Fig 1.
Schematic representing the construction of a multilayer network for use in multi-scale modularity maximization.
Duplicates of a graph are connected in a multilayer fashion to construct a 3D graph. The smallest resolution parameter γ is assigned to the first layer (x), and it is linearly increased for the neighboring layers (y, z). The topological scale coupling parameter, τ, tunes the strength of the dependence of communities across layers. Since the community assignments are dependent on the adjacent layers, nodes that display high clustering over neighboring topological scales are identified as a single community spanning several scales. In this schematic, the large communities identified at initial layers progressively break into smaller sub-communities, revealing the hierarchical community organization of the graph.
Fig 2.
Uncovering hierarchical community structure in a synthetic graph.
(A) Graphs can display heterogeneity in hierarchical community structure. To gain intuition regarding the utility of our method for characterizing these sorts of graphs, we design a synthetic graph such that each node is part of a small cluster composed of 3 nodes, a medium sized cluster composed of 9 nodes, and a large cluster composed of 27 nodes. Topological heterogeneity is introduced by adding gradients in the values of edge weights such that not all clusters of a given size have the same average weight. (B) We seek to uncover the hierarchical structure in this synthetic graph. First, we use the more traditional approach of maximizing a single-layer modularity quality function [53] with the Newman-Girvan null model [57] using a Louvain-like locally greedy algorithm [54]. We sweep the resolution parameter between 1 and 16, and identify communities independently at each γ value. The limitation of this approach is that there is no guarantee that communities at one resolution correspond to communities at another resolution. (C) To overcome this limitation, we next seek to uncover the hierarchical structure in this synthetic graph using a multi-scale approach built on multi-scale community detection [31]. We find that the hierarchical community detection uncovers the true underlying community organization as we vary the value of the structural resolution parameter (γ ∈ [0, 12], τ = 0.05, an inter-layer γ increment of 0.05, and 241 layers). Moreover, the γ value at which a community is detected tracks the mean edge weight of the community; stronger communities are identified at larger γ values, and weaker communities are identified at smaller γ values. (D) For comparison, we created a null graph without any clear hierarchical community structure by randomly shuffling the edges of the graph in panel (A), while preserving the weight, degree, and strength distributions (see [63] for details). Panels (E) and (F) show the communities detected in this null graph when using the parameter sweep method and the multi-scale community detection method, respectively. Note that the lack of hierarchy in the null graph is similarly echoed in the multi-scale community structure displayed in panel (F), marked by predominantly small and unstable communities across topological scales.
Fig 3.
Application of multi-scale community detection to subject-level and group-level structural brain networks.
(A) An example structural connectivity matrix from one subject, in which each element linking a pair of brain regions represents the number of streamlines reconstructed between those two areas. To better visualize its structure, we apply a log transformation (log(SC + 1)/max(log(SC + 1))). (B) The consensus partition representing the multi-scale community structure for the matrix in panel (A). To enhance the visual detection of communities, we have represented all singleton communities with the same gray color. (C) The group-level consensus partition representing the multi-scale community structure for the structural matrix, which is defined as the consensus over all participants’ partitions. Here, again, to enhance visual clarity, we color the singleton communities in the same gray color. (D) The average number (out of 600 total nodes) as well as the average size (expressed as the percentage of total nodes) of the non-singleton communities calculated across layers, which in our case represent γ increments. In these analyses, we used γ ∈ [0.0133, 1], an inter-layer γ increment of 0.0133, and 75 layers.
Fig 4.
Application of multi-scale community detection to subject-level and group-level resting state functional brain networks.
(A) An example rsfMRI connectivity matrix from one subject, in which each element linking a pair of brain regions represents the pairwise wavelet coherence between regional time series. (B) The consensus partition representing the multi-scale community structure for the matrix in panel (A). To enhance the visual detection of the communities, we have represented all singleton communities with the same gray color. (C) The consensus partition representing the multi-scale community structure for the group-level functional matrix, which is defined as the average connectivity matrix across participants. Here, again, to enhance clarity, we color the singleton communities in the same gray color. (D) The average number (out of 600 total nodes) as well as the average size (expressed as the percentage of total nodes) of non-singleton communities calculated across layers, which in our case track γ increments. In these analyses, we used a structural resolution parameter γ ∈ [0.95, 1.7], an inter-layer γ increment of 0.01, and 75 layers.
Fig 5.
Local topological scales of hierarchical community organization in structural and functional brain graphs.
The black lines show the average number of communities to which each node is allied across different stability thresholds in the hierarchical community organization of SC (A-B) and FC (C-D) graphs, at both the subject level (A,C) and the group level (B,D). Shaded areas show standard deviation. We define the stability threshold as the percent range of γ values over which a node is stably allied to a given community, here shown along the x-axis.
Fig 6.
Reliability of inter-layer multi-scale community allegiance of brain regions in SC and FC graphs.
(A) The percentage of brain regions from the SC (blue) and FC (red) graphs containing inter-layer edges with significant probability at the group-level across layers, which in our case track γ increments. The probability of identifying the SC inter-layer edges falls below chance for the second half of the γ range values. These results suggest that the estimated stability and the hierarchical structure of the higher γ layers are unreliable. Inter-layer coupling weights were (τ = 0.50). The probability of identifying the FC inter-layer edges was above the significance level for a small range of γ values. Since the vast majority of the group-level consensus inter-layer edges are not significant, the identified communities frequently switch across layers (as seen in (Fig 4C)). (B-C) The regions from the SC (panel (B)) and FC (panel (C)) graphs containing inter-layer edges with significant probability at the group-level are overlayed on the brain and color-coded to represent the ratio of the significant inter-layer edges (i.e., by dividing by the total number of inter-layer edges).
Fig 7.
Similarity between the hierarchical community structure of structural and functional brain graphs.
(A) The similarity between the allegiance matrices of the hierarchical community structures of the FC and SC graphs. Specifically, we calculate the Pearson correlation coefficient between allegiance matrix elements of SC at a given γ and the allegiance matrix elements of FC at a given γ, for all possible γ pairs, allowing us to identify the alignment that results in the highest similarity between their allegiance matrices (marked by in the plot). Note that a direct comparison between FC and SC allegiance matrices across corresponding layers shows relatively small correlation values (first row). Nevertheless, shifting the SC matrix towards higher layers yields higher correlation values and thus a better allegiance between FC and SC matrixes across topological scales. The dashed white lines in panel (B) highlight the layers where the SC and FC realignment yield highest similarity values. (C) The thresholded allegiance matrices of layers highlighted by dashed lines in panel (B) of the FC and SC hierarchical communities. FC and SC allegiance matrices are ordered identically based on their original node labels.
Fig 8.
Similarity between the allegiance matrices of the multiplex SC-FC graph, the FC graph alone, and the SC graph alone, as a function of the topological scale for the group.
Using a (κ = 1), we calculated the Pearson correlation coefficient for all layers and all layer shifts between the allegiance matrices of (A) the multiplex SC-FC graph and the SC graph, (B) the multiplex SC-FC graph and the FC graph, and (C) the SC graph and the FC graph outside of the multiplex formulation. Layers and shifts with maximum correlation values are marked by . The layer numbers that yield maximum correlation values are also presented on the plots (white). (D-F) Brain overlays show the communities identified at layers with maximal correlation values (panels (A-C)). The color bars in each panel (right hand side) represent the color-coded communities. The multiplex SC-FC communities at low γ layers are very similar to SC communities at slightly higher γ layers (green boxes). Although FC communities overall show smaller similarity to the muliplex SC-FC communities, the multiplex SC-FC communities at higher γ layers show relatively higher similarity to FC communities at high γ layers (red boxes). The smallest similarity is observed between FC and SC graphs, peaking around a coarse scale with small γ values for both modalities (cyan boxes).