Structural coarse-graining enables noise-robust functional connectivity and reveals hidden inter-subject variability

TL;DR

The paper tackles the challenge of estimating reliable functional connectivity from temporally limited neuroimaging data, where $T$ is small relative to network size $N$. It introduces a two-step framework that first applies diffusion-based structural coarse-graining via the Laplacian Renormalization Group (LRG) to enforce $T > N'$, then employs Random Matrix Theory (RMT) spectral filtering against the Marchenko–Pastur bulk to isolate signal from noise, yielding robust FC at mesoscale resolutions. Across synthetic benchmarks and real human connectome data, this approach preserves large-scale structure while uncovering hidden inter-subject variability and revealing distinct functional organization modes that standard pipelines obscure. The method provides a practical route to reliable population-level brain networks under realistic scan durations and offers a principled way to separate noise-driven artifacts from reproducible, subject-specific patterns of variability, with implications for cognition and pathology. Mathematical constraints such as $T > N'$ and spectral boundaries defined by $ ext{MP}$ theory anchor the framework in well-established statistical physics principles.

Abstract

Functional connectivity estimates are highly sensitive to analysis choices and can be dominated by noise when the number of sampled time points is small relative to network dimensionality. This issue is particularly acute in fMRI, where scan resolution is limited. Because scan duration is constrained by practical factors (e.g., motion and fatigue), many datasets remain statistically underpowered for high-dimensional correlation estimation. We introduce a framework that combines diffusion-based structural coarse-graining with spectral noise filtering to recover statistically reliable functional networks from temporally limited data. The method reduces network dimensionality by grouping regions according to diffusion-defined communication. This produces coarse-grained networks with dimensions compatible with available time points, enabling random matrix filtering of noise-dominated modes. We benchmark three common FC pipelines against our approach. We find that raw-signal correlations are strongly influenced by non-stationary fluctuations that can reduce apparent inter-subject variability under limited sampling conditions. In contrast, our pipeline reveals a broader, multimodal landscape of inter-subject variability. These large-scale organization patterns are largely obscured by standard pipelines. Together, these results provide a practical route to reliable functional networks under realistic sampling constraints. This strategy helps separate noise-driven artifacts from reproducible patterns of human brain variability.

Figures (4)

• Figure 1: Spectral noise analysis of random signals. (a) Eigenvalue distribution of the Pearson correlation matrix for $N$ non-interacting random walks (see legend for system size). Black dashed line indicates $P(\lambda)\sim\lambda^{-1.5}$. (b) Eigenvalue distribution for the Pearson correlation matrix of the derivative of the dynamical evolution for $N=10^4$ random walks. The dashed line shows the expected Marchenko-Pastur distribution. Inset: Mean off-diagonal absolute Pearson correlation versus system size for both cases; dashed line shows $|C_{ij}|\sim1/\sqrt{N}$, as expected from the central limit theorem. All curves are averaged over $10^2$ realizations.
• Figure 2: Validation of RMT pipeline on a synthetic network model. (a) Kuramoto order parameter, $R$, as a function of coupling $K$ for an Erdős-Rényi (ER) network and the human connectome (HC). The HC's heterogeneous structure (used in b-d) creates a complex transition, providing a non-trivial benchmark for emergent correlations. (b) Eigenvalue distributions for the HC. When weakly coupled ($K=0.1$, inset), the spectrum matches the Marchenko-Pastur (MP) law (dashed line), indicating a successful identification of pure noise. In the coupled regime ($K=3.6$), "signal" eigenvalues (red) clearly emerge, deviating from the MP noise bound. (c)-(d) Filtered correlation matrices (a common colorbar is used for both matrices). (c) The ER network shows weak structure, while (d) the HC matrix, reconstructed only from the "signal" eigenvalues identified in (b), reveals a strong, non-trivial modular structure. This validates the pipeline's ability to isolate emergent function from noise.
• Figure 3: Multiscale coarse-graining and spectral filtering of empirical brain networks.(A) Sketch of the Laplacian Renormalization Group (LRG). The LRG is used to hierarchically coarse-grain the human structural connectome from $N = 2165$ nodes to reduced networks of $N' = 600$, $300$, and $60$ nodes. (B) For each reduced network, resting-state fMRI time series ($T = 652$ points) are averaged within the corresponding supernodes to generate representative signals. (C) The eigenvalue probability distributions of the resulting correlation matrices are analyzed. The Marchenko–Pastur law (red curve) defines the theoretical boundary for noise, allowing for the identification and removal of noise-driven eigenvalues. (D) Structural connectivity matrices, derived from diffusion MRI, are shown at each coarse-grained resolution. (E) Corresponding coarse-grained functional correlation matrices are shown for each resolution, organized by the same structural partitions as in (D). Collectively, this workflow integrates structural coarse-graining (A, D) with functional data averaging (B, E) and spectral filtering (C) to produce robust, reduced network models where the number of time points ($T$) is proportional to the network size ($N'$), i.e., $T \sim N'$.
• Figure 4: LRG--RMT pipeline reveals hidden functional organizational modes and their neuroanatomical signatures.(A) Distributions of pairwise inter-subject distances computed from LRG-derived networks at an intermediate resolution ($N' \approx 300$). Functional connectivity (FC) estimated from raw BOLD signals (1) yields a narrow, unimodal distribution ($\mu = 107.5$), indicating limited apparent inter-subject variability. Using temporal derivatives (2) shifts the distribution toward higher mean distances but remains weakly structured. In contrast, the LRG--RMT pipeline (3) produces a broad, multimodal distribution ($\mu = 119.1$), revealing previously obscured population structure. (B) Hierarchical clustering of the RMT-filtered FC matrices identifies two dominant functional organizational modes (Mode A and Mode B). While both modes share the same underlying structural modularity (modules M1--M5), they differ in their patterns of inter-modular functional organization. The difference matrix ($A - B$) highlights pronounced inter-modular anticorrelations in Mode B (red boxes), which are attenuated in Mode A. (C) Spatial correspondence between LRG-derived structural modules and canonical resting-state networks (RSNs), demonstrating alignment with established functional systems (e.g., Visual, Default Mode, and Salience networks). (D) Neuroanatomical projection of the five structural modules (M1--M5) identified by LRG. Mode A: $n=96$; Mode B: $n=40$.