Top left: a spring embedded layout of the areal graph at 4% tie density visualizing the graph and the basis for subgraphs. Top right: for both cohorts, plots are shown of the areal assignments into subgraphs (colors) at tie densities from 10% down to 2% in 1% steps. ROI ordering is identical, and all subgraphs with fewer than 4 members are colored white. The standard measure of subgraph similarity, normalized mutual information, between node assignments of the cohorts at identical tie densities ranged from 0.86-0.92, indicating highly similar patterns across cohorts (1 = identical assignments, 0 = no information shared between assignments). Bottom: subgraphs from three thresholds are shown for the areal (spheres) and modified voxelwise graphs (surfaces). Note the similarity of subgraph assignments beteween networks, despite the great difference in network size and cortical coverage, even in different subjects (main vs replication cohorts). All areal subgraphs with fewer than 4 members are colored white, and all modified voxelwise subgraphs with fewer than 100 voxels are colored white. Areal networks are shown at 10%, 3%, and 2% tie density (r > 0.16, 0.30, and 0.33), and modified voxelwise networks are shown at 5%, 2%, and 0.5% tie density (r > 0.16, 0.23, and 0.31).

Select subgraphs from the modified voxelwise analysis are presented from a dorsal view for both cohorts and for an additional single subject. Cohort subgraphs are taken from the 2% tie density analysis and subgraphs in the individual are taken from a 0.5% tie density analysis. The overall NMI between cohort assignments at this threshold was 0.71, and NMI values between subgraphs from different cohorts are shown in the matrix to the right. Additional views of this data and replications of subgraphs from other thresholds are found in .

At left, the task-defined locations of four established functional systems. The next three columns display, for the main cohort, the single subgraph that best corresponds to each functional system under the four graph definitions. Circles are placed around small portions of subgraphs that might otherwise be overlooked (there are small green regions within green circles). Data from a single threshold tailored to each graph are shown. The threshold was the next-to-highest threshold that each graph can achieve before the graph becomes severely fragmented (defined by the giant component containing fewer than 50% of the nodes in the graph). Tailored thresholds were 3% for the areal graph, 5% for the AAL-based graph, and 2% for both voxel-based graphs. Correspondence between these functional systems and subgraphs is good for the areal and modified voxelwise graphs, intermediate for the voxelwise graph, and poor for the AAL-based graph. Note especially the correspondence between areal (spheres) and modified voxelwise (surface) subgraphs, despite great differences in network size (N = 264 vs N = 40,100). See Figure and for more comprehensive and quantitative presentations of subgraph assignments. Images in the left column are modified from (; ; ; ).

Left: Visual (blue), auditory (pink), and hand (cyan) and face (orange) sensory-somatomotor (SSM) subgraphs are shown for the areal network at 2% (spheres) and the modified voxelwise network at 0.5% tie density (surface). Below, the mean correlations in the main cohort between auditory processing (pink, MNI: −38 −33 17) and hand (cyan, −40 −19 54) and face (orange, −49 −11 35) regions are shown. Auditory-face correlations are significantly higher than auditory-hand correlations in both cohorts (p < 0.001, two-sample two-tail t-test). Bottom, slices from the 4% tie density modified voxelwise analysis, with labels on relevant thalamic nuclei (numbers are z coordinates). Middle: Two cingulo-opercular subgraphs shown from the 3% areal (spheres) and 2% tie density modified voxelwise analysis (surface). Middle, published ROIs (cingulo-opercular task control: (); salience: ()) or modified voxelwise subgraphs, with an overlaid heat map of on-cue meta-analysis activation. On-cue activity localizes to the purple subgraph. Bottom, very strong fc-Mapping gradients are displayed separating the black and purple subgraphs, indicating that they possess distinct rs-fcMRI signals. Right: At top, three unknown subgraphs from the 0.5% tie density modified voxelwise analyses are shown. The salmon subgraph (gray in all other figures, here salmon for contrast) is reproduced with a 2% areal subgraph overlaid as spheres, and the strongest activations from the memory retrieval meta-analysis are shown below. The light blue subgraph is also reproduced and the coordinates of a putative functional system from () are overlaid as tan spheres.

At left, the “task+ system” of (). At right, three subgraphs from the 0.5% tie density modified voxelwise analysis. The “task+ system” is composed of at least three subgraphs, corresponding to the fronto-parietal task control, cingulo-opercular task control, and dorsal attention systems.

At top, the subgraphs, local efficiencies, and participation coefficients for all nodes in the areal network over a range of thresholds are shown. The local efficiency of each node indicates the extent to which a node is embedded in a richly connected local environment. High (hot color) values indicate a richly connected local environment. The participation coefficient of each node indicates the extent to which a node has ties to other subgraphs. Here, low (cool color) values indicate that nodes are connected almost exclusively to members of their own subgraph. One-factor ANOVAs indicate a significant effect of subgraph at all thresholds for both indices (all with p < 10⁻⁶), and post-hoc t-tests indicate that the cyan, blue, and red subgraphs have significantly higher local efficiencies and lower participation coefficients at most or all thresholds than the yellow subgraph. Node assignments for a single threshold (4% tie density) are shown on a brain and in a spring embedded layout, and the local efficiencies and participation coefficients of relevant subgraphs at this threshold are shown. Note that local efficiency is independent of subgraph assignment, whereas participation coefficients depend upon subgraph assignment.

In charts, particular subgraphs at a single threshold are selected, the spatial boundaries of that subgraph are found, and the distribution of spatial interfaces (en face voxels) to other subgraphs are calculated. The most frequent interfaces are plotted as percents of the total subgraph interface volume. Motifs are inferred by finding instances where subgraphs interfacing with a subgraph are themselves very unlikely to interface. For instance, in the top chart, the light blue subgraph interfaces most frequently with the yellow and red subgraphs, but red is only 3.6% of yellow’s interface, and yellow is only 2.6% of red’s interface. Below each chart, plots of relevant subgraphs on brain surfaces visually demonstrate the repeated spatial patterns of subgraphs. Data from the modified voxelwise analysis at 1% tie density in the replication cohort are presented.