Edge-Level FC-SC Correlations

• Edge-level FC-SC correlation is a measure that quantifies the relationship between functional and structural connectivity for individual brain region pairs across subjects.
• Random effects models decompose variance into global, edge, subject, and interaction components, highlighting the limited observable coupling at individual edges.
• Empirical findings show very weak edge-level correlations (~0.011) compared to network-level metrics, emphasizing challenges for reliable subject-specific biomarkers.

Edge-level FC-SC correlations quantify the association between functional connectivity (FC) and structural connectivity (SC) for individual edges (region pairs), typically measured across a population of subjects. Unlike network-level analyses—where the focus is on the aggregate correlation between entire FC and SC matrices within a subject or at the group average—edge-level approaches probe the relationship for each specific connection, revealing granular insight into the functional-structural coupling of the brain. Recent developments have brought statistical, network-theoretic, and dynamical system perspectives to the formalization, measurement, and interpretation of these correlations.

1. Definitions and Statistical Formulation

Edge-level FC-SC correlation refers to the computation, for a fixed pair of brain regions (i, j), of the correlation between FC and SC across subjects. If $C_{ij}^s$ and $\tau C_{ij}^s$ denote the functional and structural connectivity for the edge (i, j) and subject $s$, the correlation is: $$\mathrm{Corr}(C_{ij}^s,\, \tau C_{ij}^s~|~ij) = \frac{\sum_s (C_{ij}^s - \bar{C}_{ij})(\tau C_{ij}^s - \bar{\tau C}_{ij})}{\sqrt{\sum_s (C_{ij}^s - \bar{C}_{ij})^2}\, \sqrt{\sum_s (\tau C_{ij}^s - \bar{\tau C}_{ij})^2}},$$ where $\bar{C}_{ij}$ and $\bar{\tau C}_{ij}$ are the across-subject means for edge $(i,j)$ (Peng et al., 4 Aug 2025).

This approach differs fundamentally from network-level correlation, which is computed across all edges within a matrix (for a single subject or the group average), and it captures how well variability in SC across individuals explains variability in FC for a given connection.

2. Variability Sources: Random Effects Decomposition

A major advance in the analysis of edge-level FC-SC correlation is the establishment of random effects models that decompose the observed variability into systematic sources:

For FC: $$C_{ij}^s = \mu + \alpha_{ij} + \beta^s + (\eta_{ij} \cdot \omega^s) + \epsilon_{ij}^s$$ For SC: $$\tau C_{ij}^s = \tau\mu + \tau\alpha_{ij} + \tau\beta^s + (\tau\eta_{ij} \cdot \tau\omega^s) + \tau\epsilon_{ij}^s$$

Where:

• $\mu$, $\tau\mu$: global means,
• $\alpha_{ij}$, $\tau\alpha_{ij}$: main edge effects (across-region variance),
• $\beta^s$, $\tau\beta^s$: main subject effects (subject variance),
• $\eta_{ij}$, $\omega^s$ (and analogs): interaction effects between edge and subject,
• $\epsilon_{ij}^s$, $\tau\epsilon_{ij}^s$: residual noise.

Only the corresponding components (e.g., $\alpha_{ij}$ and $\tau\alpha_{ij}$) are modeled as potentially correlated across modalities, with these correlations denoted $$\rho_{(\alpha)}, \rho_{(\beta)}, \rho_{(\eta)}, \rho_{(\omega)}$$ (Peng et al., 4 Aug 2025).

The decomposition demonstrates that strong network-level FC-SC correlations are dominated by covariance between edge effects across FC and SC. At the edge level, the subject- and interaction-level variability is minor, fundamentally limiting the observable edge-level FC-SC coupling.

3. Empirical Findings on Edge-Level Correlations

Comprehensive analysis reveals that edge-level FC-SC correlations are typically very weak. Across all empirical edges assessed in a large multi-subject cohort:

• The mean edge-level correlation is $\sim 0.011$ (std $\sim 0.043$).
• Over 96% of edge-wise correlations are statistically insignificant after correction for multiple comparisons (Peng et al., 4 Aug 2025).

These values contrast starkly with moderate network-level correlations observed in the same data (mean subject-level $\sim 0.23$, group-level $\sim 0.33$), emphasizing a clear dichotomy in structure-function coupling depending on the aggregation scale. The difference is attributable to the partitioning of variance: most SC variability is across edges (not subjects), so edgewise cross-subject variability is limited.

4. Interpretation and Implications

The random effects modeling clarifies that network-level FC-SC correlations primarily reflect edge-to-edge variability (i.e., some anatomical connections are consistently stronger and also show stronger functional co-activation across the population). However, when focusing on a single connection across subjects, the structurally meaningful variance is overwhelmed by measurement noise and unmodeled biological variability, resulting in weak edge-level associations.

From an interpretive standpoint, this places strong constraints on the use of edge-level FC-SC correlation for biomarker discovery or subject-specific inference. Compared to network-level analyses (which are more robust due to signal averaging across many edges), per-edge approaches are less reliable given current data properties and neuroimaging variability profiles.

5. Methodological Considerations and Extensions

The random effects framework quantitatively exposes how network construction choices—such as deterministic vs. probabilistic tractography, fractional anisotropy weighting, and parcellation scale—alter the variance components and, thus, impact both edge- and network-level FC-SC correlations (Peng et al., 4 Aug 2025). A plausible implication is that methodological optimization should be sensitive to the analytic scale of interest.

Further, while edge-level analysis in current models excludes indirect structural paths (e.g., polysynaptic connections), future work is likely to focus on communicability or network-theoretic metrics that incorporate multi-step SC influences. This may increase the functional explanatory power of SC at the edge level.

6. Practical and Clinical Significance

Although the low edge-level FC-SC correlation suggests limits on edge-wise structure-function mapping in healthy populations, changes in the variance partitioning (e.g., altered subject or interaction effects) could become informative biomarkers in certain clinical or developmental contexts (Peng et al., 4 Aug 2025). Analysis of how these components vary across disease states or with interventions could yield new translational neuroimaging insights.

For instance, if neurological or psychiatric disorders shift the balance between subject and edge effects, edge-level correlations might become more prominent or variable, supporting novel diagnostic or prognostic applications.

7. Comparative Overview: Edge- vs. Network-Level Coupling

The table below summarizes key differences:

Dimension	Network-Level FC-SC Correlation	Edge-Level FC-SC Correlation
Aggregation	Across all edges, per subject	Across all subjects, per edge
Typical Value	Moderate (0.23–0.33)	Very weak (≈0.011)
Dominant Variance	Edge effects	Subject/interactions
Statistical Power	High	Low (96% insignificant)
Interpretability	Robust, population coupling	Limited, individual links

This contrast highlights why network-level structure-function association is observed consistently, while granular edge-level mapping is generally weak under standard paradigms.

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In summary, edge-level FC-SC correlation analysis provides a fine-grained but statistically weak measure of structure-function coupling in the human brain, fundamentally constrained by limited subject-wise variance at individual connections. Random effects modeling rigorously elucidates these variance sources and quantifies why network-level correlations remain robust while edge-level correlations are predominantly weak and unreliable in current neuroimaging datasets (Peng et al., 4 Aug 2025). This framework informs future directions for both methodological development and clinical application in multimodal connectomics.