The Effect of Modular Degeneracy on Neuroimaging Data

Abstract

Introduction: The concept of community structure, based on modularity, is widely used to address many systems-level queries. However, its algorithm, based on the maximization of the modularity index Q, suffers from degeneracy problem, which yields a set of different possible solutions. Methods: In this work, we explored the degeneracy effect of modularity principle on resting-state functional magnetic resonance imaging (rsfMRI) data, when it is used to parcellate the cingulate cortex using data from the Human Connectome Project. We proposed a new iterative approach to address this limitation. Results: Our results show that current modularity approaches furnish a variety of different solutions, when these algorithms are repeated, leading to different number of subdivisions for the cingulate cortex. Our new proposed method, however, overcomes this limitation and generates more stable solution for the final partition. Conclusion: With this new method, we were able to mitigate the degeneracy problem and offer a tool to use modularity in a more reliable manner, when applying it to rsfMRI data.

Keywords: cingulate cortex; consensus; graph theory; modularity; resting-state fMRI.

FIG. 1.

Flowchart for the iterative-consensus method illustrating the procedure. (Left) Initial 1000 inputs obtained from the community_Louvain. (Middle) 100 partitions that are randomly selected and used in the consensus algorithm to generate one result, repeating it 100 times. (Right) 100 results are the final input to the consensus algorithm to generate one final partition.

FIG. 2.

Violin plot of the z-score values for the 10 subgroups (20 subjects in each subgroup) for each gamma. The individual data points represent the subgroups. The mean z-scores averaged across the 10 subgroups, for each gamma, are represented by the horizontal lines and are displayed under each plot.

FIG. 3.

Left and right views of the cingulate cortex with the subdivisions calculated with the iterative-consensus method for gamma = 1.2, using the low-resolution (4 mm isotropic-top) and the high-resolution (2 mm isotropic-bottom) data. Each module is represented by a different color and a respective number.

FIG. 4.

Functional connectivity (FC) maps for the five subdivisions of the cingulate cortex, which were obtained with the iterative-consensus method for gamma = 1.2, using the high-resolution data (2 mm isotropic). The left and right view of the cingulate cortex with its subdivisions is shown on the right. Each module is represented by a different color. The sagittal and axial views of the FC maps, for each subdivision, are shown at the middle and left side of the figure, respectively. Each row displays the FC maps of the subdivision represented by the color on the right. Significance: uncorrected p value (p ≤ 0.001), corrected p value (p_corr < 0.05). Radiological convention.