Multiresolution Consensus Clustering in Networks

Abstract

Networks often exhibit structure at disparate scales. We propose a method for identifying community structure at different scales based on multiresolution modularity and consensus clustering. Our contribution consists of two parts. First, we propose a strategy for sampling the entire range of possible resolutions for the multiresolution modularity quality function. Our approach is directly based on the properties of modularity and, in particular, provides a natural way of avoiding the need to increase the resolution parameter by several orders of magnitude to break a few remaining small communities, necessitating the introduction of ad-hoc limits to the resolution range with standard sampling approaches. Second, we propose a hierarchical consensus clustering procedure, based on a modified modularity, that allows one to construct a hierarchical consensus structure given a set of input partitions. While here we are interested in its application to partitions sampled using multiresolution modularity, this consensus clustering procedure can be applied to the output of any clustering algorithm. As such, we see many potential applications of the individual parts of our multiresolution consensus clustering procedure in addition to using the procedure itself to identify hierarchical structure in networks.

Determining the Hierarchical Architecture of the Human Brain Using Subject-Level Clustering of Functional Networks

“…Several methods have been used to address this (Traag et al, 2011;Reichardt and Bornholdt, 2006;Fortunato, 2010;Nicolini et al, 2017). Here we adopt a recursive hierarchical approach to recover the community structure at multiple scales (Jeub et al, 2018;Sales-Pardo et al, 2007). After detecting the first-level (i.e., coarsest) communities in the initial run of the algorithm, we define each detected community as a separate subgraph and run the algorithm recursively on each.…”

“…Our multiresolution co-classification matrices embed information from the different hierarchical levels of community structures detected from the clustering results at the level of the subjects' networks. The recently developed method that we adopted extends the classical formulation of consensus reclustering (Lancichinetti and Fortunato, 2012) by allowing a hierarchical multiresolution output and has built-in tests for statistical significance (Jeub et al, 2018). Here, the quality function was modularity-like as in Eq.…”

“…This means that the null network only receives contributions from nodes that are less frequently coclassified together compared to the local permutation model, at a statistical significance level α. More detail can be found in Jeub et al (2018).…”

“…The same recursive principle and Louvain implementation described above were used here. To ensure a stable output, 100 iterations are used at each level (cases of output instability are dealt by meta-reclustering as in Lancichinetti and Fortunato (2012); Jeub et al (2018)); the threshold for statistical significance is set at α = 0.05.…”

“…We hypothesized that determining the mesoscale architecture of the average human cortex can be better achieved by first mapping the mesoscale in each subject and then arriving at a meaningful central tendency, rather than by mapping the mesoscale of an averaged brain network. To this aim, we perform the clustering at the subject level followed by a reclustering procedure to reach a population consensus mesoscale architecture (we refer to this as subject-derived consensus) (Jeub et al, 2018;Lancichinetti and Fortunato, 2012). We compare this subject clustering-reclustering framework to the traditional group-representative method (we refer to it as group-derived consensus) to assess which results in solutions that map best to the individuals' data.…”