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# Graph Slepian Framework for Guided Filtering With Application to Neuroimaging

Sébastien Dam, Julie Coloigner, Dimitri van De Ville

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## To cite this version:

Sébastien Dam, Julie Coloigner, Dimitri van De Ville. Graph Slepian Framework for Guided Filtering With Application to Neuroimaging. IEEE Transactions on Biomedical Engineering, 2025, pp.1-10. 10.1109/TBME.2025.3628058. hal-05353573

HAL Id: hal-05353573

https://inria.hal.science/hal-05353573v1

Submitted on 7 Nov 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Graph Slepian Framework for Guided Filtering with Application to Neuroimaging

Sébastien Dam, Julie Coloigner, and Dimitri Van De Ville, Fellow, IEEE

**Abstract**— **Objective:** Graph signal processing (GSP) has enabled new approaches for jointly analyzing graphs and graph signals. Various classical operations, such as the Fourier transform and filtering, have been extended to this setting, along with more advanced constructs including Slepian functions. The latter provides a basis for bandlimited graph signals that are maximally concentrated in a given subgraph. Here, we propose a novel approach that introduces complex values to encode several subgraphs, enabling a richer analysis of how graph signals are expressed. **Methods:** The motivating application from neuroscience is to jointly analyze brain graphs obtained from diffusion-weighted magnetic resonance imaging (MRI), with brain graph signals from functional MRI. The brain activity measured by the latter is constrained by the underlying brain graph. Complex-valued graph Slepians constructed with prior knowledge from well-known task-positive and -negative functional networks can then reflect how activity is dynamically reorganizing. **Results:** The feasibility of the approach is demonstrated using synthetic data first, and then applied to data from the Human Connectome Project, revealing patterns of brain network interactions. Results are currently limited to two subgraphs, but future work will explore more extensive graph configurations. **Conclusion:** Slepian functions offer new ways to decode graph signals lying on top of a graph structure. **Significance:** This confirms that the proposed method provides a new representation for studying brain activity constrained by the brain's structural connectivity.

**Index Terms**— Graph signal processing, network neuroscience, magnetic resonance imaging, brain connectivity, Slepian function.

## I. INTRODUCTION

NETWORK science provides a highly refined perspective to accommodate complex systems, capturing organizational principles that emerge from graph representations. Conventional approaches are rooted in graph theory, using metrics such as local clustering coefficient, efficiency, and centrality measures. More recently, graph signal processing (GSP) has emerged, acknowledging the existence of graph signals on top of the graph. Classical signal processing operations including

the Fourier transform or filtering can therefore be revisited for graph domains.

Modern techniques such as magnetic resonance imaging (MRI) have enabled the study of large-scale brain networks. Diffusion MRI allows for probing the white-matter organization in terms of large fiber bundles and summarizing this information into a structural connectome; i.e., a graph where nodes are brain regions and edge weights reflect structural connectivity (SC) estimated by tractography [1], [2]. Functional MRI (fMRI) measures indirectly brain activity through the blood-oxygenation-level-dependent (BOLD) signals. Together, such multimodal data provide a comprehensive view of the brain's wiring diagram (graph) along with activity traces (graph signals).

The GSP framework has been tailored for neuroimaging data [3], [4], in particular by leveraging the spectral domain for defining and implementing graph filters. These have turned out to be useful for quantifying brain structure-function coupling [5], performing brain fingerprinting and task decoding [6], classifying brain diseases [7]–[9], denoising of brain signals [10], and upscaling to voxel-wise brain graphs [11], [12]. In general, there is a growing interest in understanding how the underlying brain anatomy supports spatially and temporally varying patterns of functional activity [13], [14].

Recent advances in GSP have proposed approaches to focus on a specific set of nodes without cutting out the subgraph. Slepian functions maximize energy concentration within a pre-defined interval via a selection matrix, while being bandlimited [15], [16]. The extension to graphs has led to graph Slepians [17], [18], offering a way to reconcile spectral bandwidth with the choice of a subgraph. This approach has been used for analyzing the C. Elegans wiring diagram [19], [20] and fMRI [21]. One recent extension is to incorporate complex-valued entries in the Slepian selection matrix, thereby enabling the specification of multiple intervals of interest [22].

In this work, we leverage and extend the complex graph Slepian framework to propose a spectral decomposition approach of BOLD signals on localized brain structures, providing a novel way to analyze the dynamics of brain networks. We focus on two intervals of interest to extract the optimally concentrated spectral components of the signals. We then propose a novel filtering scheme that enables us to selectively examine the contributions of filtered signals in specific subgraphs, thus allowing to decipher patterns of brain network interactions. Our approach delves into the interplay between signal processing and network science, and is useful

Sébastien Dam and Julie Coloigner are with Univ Rennes, Inria, CNRS, Inserm, IRISA UMR 6074, Empenn ERL U 1228, F-35000 Rennes, France (e-mail: sebastien.dam@inria.fr; julie.coloigner@irisa.fr).

Dimitri Van De Ville is with the Neuro-X Institute EPFL, 1202 Geneva, Switzerland, and also with the Department of Radiology and Medical Informatics, University of Geneva, Switzerland (e-mail: dimitri.vandeville@epfl.ch).
for studying the brain network organization that underlies cognition, behavior, and disease.

To validate our framework, we first conduct simulations to assess its accuracy and robustness with a setting containing ground truths. We then explore our approach to real-world neuroimaging data from the Boston Adolescent Neuroimaging of Depression and Anxiety (BANDA) dataset, focusing on depression as a case study [23]. Depression has been associated with alterations in large-scale brain networks, including disruptions in the task-positive network (TPN), task-negative network (TNN), and the interactions between them [24]–[26]. By jointly analyzing graph signals with structural and functional information, we aim to uncover novel patterns of connectivity disruptions in depression, shedding light on the node-level and network-level mechanisms underlying affective and cognitive dysfunction.

## II. GRAPH SLEPIANS

### A. Graph Setting

Let $\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathbf{A})$ be an undirected graph with $N$ nodes, where $\mathcal{V}$ is the set of nodes and $\mathcal{E}$ is the set of undirected edges in $\mathcal{G}$. The matrix $\mathbf{A}$ is a symmetric matrix representing the weighted adjacency matrix where each entry is the connection strength between two nodes. One can define the graph normalized Laplacian matrix as

$$L = I - D^{-1/2}AD^{-1/2}, \quad (1)$$

where $D$ is the diagonal matrix of node degrees, i.e., $d_{ii} = \sum_j a_{ij}$. Since $L$ is real, symmetric, and positive semi-definite, it can be diagonalized via its eigendecomposition as $L = U\Lambda U^\top$, where the columns of $U = [u_1, u_2, \dots, u_N]$ are the orthonormal eigenvectors, and $\Lambda$ is the diagonal matrix that stores the corresponding non-negative eigenvalues $\Lambda = \text{diag}(\lambda_1, \lambda_2, \dots, \lambda_N)$. The $k^{\text{th}}$ eigenvalue $\lambda_k$ reflects a notion of frequency of the corresponding eigenvector $u_k$, which can be seen as a structural harmonic.

We consider a graph signal $x \in \mathbb{R}^N$, where $x_n$ denotes the signal value at node $n$. The signal $x$ resides on the nodes of the graph $\mathcal{G}$, thus linking the graph structure to the signal. The latter can then be projected onto the structural harmonics through the graph Fourier transform to perform a frequency analysis of the signal, as

$$\hat{x} = U^\top x,$$

and conversely be perfectly recovered through its inverse form, as

$$x = U\hat{x}.$$

### B. Conventional Graph Slepians

Slepian functions were originally proposed in the seminal works of Slepian et al. [15], [16], which tackled the fundamental problem of optimally concentrating a signal jointly in temporal and spectral domains. In graph Slepians, the goal is to find a bandlimited graph signal with maximal energy concentration in a subset of nodes $S$ [17], [18], i.e., maximizing

$$\frac{\sum_{k \in S} [g(k)]^2}{\sum_{k=1}^{N} [g(k)]^2}, \quad (2)$$

for any bandlimited graph signal $g$.

Let $M \in \mathbb{C}^{N \times N}$ be the diagonal selection matrix encoding the presence or absence of a node in $S$, that is, $M_{k,k} = \delta_{k \in S}$, $k = 1, \dots, N$, where $\delta$ is the Kronecker delta. A truncated version of the Fourier basis $U_K \in \mathbb{C}^{N \times K}$ is defined by restricting the spectrum to the first $K < N$ eigenvectors with smallest eigenvalues $\lambda$, such that any bandlimited graph signal can be written as $g = U_K \hat{g}$. Then, the problem (2) reverts to optimizing the Rayleigh quotient

$$\mu = \frac{\hat{g}^\top C \hat{g}}{\hat{g}^\top \hat{g}}, \quad (3)$$

where $C = U_K^\top M U_K$ is the concentration matrix and $\top$ denotes the transpose operator. This problem can be solved by the eigendecomposition of $C$ as $C\hat{s}_k = \mu_k \hat{s}_k$, $k = 1, \dots, K$, where $\mu_k$ is the energy concentration of the Slepian vector $s_k = U_K \hat{s}_k$ in $S$. We assume that they are ordered in ascending eigenvalue amplitude ($\mu_0 \le \mu_1 \le \dots \le \mu_K$), so that $\hat{s}_K$ is the optimal (maximally) concentrated solution. Notice that the Slepian vectors are orthonormal over the entire graph as well as orthogonal over the subset $S$; i.e., we have $s_k^\top s_l = \delta_{k-l}$ as well as $s_k^\top M s_l = \mu_k \delta_{k-l}$. Furthermore, the eigenvalues of $C$ range between 0 for Slepian vectors having all their energy outside of $S$ and 1 for those being fully concentrated in $S$.

### C. Graph Slepians for Multiple Intervals

Despite being well-established, the extension of Slepian functions to handle more than one interval of interest on conventional domains has only recently been proposed [27] by allowing positive (1), zero, or negative (-1) weights associated with different intervals, leading to augmented Slepians. Then, Liégeois et al. [22] took this to the graph setting and further allowed the selection matrix to contain complex-valued entries, thereby enabling to encode multiple subgraphs. Building on the diagonal selection matrix previously defined, a set of Slepian vectors $s_k$, $k = 1, \dots, K$ are then obtained for each subgraph.

Specifically, for $P$ subgraphs of interest $S^1, \dots, S^P$, the diagonal entries of the selection matrix are defined as

$$M_{k,k} = \rho_i e^{j\phi_i}, \text{ for } k \in S^i,$$

which assigns a complex weight to nodes of the subgraph $S^i$. The optimization problem then writes

$$\frac{\sum_{i=1}^{P} \left( \rho_i e^{j\phi_i} \sum_{k \in S} g(k) \bar{g}(k) \right)}{\sum_{k=1}^{N} g(k) \bar{g}(k)}, \quad (4)$$

Similar to the conventional case, the eigendecomposition of the concentration matrix $C$ results in Slepian vectors and eigenvalues as in Eq. (3). This time, the Slepian eigenvalues are distributed in the complex domain, where the Slepian vectors are more concentrated in a subgraph $S^i$ for eigenvalues close to $\rho_i e^{j\phi_i}$. Therefore, for a given Slepian vector $s_k$, we
will rely upon the phase of its corresponding eigenvalue $\mu_k$; i.e., $\angle(\mu_k) \in [\phi_i - \delta, \phi_i + \delta]$, where $\delta$ is a single phase tolerance parameter to be defined, then $s_k$ is associated with $S^i$. For instance, we can consider the simple case where the weights of the selection matrix are the roots of unity, i.e., $\rho_i = 1$ and $\phi_i = \frac{2(i-1)\pi}{P}$, $i = 1, \dots, P$. In this case, any Slepian eigenvalues $\mu_k$ having a phase in $[\phi_i - \delta, \phi_i + \delta]$ will have their corresponding Slepian vector associated with the interval $S^i$.

### III. METHODOLOGY

#### A. Problem Statement

Let $X \in \mathbb{R}^{N \times T}$ be a matrix of time-dependent graph signals on the graph $G$ with $N$ nodes. The goal is to extend the complex Slepian framework to propose a filtering scheme of graph signals $X$ based on the complex Slepian basis. In particular, we go for two subgraphs $S^1$ and $S^2$ encoded by real and imaginary selection entries, respectively:

$$M_{k,j} = \begin{cases} 1, & k \in S^1, \\ j, & k \in S^2. \end{cases} \quad (5)$$

We propose a block-stochastic graph [28] to illustrate this construction. Specifically, as shown in Fig. 1a, we choose $N = 90$ nodes that are divided in 5 communities $C^i$, $i = 1, \dots, 5$. The nodes were assigned as follows: $C^1 = [1, 20], C^2 = [21, 40], C^3 = [41, 50], C^4 = [51, 70], and C^5 = [71, 90]$. We then selected the two subgraphs of interest using intersections of communities: $S^1 = \bigcap_{i=1}^3 C^i$ and $S^2 = \bigcap_{i=3}^5 C^i$, so that $C^3$ belongs to both subgraphs. A bandwidth $K = 10$ is chosen as well as a phase tolerance $\delta = 0.3$. The distribution of the complex eigenvalues obtained from the concentration matrix are shown in Fig. 1b. All graph Slepians are highly concentrated at 1 and $j$, except two that are transitioning between the two subgraphs of interest.

#### B. Filtering Based on Graph Slepians

The first step is to represent a graph signal $x_t$ in the spectral domain spanned by the complex Slepian vectors. Let $S = [s_1|s_2|\dots|s_K] \in \mathbb{C}^{N \times K}$ be the matrix containing the complex Slepian vectors. We define the complex Slepian transform (CST) as the projection of $x_t$ onto the complex Slepians:

$$\hat{x}_t = S^H x_t = [\langle s_i, x_t \rangle]_{i=1,\dots,K} \quad (6)$$

where $S^H$ is the Hermitian transpose. Notice that the vector norm is not preserved through this transform, despite the columns of $S$ being orthonormal, because there are less Slepian vectors than dimensions ($K < N$). Therefore, the bandwidth $K$ acts on the amount of information that is retained from the signal. Notwithstanding the dimensionality reduction, the CST enables the signal to be analyzed in terms of its localization. As such, the Slepian coefficients can be interpreted according to the concentration values $\mu_k$. For our particular choice here, the real part $\Re(\mu_k)$ and the imaginary part $\Im(\mu_k)$ represent the energy concentration in $S^1$ and $S^2$, respectively.

Given the dimensionality reduction, the reconstruction $\tilde{x}_t = S\hat{x}_t$ will not be perfect. However, we can benefit from the

<figure><img src="image_4.png" /><figcaption>Fig. 1: Example of complex Slepian framework. (a) A block-stochastic graph with $N = 90$ nodes divided in 5 communities. Connections between nodes within the same community are created with probability 0.5. Inter-community connection probabilities are defined as follows: $p_{12} = p_{3k} = p_{45} = 0.4$, for $k \in \{1, 2, 4, 5\}$, and $p_{14} = p_{15} = p_{24} = p_{25} = 0.02$, where $p_{ik}$ denotes the connection probability between nodes in communities $C^i$ and $C^k$. (b) The eigenvalues of the concentration matrix color-coded according to their phases: green for $S^1$ and red for $S^2$. Unlabeled eigenvalues are colored in gray.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2: Ground truth temporal signals from synthetic data in $S^1$ (green) and in $S^2$ (red).</figcaption></figure>

CST representation to specify a filter in that domain that will turn over the complex values of the CST coefficients encoding their localization, to the complex values in the original domain. In particular, we propose the spectral filter weights for all coefficients $k = 1, \dots, K$:

$$W(k) = \begin{cases} 1, & \text{if } \Re(\mu_k) > \alpha \text{ and } \Im(\mu_k) < \alpha, \\ j, & \text{if } \Re(\mu_k) < \alpha \text{ and } \Im(\mu_k) > \alpha, \\ 1 + j, & \text{if } \Re(\mu_k) > \alpha \text{ and } \Im(\mu_k) > \alpha, \\ 0, & \text{otherwise,} \end{cases} \quad (7)$$

where $\alpha$ is the cut-off concentration.

The filtered version of the graph signal $x_t$ can then be
obtained as

$$y_t = SWS^H x_t, \quad (8)$$

where $W$ is a diagonal matrix with its entries based on the spectral window $W$. Eq. (8) is reminiscent of the graph Fourier filtering theorem that reduces graph filtering to coefficient-wise multiplication in the spectral domain [29]. Here, we first take the CST of the input signal ($S^H x_t$), then multiply it with the diagonal filtering matrix $W$, and finally apply $S$ to transform the output back to the original domain. The filtered signal is thus computed such that it is bandlimited to a specific spectral range, maximizing energy concentration within the intervals and backprojecting that information to the real and imaginary parts of the reconstructed signal, respectively. Therefore, we can also consider that the complete operation can be taken apart as

$$y_t^R = \Re(SWS^H)x_t, \quad (9)$$

$$y_t^I = \Im(SWS^H)x_t. \quad (10)$$

Finally, we introduce the network interactions balance (NIB) as the difference between the norms of these two signal contributions:

$$n_t = \|y_t^R\| - \|y_t^I\|. \quad (11)$$

### C. Validation With Synthetic Data

To evaluate the validity of the filtering design, we simulate graph data with a known ground truth. In particular, we consider the same block-stochastic graph that was previously described in Sect. III-A and illustrated in Fig. 1a. Additionally, we generate the ground truth signals $t_1$ and $t_2 \in \mathbb{R}^T$ of $T = 406$ time points for the two subgraphs $S^1$ and $S^2$, as depicted in Fig. 2. Inspired by the fMRI setting, the signals are created in the low-frequency ranges of 0.03-0.05 Hz for $S^1$ and 0.01-0.03 Hz for $S^2$, with added Gaussian noise. These signals are then convolved with a canonical hæmodynamic response function modeled as a normalized double-exponential decay function [30], scaled to represent a maximum signal change of 6%. Subsequently, the ground truth signals are expanded into corrupted signals of dimensionality $N \times T$ with spatial variability. Specifically, amplitude scaling is applied to individual regions, introducing regional heterogeneity within the neighboring regions. A block structure is enforced, ensuring strong correlations within $S^1$ or $S^2$, and anti-correlations between signals of the two subgraphs. Spatial smoothing was applied across regions using a Gaussian kernel. Finally, the data is corrupted with additive white Gaussian noise with standard deviation $\sigma = 1/\text{SNR}$ for $\text{SNR} = 30$.

### D. Experiments Using fMRI Data

To extend our work to the fMRI setting, we use the depressed cohort of the BANDA dataset from the Human Connectome Project [23]. We consider 59 depressed adolescents from the cohort and collect brain imaging data of T1-weighted MRI, rs-fMRI and DWI for each of them. Table I details the acquisition parameters for each modality.

<table><caption>TABLE I: BANDA acquisition parameters</caption><thead><tr><th>Parameter</th><th>T1-weighted</th><th>Diffusion MRI</th><th>Functional MRI</th></tr></thead><tbody><tr><td>TE (ms)</td><td>2.18</td><td>89.2</td><td>37</td></tr><tr><td>TR (ms)</td><td>2400</td><td>3230</td><td>800</td></tr><tr><td>FoV (mm)</td><td>256 × 240<br/>× 167</td><td>210 × 210<br/>× 138</td><td>208 × 208<br/>× 144</td></tr><tr><td>Volumes #</td><td>1</td><td>198</td><td>420</td></tr><tr><td>b-values (s/mm²)</td><td>—</td><td>1500/3000</td><td>—</td></tr><tr><td>Diffusion directions #</td><td>—</td><td>183</td><td>—</td></tr></tbody></table>

We preprocess the data using QSIPrep [31] and fMRIPrep [32]. The anatomical preprocessing workflows contain intensity non-uniformity correction of the T1-weighted image, skull-stripping, and brain tissue segmentation of cerebrospinal fluid, white-matter and gray-matter.

The DWI images are denoised using MP-PCA [33], then corrected for Gibbs unringing using local subvoxel-shifts, B1 field inhomogeneity using ANT's N4 algorithm, head motion and Eddy current using FSL's eddy [34]. Finally, the whole-brain cortical and subcortical Tian-Schaefer atlas [35], [36] with 416 regions, including 400 cortical regions and 16 subcortical regions, is adopted to construct SC matrices for each subject. Whole-brain probabilistic tractography including 10 million fibers is performed, using a spherical deconvolution approach and the Spherical-deconvolution Informed Filtering of Tractograms (SIFT2) [37].

For fMRI images, head-motion parameters (transformation matrices and six corresponding rotation and translation parameters) are estimated using FSL's mcflirt. A fieldmap is estimated based on two echo-planar imaging references with opposing phase-encoding directions, with FSL's topup. BOLD signals are also slice-time corrected using 3dTshift from AFNI. The same brain atlas is adopted and regional mean signals are estimated by averaging the BOLD signals over all voxels within each ROI.

### E. Experimental Framework

In the SC graphs, we partition the ROIs into two networks to explore their relevance for understanding depression. The TPN includes regions associated with cognitive control, attention, and emotion regulation, encompassing the dorsal attention network, the salience ventral attention network, and the control executive network, as well as subcortical structures such as the amygdala, globus pallidus, caudate, nucleus accumbens, putamen and thalamus [24], [38]. These regions are critical for goal-directed behavior, reward processing, and the regulation of emotional responses, all of which are commonly disrupted in depression [39], [40]. Conversely, the TNN consists of regions within the default mode network (DMN) and the hippocampus, which are involved in self-referential thought, introspection, and memory processes. Hyperactivity in the TNN regions has been strongly linked to rumination and maladaptive self-focused thinking in depression [26], [41], while the hippocampus is frequently affected in its structural integrity and functional capacity [42], [43]. By analyzing the
<figure><img src="image_6.png" /><figcaption>Fig. 3: Visualization of the real and imaginary parts of the complex Slepian eigenvalues. (a) Synthetic graph. (b) Structural connectivity graphs derived from diffusion MRI. For the real data, the solid lines indicate the mean across subjects, while the shaded regions indicate the standard error mean (SEM).</figcaption></figure>

organization and interaction of these networks within the SC graphs, we aim to uncover the neural mechanisms that underpin the cognitive and emotional dysregulation characteristic of depression [25], [44]. This approach not only provides insights into the pathophysiology of the disorder, but also can identify potential therapeutic targets based on network (in)balance.

For the synthetic and real fMRI data, we use a bandwidth of $K = 10$ and $K = 132$, accounting for 11.1% and 31.7% of the full bandwidths, respectively. The complex Slepian eigenvalues of the synthetic graph and the mean of the SC graphs across the BANDA subjects are shown in Figs. 3a and 3b, respectively. A clear separation of the concentrations of the real and imaginary parts between the two intervals can be seen for the synthetic graph, with an intersection between the harmonic #3 and #4, which is coherent with the nodes assignment as the cut-off between the two intervals was set to 41. In other words, for a full bandwidth, the energy concentration of the real part of the complex Slepian eigenvalues would be maximal for the nodes in [1, 40], minimal for the nodes in [41, 90], and the opposite would be observed for the energy concentration of the imaginary part. The computational complexity is based on the eigendecomposition of the graph normalized Laplacian, that is $O(N^3)$. Since each graph is decomposed for each subject, we have a runtime efficiency depending on the number of subjects. The synthetic data, subjects' identifiers for real-world data, and Python code for all analyses are provided (https://github.com/SebastienDam01/Complex_graph_Slepians), ensuring the reproducibility of our results.

## IV. RESULTS AND DISCUSSION

### A. Impact of the Filtering Step

The real and imaginary parts of the filtered signals from Eqs. (9) and (10) are illustrated in Fig. 4 for three nodes. For the node belonging to $S^1$, the real part of the filtered signal reflects well the ground truth signal $t_1$ (see Fig. 2), while the amplitude of the imaginary part is close to zero. For the node belonging to $S^2$, this behavior is exhibited in the opposite way. Then, for the node that belongs to both subgraphs, the real and imaginary parts of the filtered signal show $t_1$ and $t_2$, respectively. Their amplitudes are approximately half as high as those of the ground truth signals. In fact, it represents the transition point where these respective amplitudes are approximately equal.

These results suggest that the encoding of real and imaginary values in the selection matrix, as defined in Eq. (5), leads to a separation of the contributions of the real and imaginary parts of the filtered signals of nodes in either $S^1$ or $S^2$, or both. If we had defined +1 and -1 values in the selection matrix and for the spectral filter weights in the diagonal filtering matrix $W$ instead, we would have focused only on the real parts of the filtered signals, since these would be real too. On the complex unit circle, this would mean that the Slepian eigenvalues are distributed along the horizontal axis, with phases being either 0 or $\pi$. Therefore, with our representation, we can obtain an orthogonal representation of the eigenvalues, directly enabling the identification of contributions from both subgraphs independently.

### B. Dichotomy of Task-Positive and Task-Negative Networks

The same analysis is performed with the real fMRI data by focusing on a single ROI belonging to each of the two networks: the lateral prefrontal cortex (PFCI) and the posterior cingulate cortex (PCC), which are part of the control executive network in the TPN, and the DMN in the TNN, respectively. We also calculate the Pearson correlations between the filtered signals and the raw fMRI signal with incremental bandwidth, among all subjects for these ROIs. We then compute the mean value of the correlations to study how well the raw fMRI signal is reconstructed across bandwidths. As can be seen in Figs. 5b and 5d, the mean Pearson correlation of the real part of the filtered signal in the PFCI almost reaches 1 as the bandwidth increases. In contrast, the imaginary part fluctuates around zero, showing no correlations with the raw fMRI signal. The opposite phenomenon is observed for the filtered signal in the PCC. In Figs. 5a and 5c, the filtered signals $y_t^R$ and $y_t^I$ show accurate reconstruction patterns of a single subject's raw fMRI signal as expected. In particular, since the PFCI is part of the TPN, $y_t^R$ reconstructs well its raw signal, while the amplitude of $y_t^I$ is almost null. We note that the amplitude of $y_t^I$ is close to the raw fMRI signal, showing only slight variations due to the bandwidth.
<figure><img src="image_7.png" /><figcaption>Fig. 4: Filtering scheme on synthetic data: the graph is represented by its nodes given their belonging to subgraphs $S^1$ or $S^2$, or both, and the dash-dot lines correspond to the filtered signals of the ground truth signals (see Fig. 2) for three nodes. The latest belong to $S^1$, both, and $S^2$, as depicted by the colored arrows.</figcaption></figure>

### C. Beyond the Dichotomy: Patterns of Network Interactions

To evaluate the reconstruction per node (or ROI), we introduce the NIB, computed as the difference of norms across time between the real and imaginary parts of the filtered signal, as in Eqs. (9) and (10). Fig. 6 shows the norms of both parts of the filtered signal. These highlight, on the one hand, the reconstruction accuracy relative to the norm of the raw fMRI signal. On the other hand, we can observe how the norms of $y_t^R$ and $y_t^I$ are intertwined between the nodes #41 and #50, which are the ones that belong to both intervals. This demonstrates the subtlety of our framework to reveal the interactions occurring between $S^1$ and $S^2$ when a signal is filtered: the interactions between the two intervals grow stronger as the nodes are known to be closely related themselves. This is clearly revealed in the amplitudes of

the NIB, where the positive values indicate high localization and contribution of $y_t^R$ to reconstruct the signal in $S^1$, and conversely the negative values indicate the same for $S^2$.

### D. Network Interactions Balance Reveals Intermediate Transition Networks

The NIB is applied to the depressed cohort of the BANDA dataset. As shown in Fig. 7, this reveals a pattern consistent with the observations on the synthetic data; i.e., the NIB is high with positive values in ROIs of the TPN, and low with negative values in ROIs of the TNN. The localization map also highlights the ROIs associated with the NIB amplitudes, illustrating coherent patterns aligned with the structure of the TPN and TNN. Interestingly, certain ROIs within the TPN exhibit negative NIB values, including the left temporal occipital (ROI #111), the left orbital frontal cortex (ROI
<figure><img src="image_8.png" /><figcaption>Fig. 5: Filtering scheme on real fMRI data. Raw fMRI signal (solid line) versus real and imaginary parts of the filtered one for (a) the lateral prefrontal cortex and (c) the posterior cingulate cortex of a single subject. The mean Pearson correlations among all subjects between the filtered signals and the raw fMRI signal across bandwidths are illustrated for the PFCI and the PCC in (b) and (d), respectively.</figcaption></figure>

#149), and the posterior region of the DAN (ROI #289). The opposite phenomenon is observed in the TNN where several ROIs display positive NIB values, including the left temporal region (ROI #164), and two ROIs within the left prefrontal cortex (ROI #200 and ROI #204). This divergence from their respective reference networks suggests interactions between these networks that turn out not to be fully segregated.

## E. Intermediate Transition Networks Are Representative of Dysfunctional Mechanisms Underlying Depression

Divergence of NIB values can be extended to the ROIs that are not assigned to either network, which we refer to “NS” (not selected). To quantify the NS ROIs whose NIB diverges from 0, we compute thresholds as the first (resp. third) quartile of the NIB for the ROIs belonging to the TPN (resp. TNN). This allows us to quantify the number of subjects whose NIB for these ROIs exceeds these thresholds. We then apply a secondary thresholding step, retaining only ROIs where the count of subjects exceeds a predefined criterion ($\beta = 20$). Finally, we categorize the ROIs based on their relationship to the networks, classifying them as part of the TPN, TNN, or within transition groups, that we define as:

* NS-TPN, corresponding to NS ROIs with NIB values close to those of the TPN, suggesting potential interactions with this network;

* NS-neutral, corresponding to NS ROIs with NIB values near zero, indicating no strong alignment with either network;

* NS-TNN, with NIB values close to those of the TNN, suggesting potential interactions with this network.

By identifying the NS ROIs that exhibit alignment with either the TPN or TNN, we can better understand their functional relevance and possible involvement in transitional or interme-diary processes between these networks. As depicted in Fig. 8, the ROIs are ordered by network membership, revealing a continuous spectrum of NIB values ranging from the TPN to the TNN, interspersed with the three transition groups. Among the 164 NS ROIs, 36 ROIs are classified into the NS-TPN group and all belong to the somatomotor network (SMN), suggesting positive interactions with the TPN. Resting-state fMRI analyses have revealed that while the SMN, primarily involved in motor control and sensory processing, did not show intrinsic preference for either TPN or TNN network [24], [45], its network interactions patterns with the TPN here suggest a role in higher-order motor planning and sensorimotor integration. Another study suggests that the TPN exerts a top-down modulatory influence over the SMN, particularly through the frontoparietal network, which plays a key role in cognitive control and flexible behavior [46]. Here, the presence of SMN ROIs in the NS-TPN group may indicate disruptions in the balance between large-scale brain networks, particularly hyper-connectivity in sensorimotor regions and reduced engagement of executive control networks.

In contrast, the NS-TNN group includes 9 ROIs that mainly belong to limbic regions, such as the orbital frontal cortex and the temporal pole, which are known to play a crucial role in affective regulation and self-referential processing [47]–[49]. These regions exhibit strong interactions with the DMN. Altered connectivity between the limbic system and the DMN has been associated with maladaptive rumination, emotional dysregulation, and impaired cognitive control over negative
<figure><img src="image_9.png" /><figcaption>Fig. 6: Network interactions balance between the two intervals $S^1$ and $S^2$ on the synthetic data. The L2-norm of the real and imaginary parts of the filtered signal are plotted against the L2-norm of the raw fMRI signal (top). The difference between these two norms across time are computed to obtain the network interactions balance (bottom).</figcaption></figure>

affect [50], [51]. In depression, hyperconnectivity within the DMN, particularly in medial prefrontal and limbic regions, is considered to contribute to excessive self-focused thought and impaired disengagement from negative emotional stimuli [52]. The presence of limbic areas in NS-TNN suggests that they may serve as transitional hubs mediating dysfunctional communication between the DMN and task-related networks in depressive states.

Finally, the NS-neutral group consists of 119 ROIs, encompassing regions from the visual network as well as some portions of the limbic and somatomotor areas. Unlike the NS-TPN and NS-TNN groups, which show connectivity patterns that link them to specific functional domains, the NS-neutral ROIs appear to be functionally nonspecific or weakly integrated into either network. This may indicate that these regions serve a more modulatory or context-dependent role, potentially engaging in transient interactions depending on task demands or cognitive states. Alternatively, they could correspond to ROIs with lower functional specialization or weaker connectivity profiles, making them less relevant for the major functional roles observed in other brain networks. The presence of these NS-neutral regions highlights the complexity of network organization and the potential for intermediate networks that do not conform specifically to the traditional TPN-TNN dichotomy, but may play a crucial role in network-level dysfunctions observed in mood disorders such as depression.

## F. Limitations And Perspectives

This study comes with a number of limitations. The framework is restricted to two intervals of interest. This choice was motivated by the potential conflictual phase interactions of the Slepian coefficients one may find in a setting with more than two intervals encoded by non-orthogonal angles in the

<figure><img src="image_10.png" /><figcaption>Fig. 7: Network interactions balance between the task-positive and -negative network on the depressed cohort of the BANDA dataset. The solid line indicates the mean across subjects, while the shaded regions indicate the standard error mean (SEM). The ROIs are either subcortical, unlabeled regions, or belong to the task-positive or -negative network, as indicated by the colored crossbars at the top of the plot. The color-coded localization map is also displayed to identify the regions associated with the amplitudes of the NIB.</figcaption></figure>

complex domain. For instance, for three intervals, one could modify the encoding of the selection matrices such that they are the roots of unity. The resulting Slepian coefficients distributions in the complex unit circle would exhibit overlapping regions, leading to ambiguous interpretations of phase separation. This could introduce instability in the filtering process and reduce the effectiveness of the spectral decomposition in capturing locally concentrated eigenmodes. Thus, one might need to operate on the case where Slepian coefficients have a similar phase, potentially by adding conditions on the spectral filter weights matrix. Next, the sample size was relatively small for the depressed cohort, which may limit the generalizability of our findings to the broader population of individuals with depression.

In terms of perspectives, the explorations of fMRI data using new analytical tools opens interesting perspectives in a clinical point of view, allowing future studies to replicate and investigate these findings for diagnosis and prognosis.

## V. CONCLUSION

Our proposed graph filtering scheme provides a novel approach to study graph signals lying on top of a graph structure. By leveraging complex graph Slepians, we enable a localized spectral decomposition that isolates graph signals within two predefined subgraphs, offering a refined way to extract eigenmodes that are highly locally concentrated. Our framework extends graph filtering to identify interactions between network components, revealing how localized signal dynamics emerge from underlying structural constraints. Through simulations, we demonstrated the robustness of our method in capturing
<figure><img src="image_11.png" /><figcaption>**Fig. 8:** Network interactions balance ordered by network belonging: the regions in the task-positive network (TPN, in green), the unassigned regions with values close to those of the TPN (NS-TPN, in light blue), the unassigned regions with values near zero (NS-neutral, in blue), the unassigned regions with values close to those of the task-negative network (NS-TNN, in orange), and the regions in the TNN (in red). The color-coded localization map is also displayed to identify the regions belonging to each of these networks.</figcaption></figure>

localized network phenomena. Applied to real fMRI data, our approach uncovered patterns of network interactions in depression, highlighting connectivity patterns that may be associated with it. In general, our findings emphasize the importance of spectral methods in understanding the organization of brain networks. By incorporating anatomically constrained spectral representations and prior subgraph selection, this work bridges network science and signal processing, advancing the study of brain connectivity. The proposed framework opens new research avenues for investigating the interplay between brain structure and function, particularly in clinical applications where network disruptions play a central role. Future work will explore applications towards these directions, further enhancing our ability to decode complex brain processes in health and disease.

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## ACKNOWLEDGMENT

This research was supported in part by the financial supports of the Britanny Region, as well as Rennes Métropole and GdR IASIS to fund Sébastien Dam’s stay for his mobility at EPFL, Campus Biotech, Switzerland.

Data were provided [in part] by the Boston Adolescent Neuroimaging of Anxiety and Depression (BANDA) Consortium’s Human Connectome Project, supported by 1U01MH108168 (PIs: Susan Whitfield-Gabrieli, John Gabrieli).