Date Published: May 9, 2019
Publisher: Public Library of Science
Author(s): Arian Ashourvan, Qawi K. Telesford, Timothy Verstynen, Jean M. Vettel, Danielle S. Bassett, Filippo Radicchi.
Community detection algorithms have been widely used to study the organization of complex networks like the brain. These techniques provide a partition of brain regions (or nodes) into clusters (or communities), where nodes within a community are densely interconnected with one another. In their simplest application, community detection algorithms are agnostic to the presence of community hierarchies: clusters embedded within clusters of other clusters. To address this limitation, we exercise a multi-scale extension of a common community detection technique, and we apply the tool to synthetic graphs and to graphs derived from human neuroimaging data, including structural and functional imaging data. Our multi-scale community detection algorithm links a graph to copies of itself across neighboring topological scales, thereby becoming sensitive to conserved community organization across levels of the hierarchy. We demonstrate that this method is sensitive to topological inhomogeneities of the graph’s hierarchy by providing a local measure of community stability and inter-scale reliability across topological scales. We compare the brain’s structural and functional network architectures, and we demonstrate that structural graphs display a more prominent hierarchical community organization than functional graphs. Finally, we build an explicitly multimodal multiplex graph that combines both structural and functional connectivity in a single model, and we identify the topological scales where resting state functional connectivity and underlying structural connectivity show similar versus unique hierarchical community architecture. Together, our results demonstrate the advantages of the multi-scale community detection algorithm in studying hierarchical community structure in brain graphs, and they illustrate its utility in modeling multimodal neuroimaging data.
Hierarchical organization is a common motif in information processing systems . The local embedding of similar processing units within groups that are then iteratively combined into larger and larger subsystems  provides a unique solution to the problem of balancing information segregation (within a group at a single scale) and integration (between groups across multiple scales) . Such an organization is observed in very large-scale computer circuits and computing architectures [4, 5], cellular communication systems , and social messaging systems . Across these various real-world information processing systems, hierarchical organization can additionally offer robustness to damage [8, 9], and a complex and diverse repertoire of system functions [10, 11] that promote optimal and efficient information processing [12, 13] and transmission.
The human brain is a complex system that can be fruitfully represented as a graph or network in which brain regions correspond to network nodes and structural or functional connections between regions correspond to network edges [20, 66]. Recent observations have pointed to the fact that both structural and functional brain networks may have community structure : the presence of densely interconnected groups of regions that might support specific cognitive functions [68–70]. Moreover, evidence suggests that these communities exist over multiple topological scales , with larger communities potentially being composed of smaller communities . Yet a comprehensive characterization of this putative hierarchical community structure in structural and functional brain graphs has remained difficult largely due to inadequacies in existing analytical paradigms and computational tools. Here we address these limitations by exercising a multi-scale community detection algorithm , and applying it to structural brain networks estimated from diffusion imaging and to functional brain networks estimated from resting state fMRI. Using novel statistics including community stability and inter-scale reliability, we show that structural brain graphs display a wider range of topological scales than functional graphs. We also illustrate the utility of this method in examining multimodal graphs that combine both structural and functional connectivity information. Our work illustrates the practical utility of multi-scale community detection in revealing hierarchical community structure in brain graphs, and opens the door for future investigations of this structure in cognition, development, aging, and disease.
In this work, we examined a multi-scale community detection algorithm and its advantages for uncovering the hierarchical organization of synthetic and real world graphs. By assuming dependence between the adjacent topological scales, the multi-scale algorithm links the communities persisting over several scales, thereby effectively uncovering hierarchical community organization in graphs. We demonstrated the statistical robustness of this hierarchical organization by defining notions of community stability and inter-scale reliability. After exercising the method on synthetic graphs, we next examined and compared the hierarchical community organization of structural brain graphs and functional brain graphs estimated from diffusion imaging and resting state functional MRI, respectively. Compared to the functional graphs, the structural graphs displayed a higher degree of topological heterogeneity with a more pronounced hierarchical organization as evidenced by a higher average number of stable communities across topological scales. With the exception of basal ganglia-thalamo-cortical circuitry, the structural communities across topological scales tended to be spatially localized, where nodes within a community were located in close physical proximity to one another. Interestingly, functional communities displayed weak similarity to structural communities at coarse topological scales, and this dissimilarity became more pronounced at finer topological scales as spatially distributed functional communities emerged. These statistical differences between the spatially distributed functional communities and spatially localized structural communities were also apparent in an explicit multi-modal extension of our method, which performs a joint optimization of modularity across a multiplex network composed of both functional and structural layers. Taken together, these results illustrate the practical utility of multi-scale community detection in revealing hierarchical community structure in single-modality and multi-modality brain graphs.
The single layer modularity quality function has been generalized to multi-slice networks to identify communities in multiplex or time-dependent networks. Formally, the multi-slice modularity quality function can be defined as
where the adjacency matrix of layer l has components Aijl, the element Pijl gives the components of the corresponding layer-l matrix for the null model, γl is the structural resolution parameter of layer l, gil gives the community assignment of node i in layer l, gjr gives the community assignment of node j in layer r, ωjlr gives the connection strength (i.e., an inter-layer coupling parameter) from node j in layer r to node j in layer l, the total edge weight in the network is μ=12∑jrKjr, the strength (i.e., weighted degree) of node j in layer l is Kjl = kjl + Cjl, the intra-layer strength of node j in layer l is kjl = ∑iAijl, and the inter-layer strength of node j in layer l is kjl = ∑rωjlr.
Here we provide synthetic examples of graphs where each node can be identified locally within a community at three different topological scales. Next we create variations in the hierarchical organization of the graphs by systematically introducing edge strength inhomogeneities. As seen in S4 Fig, multi-scale communities and the relative stability of communities across scales clearly uncovers the planted relationships and inhomogeneity profile across nodes.
Here we use principal components analysis (PCA) to assess the stability profiles of nodes across γ increments (i.e. layers) and measure the topological heterogeneity of the graphs. We used the number of components that account for more than 95% of the variance in the stability matrices as a proxy for topological heterogeneity. Low numbers indicate that most nodes display similar stability profiles and therefore the graph is relatively topologically homogeneous. Conversely, higher numbers indicate that most nodes display diverse stability profiles and therefore the graph is relatively topologically heterogeneous (S3 Fig).
The FC and SC communities share similar community organization, and joint-optimization of FC and SC graphs (i.e. SC-FC multiplex graphs) can in theory be used to evaluate these similarities. That said, the community organization of the SC-FC multiplex graph is highly dependent on the inter-modality coupling parameter, κ. In S9 Fig, we demonstrate that at smaller κ values the FC-SC graph yields two separate community structures for FC and SC components of the graph. However, for higher κ values they both share the same hybrid hierarchical community structure. The direct comparison between the community allegiance matrices of the FC, SC, and SC-FC graphs provided in S10 Fig shows the effect of increasing inter-modality coupling parameter on the SC-FC community structure.
Here we provide a brief note on visualization. Sorting the nodes of the adjacency matrices based on their community allegiance allows us to visualize communities of densely connected nodes. Nevertheless for the multi-scale communities, the order of the nodes can change depending on the topological scale. In an effort to by-pass this limitation and enhance the visualization of these communities we sort the nodes based on the similarity of their community assignments across scales. Specifically, we perform optimal leaf ordering (optimalleaforder.m) for hierarchical clustering (linkage.m) using the distances (pdist.m) calculated between the community assignments of each pair of nodes. All multi-scale community plots in this manuscript were generated using this node sorting algorithm.
Here we provide the group consensus multi-scale community results for the structural (S5 Fig) and functional (S7 Fig) graphs. One salient feature of the group consensus SC multi-scale community is that the communities are overwhelmingly made up of neighboring brain structures across the entire range of topological scales. To highlight the spatial proximity of the communities of the structural connectivity graphs, we identified (bwconncomp.m) and only presented the communities with more than one cluster in S6 Fig while removing all the other communities with only one cluster of brain regions. Next, we tested the statistical significance of these observations via permutation test (N = 1000) across γ increments (S12 Fig). Our results demonstrate that the number of SC communities with more than one cluster is significantly (p < 0.01, Bonferroni corrected for multiple comparisons) smaller than that of the null distribution (generated by changing the assignment of nodes to communities uniformly at random) for all increments of γ (except layer #3). Unlike structural graphs, functional graphs fail to yield group level consensus results for a large number of brain regions, including several subcortical and cortical structures (S7 Fig). We highlight these structures separately in S8 Fig. The distribution of the edges in the structural and functional connectivity matrices are notably different. While the distribution of SC edges are extremely heavy-tailed, the FC edges are close to a Gaussian distribution (S2 Fig). Source: http://doi.org/10.1371/journal.pone.0215520