Network-Level FC-SC Correlations in Brain Networks

• Network-level FC-SC correlation is the statistical association between functional and structural connectivity, capturing how anatomical links constrain dynamic interactions.
• Quantitative frameworks, including Pearson correlation and random effects modeling, show that main edge effects drive robust network-level coupling with typical values around 0.2–0.33.
• Methodological choices such as matrix construction and edge selection critically affect FC-SC estimates, guiding reliable inferences in brain network science.

Network-level FC-SC correlations describe the statistical association between patterns of functional connectivity (FC; often defined via temporal correlations in neural activity among brain regions) and structural connectivity (SC; typically derived from anatomical or tractography-based measures quantifying physical links) across a network. This concept is central to understanding how the fixed anatomical substrate constrains or enables the dynamic functional interactions observed in neural, biological, and artificial networks. The following sections delineate definition, quantitative frameworks, methodological considerations, sources of variability, modeling strategies, and implications for neuroscience and network science.

1. Conceptual Definition and Quantitative Measures

Network-level FC-SC correlation refers to the statistical relationship, usually assessed by the Pearson correlation coefficient, between the set of edge weights defined by the functional connectivity matrix $\mathbf{FC}$ and those from the structural connectivity matrix $\mathbf{SC}$, considered over all pairs of nodes (edges) in a given network. The typical computation aggregates across all edges (possibly restricted to supra-threshold or nonzero SC values), thus providing a single correlation coefficient per network or subject:

$$\text{Corr}^{(s)} = \mathrm{Corr}(\{FC_{ij}^{(s)}\}_{(i,j)\in\Omega}, \{SC_{ij}^{(s)}\}_{(i,j)\in\Omega})$$

where $(i,j)$ indexes edges, $\Omega$ is the set of considered connections, and $s$ indexes subjects (Peng et al., 4 Aug 2025). This metric provides an overall summary of the degree to which anatomical connectivity predicts functional interactions.

Distinctions are crucial:

• Network-level correlation: Across all edges within a subject, how well does the spatial pattern of FC mirror that of SC?
• Edge-level correlation: For each edge, what is the correlation across subjects between FC and SC at that edge?

The network-level (aggregate) correlation is an indicator of the population-mean function/structure association, whereas edge-level correlations probe variability and association for specific connections.

2. Statistical Modeling and Decomposition of Variability

A major advance in the field is the use of random effects models to dissect the sources of variability driving FC, SC, and their association (Peng et al., 4 Aug 2025). The model for FC (and analogously SC) can be formulated as:

$$FC_{ij}^s = \mu + \alpha_{ij} + \beta^s + \eta_{ij} \cdot \varpi^s + \varepsilon_{ij}^s$$

• $\mu$: global mean;
• $\alpha_{ij}$: main edge effect (across-connection differences in mean connectivity);
• $\beta^s$: main subject effect (subject-wide differences);
• $\eta_{ij} \varpi^s$: interaction capturing edge-by-subject variability;
• $\varepsilon_{ij}^s$: residual.

Analogous terms are defined for SC. This structure allows for the estimation of variance components (how much of the total variability is due to edge, subject, or interaction effects) and the estimation of cross-modality correlations for each component, e.g., $$\rho_{\alpha} = \mathrm{Corr}(\alpha_{ij}^{FC}, \alpha_{ij}^{SC})$$.

Key quantitative results:

• Main edge effects dominate variance in SC ($\sim$70.5%) and are substantial in FC ($\sim$46.3%), explaining why network-level correlations are robust.
• Main subject effects are notable for FC ($\sim$20.2%) but negligible for SC.
• Edge-level FC-SC correlations are weak (mean $\sim$0.01), while network-level correlations can reach 0.2–0.33 (Peng et al., 4 Aug 2025).

3. Methodological Considerations and Interpretation

The assessment of FC-SC correlations involves several methodological nuances:

• Matrix construction: FC is often computed from BOLD-fMRI or electrophysiological time series using Pearson or partial correlation, possibly after preprocessing or denoising. SC is derived from diffusion tractography (e.g., streamline counts, fractional anisotropy) or anatomical datasets.
• Edge selection: Inclusion of all edges, only those above certain thresholds, or only anatomically plausible links, affects the resulting correlations.
• Group- vs. subject-level: Aggregating FC/SC matrices across subjects before correlation calculation yields higher FC-SC coupling than subject-specific calculation (Peng et al., 4 Aug 2025).
• Statistical significance: Permutation tests with FDR correction are standard (Peng et al., 4 Aug 2025).
• Variability decomposition: Without explicit consideration of random effects and variance structure, apparent discrepancies between network- and edge-level correlations can arise.

4. Interpretation and Sources of Discrepancy

The divergence between network-level and edge-level FC-SC correlations reflects different dominant sources of variability:

• Network-level FC-SC correlation is primarily driven by main edge effects, encapsulating population-mean anatomical-functional coupling. These components account for broad correspondence in spatial structure.
• Edge-level FC-SC correlation (across subjects for a given connection) is typically weak. Subject effects and interaction terms, capturing individual variability not aligned between FC and SC, dominate at this level.

Structural networks (SC) show high consistency across subjects (group-to-individual correlations $\sim$0.8–0.9), while FC matrices are more individual-specific and variable. The higher consistency of SC across the population inflates group-level correlations but reduces the power of edge-wise analysis for functional-structural mapping (Peng et al., 4 Aug 2025).

5. Implications for Brain Network Science and Beyond

These findings have several implications:

• Interpretation of Coupling: Moderate network-level FC-SC correlation (0.2–0.3) is robust and reflects consistent structure-function constraints at the population mean, but individualized prediction or interpretation at single edges is unreliable owing to inherent variability.
• Population vs. Individual-Level Inference: Group-average inferences about FC-SC coupling are more meaningful and stable than edge- or subject-level claims, which are susceptible to noise and methodological artifacts.
• Framework for Comparative Studies: The random effects decomposition offers a principled approach for comparing different parcellations, preprocessing pipelines, or SC/FC quantification strategies in terms of their effects on observable FC-SC correlations.

6. Mathematical Framework and Formulas

Key formulas:

• Network-level (subject-specific) correlation:

$$\mathrm{Corr}^{(s)} = \frac{\sum_{ij} (FC_{ij}^s - \bar{FC}^s)(SC_{ij}^s - \overline{SC}^s)}{\sqrt{\sum_{ij}(FC_{ij}^s - \bar{FC}^s)^2} \sqrt{\sum_{ij}(SC_{ij}^s - \overline{SC}^s)^2}}$$

• Group-average network-level correlation:

$\mathrm{Corr}(\bar{FC}_{ij}, \bar{SC}_{ij})$

where averages are across subjects.

• Edge-level correlation (across subjects for $(i,j)$):

$\mathrm{Corr}(\{FC_{ij}^s\}_s, \{SC_{ij}^s\}_s)$

• Variance decomposition and random effects correlation: See model equations above and estimation procedures as in (Peng et al., 4 Aug 2025).

7. Summary Table: Sources of Variability and Correlation Types

Level	Main Source of Variability	Typical Correlation	Implication
Network-level	Main edge effects	Moderate (0.2–0.33)	Structure-function coupling
Edge-level	Subject/edge interactions	Weak (∼0.01)	Poor single-edge predictivity
Group-average	Population mean (edges)	Highest (∼0.33)	Most robust summary

Network-level FC-SC correlations are robust aggregate measures particularly sensitive to global spatial patterns grounded in structural connectivity, but edge-level correlations are weak due to the heterogeneous and individual-specific nature of functional dynamics. Random effects modeling enables precise parsing of these regimes, providing a comprehensive statistical toolkit for interpreting function-structure coupling in complex networks (Peng et al., 4 Aug 2025).