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The modular organization of human anatomical brain networks: Accounting for the cost of wiring
Richard F. Betzel, John D. Medaglia, Lia Papadopoulos, Graham Baum, Ruben Gur, Raquel Gur, David Roalf, Theodore D. Satterthwaite, Danielle S. Bassett
TL;DR
Brain module detection can confound spatial wiring constraints with genuine modular organization. The paper modifies modularity maximization to identify connections and modules unexpected under cost-reduction wiring, finding distinct functional, rich-club, and developmental organization.
Problem
Existing module-detection methods make it difficult to determine whether detected brain modules reflect wiring-cost reduction rather than additional organization.
Method
The paper integrates a spatial cost-reduction null model into modularity maximization to focus on long-distance connections unexplained by spatial wiring.
Results
The modified approach detects modules that differ from Newman-Girvan modules, have distinct functional fingerprints, overlap rich clubs, and include a module tracking developmental age.
Takeaways & Limitations
Accounting for wiring cost provides a complementary interpretation of brain modules and the functional roles of specific brain regions.
Takeaways & Limitations
The detected modules could not be verified for robustness with other algorithms because readily available alternatives do not accept alternative null connectivity models.
Abstract
from arXiv · showhide
Brain networks are expected to be modular. However, existing techniques for estimating a network's modules make it difficult to assess the influence of organizational principles such as wiring cost reduction on the detected modules. Here, we present a modification of an existing module detection algorithm that allows us to focus on connections that are unexpected under a cost-reduction wiring rule and to identify modules from among these connections. We apply this technique to anatomical brain networks and show that the modules we detect differ from those detected using the standard technique. We demonstrate that these novel modules are spatially distributed, exhibit unique functional fingerprints, and overlap considerably with rich clubs, giving rise to an alternative and complementary interpretation of the functional roles of specific brain regions. Finally, we demonstrate that, using the modified module detection approach, we can detect modules in a developmental dataset that track normative patterns of maturation. Collectively, these findings support the hypothesis that brain networks are composed of modules and provide additional insight into the function of those modules.
INTRODUCTION
Brain networks are modular, but short-range wiring and standard modularity methods can make spatial constraints appear as modules. The paper therefore modifies modularity maximization to focus on long-distance connections unexpected under a cost-reduction wiring rule.
- Brain networks contain internally dense, externally sparse modules across scales from synaptic networks to whole-brain white-matter networks.
- Wiring costs favor short connections, while a smaller set of costly long-distance connections may support communication and integrative hubs.
- Standard modularity can mistake spatially generated networks for modular networks because short-range connections inflate modularity scores.
- The modified null model discounts connections expected from spatial wiring and detects modules whose internal connections are unusually long-distance.
- Across anatomical datasets, the approach yields distinct functional and organizational findings, including rich-club overlap and a developmental module-age relationship.
Human DSI
The study analyzes representative human anatomical networks from adult DSI and developmental DTI data. It applies modularity maximization with spatially informed null models, using connectivity and interregional distance matrices as inputs.
- The adult DSI dataset contains 30 healthy individuals represented by networks with 1,014 brain regions and binary interregional connectivity.
- The analysis combines observed connectivity matrices with Euclidean distances between brain-region centroids to model spatial wiring constraints.
- The principal aim is to make module detection sensitive to long-distance connections not explained solely by cost reduction or geometry.
- Modularity maximization partitions nodes by comparing observed within-module connections with expectations from a selected null model.
- The study varies the resolution parameter and repeatedly applies the stochastic Louvain algorithm to obtain candidate partitions.
Selecting the resolution parameter
The analysis selects a resolution parameter using partition consistency across repeated Louvain runs, then condenses the resulting ensemble into a consensus partition. Statistical significance is assessed against randomized module assignments.
- The resolution parameter is chosen where repeated Louvain runs produce especially similar module partitions.
- Partition similarity is quantified using a z-score comparing shared same-module node pairs against chance variation.
- An association-reclustering procedure summarizes common node coassignments across the selected partition ensemble.
- Consensus modularity produces more mutually consistent partitions, and the algorithm typically converges within two or fewer iterations.
- A module is considered statistically significant when its modularity contribution exceeds the 99th percentile of a 10,000-permutation null model.
Null models
The study replaces the standard degree-based null model with a spatial cost-reduction model to test whether modules exceed expectations from short-range wiring. The spatial model matches observed density and selects parameters by likelihood.
- The Newman-Girvan null model tests whether modules arise from the observed degree sequence, whereas this study tests whether they arise from cost-reducing wiring.
- The spatial model makes connection probability decrease monotonically with interregional distance, with α controlling overall density and β penalizing connection length.
- Euclidean distance serves as a proxy for fiber length because fiber trajectories are unavailable for undetected connections, although the measures correlate at r = 0.696.
- Parameters are constrained so expected edge count matches the observed count, then the likelihood-maximizing parameter pair is selected.
Modularity maximization pipeline summary
The pipeline compares observed connectivity with standard and spatial wiring null models to define alternative modularity functions and related network metrics.
- Modularity functions: The analysis uses connectivity matrices, node locations, expected connection counts, and a resolution parameter to define NG and SPTL modularity functions.The NG model uses the Newman-Girvan null model, whereas SPTL uses a spatial null model.
- Participation coefficient: Participation coefficients quantify whether a node’s links remain within its module or are distributed across different modules.Values near one indicate uniform distribution across modules, while values near zero indicate predominantly within-module connections.
- Rich-club analysis: Rich-club analysis measures the density of connections among nodes exceeding a degree threshold and normalizes it against degree-preserving random networks.Peaks in the normalized coefficient identify candidate rich clubs.
Rich club module density
The study evaluates rich-club organization, functional overlap, subject-level consistency, and robustness of SPTL modules in human DSI networks.
- Subject consistency: The five largest consensus modules were consistently expressed across individual subjects using region-level overlap-based consistency scores.The consistency analysis matched each consensus module to the subject-level module with greatest overlap before averaging constituent regions.
- SPTL framework: The SPTL framework was designed to detect modules whose internal connection density exceeds expectations under a spatial wiring null model.This focuses module detection on residual, especially long-distance, connections rather than connections explained by spatial cost reduction.
- Functional fingerprints: Functional fingerprints of the consensus modules were correlated, indicating that the long-distance modular architecture occupies a relatively low-dimensional space.The functional fingerprints were derived from module overlap with established functional systems.
- Robustness: The analysis tested robustness to node definition, resolution parameter, tractography, and network-reconstruction settings.Supplementary analyses varied inclusion of subcortical regions, resolution, and maximum curvature angle.
Human DTI
In human DTI, SPTL detected five significant modules with patterns resembling DSI modules but combining several of them differently. Compared with NG modules, SPTL changed system co-assignment and participation profiles.
- Human DTI: Human DTI yielded five statistically significant SPTL modules at γ = 2.0, covering 137/233 brain regions.The analysis detected 39 modules at this scale, with five reaching statistical significance.
- Cross-dataset correspondence: The largest DTI module combined two inferior bilateral DSI modules, while the second-largest combined DSI modules 14 and 15.The remaining three significant modules imperfectly resembled DSI module 19 and collectively spanned both hemispheres.
- Module association: SPTL and NG produced different functional-system co-assignment patterns, especially for somatomotor and default-mode regions.SMN regions shifted from NG associations with SMN and DAN toward SPTL associations with VAN and DMN, while DMN regions became broadly co-assigned under SPTL except with CONT.
- Participation coefficient: Salience and somatomotor systems showed significant participation increases, whereas ventral attention, control, default mode, and visual systems decreased.Temporal and subcortical systems showed no participation changes; increased participation indicates more connections distributed across modules.
- Participation coefficient: The NG-versus-functional-partition comparison produced a similar pattern, with increased somatomotor participation and decreases in other systems.This comparison used the same participation-coefficient methods as the SPTL-versus-NG analysis.
Mean interregional distance
SPTL modules were more spatially distributed than NG modules, and their sizes were linked to the number of long-distance connections made by constituent regions. These modules also better recapitulated rich-club relationships and were used to track developmental changes in white-matter integrity.
- Mean interregional distance: For similarly sized communities, SPTL modules had greater spatial extent than NG modules, with a total curve difference of 49.44.Spatial extent increased with module size in both models, but SPTL modules were more distributed across space.
- Mean interregional distance: At γ = 2.6, 11 of 82 modules were singletons and 64 contained fewer than 11 nodes, while long-distance degree correlated with module size at r ≈ 0.76.The correlation was measured at a 70 mm distance threshold, with p < 10^-15.
- Relationship to rich clubs: SPTL modules had consistently greater rich-club module density than NG modules across γ values and the five examined rich clubs.This indicates that SPTL modules better recapitulated relationships among rich-club nodes.
- Development: Before confound correction, 12 modules showed age-related within-module FA changes, with the strongest association in a bilateral midline module at r = 0.48.The module comprised precuneus, posterior cingulate, and anterior cingulate cortex, with p < 10^-15.
- Development: After regressing out network, physiological, and morphological confounds, only the midline module retained an age association, attenuated to r = 0.14.The candidate confounds included binary density, global FA, signal-to-noise ratio, head motion, and total intracranial volume.
DISCUSSION
The discussion argues that a cost-reduction null model complements standard NG modularity by emphasizing unexpected long-distance connections. SPTL modules differed functionally, aligned more closely with rich clubs, and revealed developmental structure, while no single detection method should be treated as ground truth.
- DISCUSSION: The SPTL null model shifts modularity detection from short-range connections toward unexpected, costly, long-distance connections.It models the brain’s preference for short connections while permitting principled analysis of long-distance organization.
- DISCUSSION: SPTL and NG modules diverged in functional fingerprints, with somatomotor regions more integrative and default-mode, control, ventral-attention, and visual systems more segregated under SPTL.These differences were assessed through changes in participation coefficient.
- DISCUSSION: SPTL modules were more similar to rich clubs than NG modules, supporting a complementary interpretation of long-distance integrative organization.Rich clubs are described as spatially distributed hubs linked by costly connections that bridge modules.
- DISCUSSION: In developmental data, SPTL modules identified normative maturation through age-related changes in within-module white-matter integrity.The strongest surviving association after confound correction involved the midline module.
- DISCUSSION: Because brain modular organization has no known ground truth and detection methods can miss even annotated groups, results from multiple null models should be compared.The authors caution against overinterpreting any single module detection algorithm or null model.
Space-independent modules across development
The study’s spatially informed modularity framework identifies modules emphasizing long-distance connections, while developmental analyses link a precuneus–cingulate module to maturation. The approach also carries methodological limitations related to modularity resolution, consensus clustering, and validation.
- Space-independent modules across development: A precuneus–cingulate module showed significantly increasing mean FA with age after controlling for confounds, and overlapped default-mode and rich-club regions.The authors interpret the FA increase as maturation of these structures.
- Space-independent modules across development: The SPTL null model detects modules with more long-distance connections than expected, producing compositions and topographies usually distinct from those found with the NG model.Modules satisfying both density-based and long-distance criteria could nevertheless coincide across null models.
- Space-independent modules across development: The framework changes modularity by comparing observed networks with a null connectivity model based on wiring-reduction principles.This redefinition shifts attention toward structure not explained by spatial wiring constraints.
- Space-independent modules across development: Modularity maximization remains limited by resolution limits and near-optimal partition degeneracy, although consensus modules were used to address variability across runs.The resolution parameter only partially mitigates the resolution limit.
- Space-independent modules across development: Consensus clustering can propagate biases because modularity maximization is used again to recluster partitions produced by the same procedure.The authors note that consensus clustering improves community estimates but may retain shared methodological biases.
Extensions
The extensions discuss how the spatial cost-reduction framework can be applied beyond the analyzed binary structural networks. They emphasize tractography limitations, possible weighted-network extensions, and broader uses for separating spatial effects from other network properties.
- Extensions: The analyzed networks were binary, so the method discarded information about relative connection strength.The authors identify weighted networks as an important extension.
- Extensions: The model could incorporate edge weights and, in a weighted signed form, could be applied to functional connectivity networks.These extensions were proposed but not explored in the report.
- Extensions: The SPTL framework can test whether spatial wiring influences rich-club organization and the distribution of hub regions.The model explicitly accounts for properties driven by space and cost-reduction principles.
- Extensions: Diffusion imaging and tractography can miss fibers running parallel to the cortical surface and introduce algorithm-specific biases.These limitations constrain interpretation of reconstructed structural networks.
- Extensions: The SPTL model might help correct tractography biases by distinguishing short-range connections caused by wiring cost from those favored by tracking algorithms.The authors present this as a possible use rather than a demonstrated correction.
A. Including versus excluding subcortical regions
Supplementary analyses assess whether SPTL modules are robust to cortical-only versus subcortical-inclusive networks, resolution choices, and tractography curvature parameters. The reported partitions remain highly similar across these variations.
- A. Including versus excluding subcortical regions: The cortical-only and subcortical-plus-cortical partitions were highly similar, with zrand = 260.26 versus a null maximum of 3.40.The cortical-only network’s optimum occurred at γ = 2.8, compared with γ = 2.6 for the main network.
- B. Robustness to choice of resolution parameter: Consensus partitions across γ = 2.2–3.0 were qualitatively excellent matches, with most nonzero association values confined within modules.Rows and columns were ordered according to the γ = 2.6 consensus partition.
- C. Robustness to variation in max curvature angle: Tractography reconstructions are vulnerable to false positives, false negatives, and parameter-dependent variation, motivating robustness checks over reasonable curvature values.Maximum curvature angle determines the largest streamline orientation change allowed between integration steps.
- C. Robustness to variation in max curvature angle: Association matrices for the 30° and 40° cutoffs showed most nonzero elements within the reference modules and few between them.This qualitative pattern supported similarity to the modules obtained with the 35° cutoff.
D. Participation coefficients of structural versus functional partitions
The supplementary comparison evaluates participation coefficients under structural and functional partitions and situates the results within broader anatomical and functional network analyses. Both comparisons identify increased participation of the somatomotor network.
- D. Participation coefficients of structural versus functional partitions: The comparison used SPTL structural consensus coefficients against coefficients from a functional partition derived from prior work.Functional partitions provide an indirect way to assess node roles relative to functional systems.
- D. Participation coefficients of structural versus functional partitions: Both structural-versus-functional participation comparisons identified the somatomotor network as having increased participation.The analyses ranked coefficients, subtracted the two sets, and grouped changes by functional system.
- D. Participation coefficients of structural versus functional partitions: Figure S5 presents ranked functional-partition coefficients, their SPTL-minus-functional differences, and regional coefficients grouped by functional system.The figure supports comparison across coefficient rankings, differences, and systems.
- D. Participation coefficients of structural versus functional partitions: The study also provides supplementary visualizations of functional systems and detected SPTL, NG, and consensus modules across the DSI network.These figures show system topography, all 31 SPTL consensus modules, and NG modules at γ = 1.0 and γ = 2.1.