RESUMO
Modern diffusion and functional magnetic resonance imaging (dMRI/fMRI) provide non-invasive high-resolution images from which multi-layered networks of whole-brain structural and functional connectivity can be derived. Unfortunately, the lack of observed correspondence between the connectivity profiles of the two modalities challenges the understanding of the relationship between the functional and structural connectome. Rather than focusing on correspondence at the level of connections we presently investigate correspondence in terms of modular organization according to shared canonical processing units. We use a stochastic block-model (SBM) as a data-driven approach for clustering high-resolution multi-layer whole-brain connectivity networks and use prediction to quantify the extent to which a given clustering accounts for the connectome within a modality. The employed SBM assumes a single underlying parcellation exists across modalities whilst permitting each modality to possess an independent connectivity structure between parcels thereby imposing concurrent functional and structural units but different structural and functional connectivity profiles. We contrast the joint processing units to their modality specific counterparts and find that even though data-driven structural and functional parcellations exhibit substantial differences, attributed to modality specific biases, the joint model is able to achieve a consensus representation that well accounts for both the functional and structural connectome providing improved representations of functional connectivity compared to using functional data alone. This implies that a representation persists in the consensus model that is shared by the individual modalities. We find additional support for this viewpoint when the anatomical correspondence between modalities is removed from the joint modeling. The resultant drop in predictive performance is in general substantial, confirming that the anatomical correspondence of processing units is indeed present between the two modalities. Our findings illustrate how multi-modal integration admits consensus representations well-characterizing each individual modality despite their biases and points to the importance of multi-layered connectomes as providing supplementary information regarding the brain's canonical processing units.
RESUMO
In stochastic block models, which are among the most prominent statistical models for cluster analysis of complex networks, clusters are defined as groups of nodes with statistically similar link probabilities within and between groups. A recent extension by Karrer and Newman [Karrer and Newman, Phys. Rev. E 83, 016107 (2011)] incorporates a node degree correction to model degree heterogeneity within each group. Although this demonstrably leads to better performance on several networks, it is not obvious whether modeling node degree is always appropriate or necessary. We formulate the degree corrected stochastic block model as a nonparametric Bayesian model, incorporating a parameter to control the amount of degree correction that can then be inferred from data. Additionally, our formulation yields principled ways of inferring the number of groups as well as predicting missing links in the network that can be used to quantify the model's predictive performance. On synthetic data we demonstrate that including the degree correction yields better performance on both recovering the true group structure and predicting missing links when degree heterogeneity is present, whereas performance is on par for data with no degree heterogeneity within clusters. On seven real networks (with no ground truth group structure available) we show that predictive performance is about equal whether or not degree correction is included; however, for some networks significantly fewer clusters are discovered when correcting for degree, indicating that the data can be more compactly explained by clusters of heterogenous degree nodes.