter domains. Importantly, CHAMP can be used for multi-dimensional parameter spaces, such as those for multilayer networks where one includes a resolution parameter and interlayer coupling. Using the results from CHAMP, a user can more appropriately select robust community structures by observing the sizes of domains of optimization and the pairwise comparisons between partitions in the admissible subset. We demonstrate the utility of CHAMP with several example networks. In these examples, CHAMP focuses attention onto pruned subsets of admissible partitions that are 20-to-1785 times smaller than the sets of unique partitions obtained by community detection heuristics that were input into CHAMP.

Comparing Shared Patient Networks Across Payers

“…To identify higherlevel structure within each network, we employed a wellstudied community detection algorithm to optimize the quantity known as modularity. [21][22][23][24] Briefly, modularity attempts to maximize the strength of edges within communities above what would be expected under a random null model. To select the appropriate resolution at which to identify community structure, we employed the Convex Hull of Admissible Modularity Partitions (CHAMP) tool.…”

“…To select the appropriate resolution at which to identify community structure, we employed the Convex Hull of Admissible Modularity Partitions (CHAMP) tool. 24 Community structure was compared at the broadest domains that overlapped between the networks. We conducted all analyses using Python.…”

Transient crosslinking kinetics optimize gene cluster interactions

“…This approach efficiently explores clustering phenomena at multiple spatial and temporal scales, identifying at which scales clustering is most prevalent and at which scales clustering is nonexistent. To identify appropriate values for γ and ω, we used a variety of techniques including the CHAMP algorithm [54] (which utilizes fast algorithms that detect convex hulls in the (γ, ω) parameter space) and comparisons to other community-detection algorithms including the study of connected-components.…”

Assessing the robustness of cluster solutions obtained from sparse count matrices.

“…Some methods were developed for very large, binary matrices but do not always provide reliable results for smaller, weighted matrices. We additionally note that, for large networks, modularity suffers from a resolution limit (Fortunato & Barthelemy, 2007), motivating inclusion of a resolution parameter (Reichardt & Bornholdt, 2006) that may then be selected by various methods (see Weir, Emmons, Gibson, Taylor, & Mucha, 2017). However, we will ignore these issues for the small matrices considered here.…”