High-Resolution Dynamic Functional Connectivity Generation with Graph-Variate Flow Matching

Om Roy; Yashar Moshfeghi; Keith Malcolm Smith

cs.LGSep 29, 2026

High-Resolution Dynamic Functional Connectivity Generation with Graph-Variate Flow Matching

Authors: Om Roy, Yashar Moshfeghi, Keith Malcolm Smith

Organizations: University of Strathclyde

Abstract

High-resolution dynamic functional connectivity (DFC) can reveal rapidly evolving brain-network interactions, but short temporal windows yield noisy, often low-rank covariance estimates. Graph-Variate Dynamic (GVD) connectivity addresses this by modulating fast instantaneous interactions with stable trial-level support. This suppresses spurious fluctuations and emphasizes persistent, informative connections. We show that the Hadamard construction lifts low-rank instantaneous connectivity from the positive-semidefinite to the positive-definite cone, keeping high-resolution trajectories on the SPD manifold without ridge regularisation or post-hoc projection. We introduce GVD-CFM, a class-conditional generative model for high-resolution dynamic connectivity. Each trial is represented as SPD GVD matrices on a product Riemannian manifold, then mapped through a global log-Euclidean diffeomorphism and an invertible temporal DCT basis. A Transformer-based conditional flow models all spectral modes jointly and generates the full trajectory non-autoregressively in Euclidean coordinates while preserving exact correspondence with valid SPD sequences. Retaining the full DCT basis also enables decoding on denser temporal grids without retraining. Across multiple EEG motor-imagery datasets, GVD-CFM delivers the strongest overall results for held-out distributional fidelity, temporal-dynamics preservation, and synthetic-to-real classification. It also remains computationally efficient relative to strong raw-signal and direct GVD-space generative baselines. GVD-CFM therefore provides a practical framework for realistic, temporally coherent, high-resolution brain-network generation with preserved manifold structure and resolution-flexible decoding from a single trained model.

Figures & tables

Figure 1: Overview of GVD-CFM. (a) When a window has fewer samples than channels ( $n<d$ ), the window covariance $J_{b}$ is rank deficient and $\log J_{b}$ is undefined. With a positive-definite support $W$ , $\Delta_{b}=W\odot J_{b}$ is positive definite with $\lambda_{\min}(\Delta_{b})\geq\lambda_{\min}(W)\min_{i}q_{i}>0$ , so the log-Euclidean chart remains valid at native temporal resolution. (b) A trial is a point on $(\mathbb{S}^{d}_{++})^{B}$ . The chart $\Psi$ applies $z_{b}=\operatorname{svec}\log\Delta_{b}$ , featurewise standardization and the orthonormal temporal DCT-II; it is a global diffeomorphism and an isometry for its pullback metric (Remark 1 ). Flow matching along $z_{\tau}=(1-\tau)z_{0}+\tau z_{1}$ is performed in these coordinates by a Transformer over DCT modes, and decoding through $\Psi^{-1}$ returns a trajectory in which every $\widehat{\Delta}_{b}$ lies on $\mathbb{S}^{d}_{++}$ .

Method	Rel. GVD-FID $\downarrow$	Eva $\alpha$ $\uparrow$	Eva $\beta$ $\uparrow$	Eva F1 $\uparrow$	CAS AUC $\uparrow$	CAS F1 $\uparrow$
GVD-CFM	1.022 $\pm$ 0.019	0.677 $\pm$ 0.034	0.610 $\pm$ 0.029	0.613 $\pm$ 0.028	0.790 $\pm$ 0.008	0.725 $\pm$ 0.006
GVD-cVAE	0.815 $\pm$ 0.003	0.017 $\pm$ 0.005	0.004 $\pm$ 0.002	0.005 $\pm$ 0.002	0.727 $\pm$ 0.015	0.675 $\pm$ 0.010
GVD-DDPM	1.753 $\pm$ 0.005	0.089 $\pm$ 0.011	0.039 $\pm$ 0.005	0.053 $\pm$ 0.007	0.599 $\pm$ 0.014	0.560 $\pm$ 0.012
Window-DIFFEO-CFM	1.697 $\pm$ 0.005	0.005 $\pm$ 0.002	0.082 $\pm$ 0.002	0.009 $\pm$ 0.003	0.683 $\pm$ 0.003	0.634 $\pm$ 0.002
cVAE	1.416 $\pm$ 0.016	0.407 $\pm$ 0.036	0.006 $\pm$ 0.002	0.012 $\pm$ 0.003	0.631 $\pm$ 0.006	0.572 $\pm$ 0.017
JET	3.908 $\pm$ 0.274	0.005 $\pm$ 0.002	0.002 $\pm$ 0.002	0.002 $\pm$ 0.002	0.495 $\pm$ 0.031	0.349 $\pm$ 0.014

Table 1: Dataset-balanced generative performance across five EEG datasets, averaged over three generator seeds per dataset. Rel. GVD-FID and EvaGeM are evaluated jointly on trajectory positions and temporal increments. The real-data row reports the corresponding real-train/held-out-real-test reference and is excluded from generator rankings. Entries after

\pm

are the seed standard deviation of the dataset-balanced mean,

\bigl(\sum_{i=1}^{5}\sigma_{i}^{2}\bigr)^{1/2}/5

, computed from the per-dataset seed standard deviations

\sigma_{i}

. Lower is better for Rel. GVD-FID; higher is better otherwise. Best generative values are bold ; second-best generative values are underlined italics .

Method	Temp. corr. $\uparrow$	Lag-ACF $\uparrow$	Energy $\to 1$	Dyn. frac. $\to 1$	Diversity $\to 1$	Coverage $\uparrow$
GVD-CFM	0.910 $\pm$ 0.004	0.997 $\pm$ 0.001	0.953 $\pm$ 0.010	0.987 $\pm$ 0.013	0.987 $\pm$ 0.005	0.226 $\pm$ 0.005
GVD-cVAE	0.774 $\pm$ 0.014	0.995 $\pm$ 0.001	0.260 $\pm$ 0.007	0.289 $\pm$ 0.009	0.396 $\pm$ 0.008	0.128 $\pm$ 0.004
GVD-DDPM	0.059 $\pm$ 0.005	0.327 $\pm$ 0.068	1.688 $\pm$ 0.007	1.543 $\pm$ 0.008	1.367 $\pm$ 0.004	0.038 $\pm$ 0.002
Window-DIFFEO-CFM	0.012 $\pm$ 0.005	0.055 $\pm$ 0.091	1.794 $\pm$ 0.008	1.755 $\pm$ 0.006	1.189 $\pm$ 0.002	0.031 $\pm$ 0.001
cVAE	0.527 $\pm$ 0.016	0.957 $\pm$ 0.006	0.898 $\pm$ 0.006	0.981 $\pm$ 0.014	0.817 $\pm$ 0.008	0.034 $\pm$ 0.001
JET	0.385 $\pm$ 0.019	0.933 $\pm$ 0.009	1.940 $\pm$ 0.118	6.835 $\pm$ 0.338	1.501 $\pm$ 0.079	0.009 $\pm$ 0.004

Table 2: Complementary data-quality diagnostics averaged across five datasets and three seeds. Dynamic and diversity ratios have ideal value 1. Temporal quantities are evaluated against held-out real trajectories after removing each trial’s temporal mean, so the static support does not contribute to them.

\pm

as in Table 1 . Best values are bold ; second-best values are underlined italics . For ratio metrics, ranking is by proximity to 1.

Variant	Rel. GVD-FID $\downarrow$	Eva F1 $\uparrow$	CAS AUC $\uparrow$	CAS F1 $\uparrow$	Temp. corr. $\uparrow$	Lag-ACF $\uparrow$	Energy $\to 1$	Dyn. frac. $\to 1$
No DCT	1.092 $\pm$ 0.029	0.540 $\pm$ 0.028	0.772 $\pm$ 0.007	0.704 $\pm$ 0.007	0.404 $\pm$ 0.007	0.666 $\pm$ 0.069	0.973 $\pm$ 0.016	1.011 $\pm$ 0.017
DCT, spectral only	1.025 $\pm$ 0.006	0.600 $\pm$ 0.019	0.792 $\pm$ 0.002	0.721 $\pm$ 0.004	0.919 $\pm$ 0.002	0.997 $\pm$ 0.001	0.975 $\pm$ 0.011	1.010 $\pm$ 0.010
GVD-CFM (DCT + temporal branch)	1.022 $\pm$ 0.019	0.613 $\pm$ 0.028	0.790 $\pm$ 0.008	0.725 $\pm$ 0.006	0.910 $\pm$ 0.004	0.997 $\pm$ 0.001	0.953 $\pm$ 0.010	0.987 $\pm$ 0.013

Table 3: DCT and temporal-branch ablation. No DCT trains the Transformer on temporal log-svec coordinates; DCT, spectral only is GVD-CFM without the temporal branch; the last row is the full GVD-CFM. All values are dataset-balanced means over five datasets and three generator seeds; per-dataset values are in Tables 16 and 17 , and paired per-dataset tests in Table 18 . Ratios have ideal value 1.

\pm

as in Table 1 . Best values are bold ; second-best values are underlined italics .

Figure 2: Regional connectivity magnitude on BNCI2014_001. Mean absolute GVD edge magnitude between broad scalp regions for held-out real trials and generated trials. The generated data preserve the relative regional organization of the real EEG, including strong within-region Frontal/FC and Parietal/Occipital connectivity, strong Central–Centro-parietal coupling, and weaker long-range Frontal/FC–Parietal/Occipital interactions.

Appendix figures & tables28 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 3 — Figure 3: Velocity network of GVD-CFM. Only the DCT state $z_{\tau}$ is integrated. The spectral Transformer attends over all DCT modes. The temporal branch reads $z_{\tau}^{\mathrm{time}}=C_{B}^{\top}z_{\tau}$ , and its output is returned to DCT alignment by $C_{B}$ before gated fusion. Both are conditioned on $c_{\tau}$ through adaptive LayerNorm.

\mathcal{L}\leftarrow\frac{1}{NBm}\sum_{i=1}^{N}\left\|v_{\theta}(z_{\tau}^{(i)},\tau^{(i)},y^{(i)})-u_{\tau}^{(i)}\right\|_{F}^{2}

Appendix

Algorithm 1 GVD-CFM training

Figure 4: Schur-product rank lifting induced by the long-term support. The instantaneous covariance $J_{b}$ is positive semidefinite but rank-deficient because each temporal window contains fewer samples than EEG channels. The long-term support $W$ is positive definite, and the Hadamard composition $W\odot J_{b}$ preserves positive semidefiniteness while lifting the example to full rank. No ridge regularization is added to $J_{b}$ or $W\odot J_{b}$ .

Figure 5: Empirical and theoretical motivation for the temporal DCT representation. (a) Residual off-diagonal temporal covariance, $\rho_{\mathrm{off}}(K)=\|K-\operatorname{diag}(K)\|_{F}/\|K\|_{F}$ , for standardized log-GVD trajectories from BNCI2014_001. The full orthonormal DCT-II reduces residual temporal covariance coupling from $0.778$ to $0.068$ , corresponding to a $91.3\%$ reduction, while the empirical Karhunen–Loève transform (KLT) yields numerical zero. (b) Cross-mode coupling ratio of the population-optimal Gaussian conditional flow-matching field as a function of flow time $\tau$ . The exact KLT removes second-order cross-mode coupling, while DCT-II substantially reduces it relative to the original temporal coordinates; at $\tau=0.5$ , the DCT coupling ratio is $62.8\%$ lower. Together, these results show that DCT-II acts as an invertible, data-independent approximation to the temporal KLT that reduces the cross-mode dependency structure presented to the trajectory velocity model.

Basis	Rel. GVD-FID $\downarrow$	Eva F1 $\uparrow$	CAS AUC $\uparrow$	CAS F1 $\uparrow$	Temp. corr. $\uparrow$	Lag-ACF $\uparrow$	Energy $\to 1$	Dyn. frac. $\to 1$
No DCT	1.092	0.540	0.772	0.704	0.404	0.666	0.973	1.011
Random orthogonal	1.093	0.480	0.788	0.713	0.356	0.678	1.017	1.047
KLT	1.033	0.600	0.795	0.729	0.951	0.998	1.004	1.014
DCT, spectral only	1.025	0.600	0.792	0.721	0.919	0.997	0.975	1.010
GVD-CFM (DCT + temporal branch)	1.022	0.613	0.790	0.725	0.910	0.997	0.953	0.987

Appendix

Table 4: Temporal-basis ablation. Results are dataset-balanced means over five datasets and three generator seeds. All variants retain all

B=100

temporal coordinates. Random orthogonal uses a fixed random orthogonal basis and KLT the training-set empirical temporal Karhunen–Loève basis. The first four rows use the spectral-only velocity network and therefore isolate the basis; the last row is the full GVD-CFM. Ratios have ideal value 1. Best values are bold ; second-best values are underlined italics .

$M$	Rel. Fréchet	Temp. corr.	Lag-ACF	Dyn. energy	SPD	Gen $\rightarrow$ Real AUC	Gen $\rightarrow$ Real F1	Real $\rightarrow$ Real AUC	Real $\rightarrow$ Real F1
25	1.098	0.944	0.995	0.978	1.000	0.803	0.739	0.838	0.764
50	0.983	0.853	0.960	0.595	1.000	0.804	0.739	0.838	0.761
100	0.919	0.860	0.960	0.350	1.000	0.802	0.734	0.831	0.756
200	0.880	0.780	0.959	0.199	1.000	0.794	0.715	0.832	0.760
400	0.872	0.506	0.822	0.105	1.000	0.785	0.647	0.810	0.732

Appendix

Table 5: Dataset-balanced continuous-resolution results on BNCI2014_001, BNCI2014_002, BNCI2015_001, Shin2017A, and Zhou2016. A single spectral-only GVD-CFM is trained at

B_{\mathrm{train}}=25

for each dataset and decoded at the indicated target resolution

M

without retraining.

Dataset	$M$	Rel. Fréchet	Temp. corr.	Lag-ACF	Dyn. energy	Gen $\rightarrow$ Real AUC	Gen $\rightarrow$ Real F1	Real $\rightarrow$ Real AUC	Real $\rightarrow$ Real F1
BNCI2014-001	25	1.163	0.977	0.995	1.041	0.814	0.724	0.863	0.775
	50	1.021	0.875	0.977	0.622	0.824	0.748	0.866	0.784
	100	0.979	0.890	0.958	0.360	0.816	0.727	0.857	0.783
	200	0.958	0.778	0.959	0.195	0.806	0.704	0.821	0.739
	400	0.943	0.440	0.558	0.103	0.778	0.497	0.797	0.710
BNCI2014-002	25	1.008	0.911	0.999	0.964	0.791	0.746	0.797	0.721

Appendix

Table 6: Per-dataset continuous-resolution results for a single spectral-only GVD-CFM trained at

B_{\mathrm{train}}=25

. The same generated DCT coefficient tensor is evaluated at

M\in\{25,50,100,200,400\}

without retraining, while the corresponding real GVD trajectories are recomputed natively at each target resolution.

Figure 6: Regional GVD dynamics on BNCI2014_001. Mean GVD edge magnitude of five representative regional interactions over the $B=100$ windows for held-out real and GVD-CFM trajectories. The generated trajectories reproduce the early transient, the fall to a lower connectivity regime, the mid-trial reduction and the gradual recovery of the real data, and these changes are aligned across regions. Shaded regions show the variability around each mean trajectory.

Figure 7: Single-trial GVD connectivity-matrix sequence on BNCI2014_001. One generated trial and its nearest held-out real trial are shown over ten consecutive temporal intervals. Both sequences exhibit substantial time-dependent reconfiguration of channel-level connectivity rather than a static graph template.

Figure 8: Single-trial scalp node-strength sequence on BNCI2014_001. Scalp node-strength maps for the same generated trial and nearest held-out real trial across ten temporal intervals. Both sequences display spatially heterogeneous and time-varying patterns involving frontal, central, centro-parietal, and posterior electrodes. The generated sequence preserves broad sensorimotor-centered spatial dynamics without reproducing the real trial exactly.

Figure 9: Efficiency and ablation analysis of GVD-CFM. (a) Average training time and batch-32 generation latency across five datasets and three generator seeds, shown on logarithmic axes. GVD-CFM is substantially faster to train than the large raw-signal generative baselines while retaining practical sampling cost. (b) DCT ablation with the spectral-only velocity network, comparing the full orthonormal DCT-II representation against direct temporal log-svec coordinates. The DCT improves relative GVD-FID, EvaGeM F1, CAS AUC, temporal-correlation agreement, and lag-autocorrelation agreement, with the largest gains observed in temporal structure. (c) Quality–speed trade-off across generators. The horizontal axis shows batch-32 generation time, the vertical axis shows CAS AUC, and marker size is proportional to EvaGeM F1. GVD-CFM combines strong downstream utility with substantially stronger trajectory-level distributional overlap than competing methods at practical generation cost.

Figure 10: Temporal-resolution ablation of graph-variate connectivity. Weighted F1 (left) and ROC AUC (right) on the held-out evaluation split as the number of full-coverage bins increases from $B=8$ to sample resolution ( $B=512$ ). The upper axis gives the mean number of EEG samples per bin. We compare the trial-level support $W$ , the instantaneous covariance $J$ without support, the control $W\odot J^{\mathrm{Gaussian}}$ , and the real graph-variate trajectory $\Delta=W\odot J$ . The Gaussian control remains close to chance, so arbitrary temporal modulation of $W$ does not reproduce the discriminative information of the real trajectory. At fine resolutions $J$ degrades as its estimates become noisier, whereas modulation by the stable support is far more robust.

CAS AUC $\uparrow$
Representation	$B=8$	$16$	$32$	$64$	$128$	$256$	$512$
Stable support	0.841	0.838	0.840	0.833	0.837	0.827	0.807
No support	0.816	0.797	0.765	0.745	0.756	0.738	0.690
Support only	0.822	0.822	0.822	0.822	0.822	0.822	0.822
Gaussian dynamics	0.512	0.518	0.527	0.545	0.574	0.558	0.528
CAS weighted F1 $\uparrow$

Appendix

Table 7: Temporal-resolution support audit (Figure 10 ). Held-out classification of real trajectories with the CAS classifier, dataset-balanced over five datasets and three seeds, as a function of the number of windows

B

. Stable support is

W\odot J_{t}

, no support is

J_{t}+\epsilon I

with

\epsilon=10^{-6}

, support only is the static

W

and Gaussian dynamics is

W\odot J_{t}^{\mathrm{Gaussian}}

. Best value in each column is bold .

Figure 11: Temporal amplitude bias in generated GVD dynamics. Mean off-diagonal GVD edge weight over the $B=100$ windows for real training trials and generated trials. The generated sequence captures the broad temporal organization, including the early transient, but shows a persistent positive amplitude offset over much of the later trajectory.

Figure 12: Temporal alignment of the full GVD-CFM model. Mean off-diagonal GVD edge weight over all $B=100$ windows for real training trials and generated trials on BNCI2014_001. The generated trajectory follows the real sequence through the initial transient, the subsequent reduction, the intermediate fluctuations and the later recovery.

Method	$M_{\mathrm{gen}}$	$M_{\mathrm{held}}$	$\|\Delta M\|\downarrow$	$M^{2}_{\mathrm{gen}}$	$M^{2}_{\mathrm{held}}$	Near-copy $\downarrow$
GVD-CFM	0.99400	0.99501	0.00101	0.98803	0.99004	0.16155
GVD-cVAE	0.99316	0.99516	0.00200	0.98636	0.99034	0.98171
GVD-DDPM	0.99671	0.99516	0.00156	0.99344	0.99034	0.18854
Window-DIFFEO-CFM	0.99678	0.99516	0.00162	0.99357	0.99034	0.00000
cVAE	0.99555	0.99516	0.00040	0.99113	0.99034	0.01807
JET	0.99298	0.99516	0.00218	0.98605	0.99034	0.00000

Appendix

Table 8: Nearest-neighbor memorization analysis.

M=d_{1}/d_{2}

compares the first and second nearest same-class training neighbors. The held-out-real reference provides the target local-neighbor geometry; for GVD-CFM it is computed in the same run as its samples.

Metric	GVD-CFM	GVD-DDPM	GVD-cVAE
Relative GVD-FID + increments	$1.0158\pm 0.0692$	$1.7955\pm 1.5328$	$0.8131\pm 0.0789$
Relative GVD-FID, positions only	$1.0040\pm 0.0453$	$1.7354\pm 1.2118$	$0.9014\pm 0.0708$
EvaGeM F1 + increments	$0.6086\pm 0.2440$	$0.0480\pm 0.0372$	$0.0028\pm 0.0034$
EvaGeM F1, positions only	$0.6442\pm 0.1778$	$0.0046\pm 0.0058$	$0.0035\pm 0.0048$
CAS AUC	$0.7920\pm 0.0973$	$0.6071\pm 0.1317$	$0.7334\pm 0.0983$
CAS weighted F1	$0.7240\pm 0.0963$	$0.5759\pm 0.1020$	$0.6766\pm 0.0840$

Appendix

Table 9: Dataset-balanced results over five EEG datasets and three generator seeds per dataset with generated log-eigenvalue clipping disabled. GVD-CFM is the full model with the temporal branch. Values are mean

\pm

standard deviation across datasets after averaging generator seeds within each dataset.

Method	Dataset	Rel. GVD-FID $\downarrow$	Eva $\alpha$ $\uparrow$	Eva $\beta$ $\uparrow$	Eva F1 $\uparrow$	CAS AUC $\uparrow$	CAS F1 $\uparrow$
GVD-CFM	BNCI2014_001	1.006 $\pm$ 0.014	0.897 $\pm$ 0.057	0.836 $\pm$ 0.025	0.864 $\pm$ 0.027	0.789 $\pm$ 0.015	0.707 $\pm$ 0.007
	BNCI2014_002	0.977 $\pm$ 0.016	0.837 $\pm$ 0.046	0.666 $\pm$ 0.055	0.740 $\pm$ 0.022	0.780 $\pm$ 0.015	0.709 $\pm$ 0.005
	BNCI2015_001	0.978 $\pm$ 0.005	0.842 $\pm$ 0.050	0.719 $\pm$ 0.090	0.775 $\pm$ 0.073	0.746 $\pm$ 0.011	0.681 $\pm$ 0.002
	Shin2017A	1.190 $\pm$ 0.084	0.369 $\pm$ 0.032	0.696 $\pm$ 0.049	0.482 $\pm$ 0.039	0.672 $\pm$ 0.029	0.622 $\pm$ 0.025
	Zhou2016	0.959 $\pm$ 0.033	0.440 $\pm$ 0.143	0.133 $\pm$ 0.081	0.203 $\pm$ 0.110	0.965 $\pm$ 0.003	0.905 $\pm$ 0.014
	5-dataset avg.	1.022	0.677	0.610	0.613	0.790	0.725

Appendix

Table 10: Full per-dataset generative performance corresponding to Table 1 . Entries are mean

\pm

standard deviation over three generator seeds. The real-data reference uses real training trajectories against held-out real test trajectories for EvaGeM, and the corresponding TRTR classifier reference for CAS. Real-data references are excluded from generator rankings. Best generator values within each dataset are bold ; second-best generator values are underlined italics .

Method	Dataset	Rel. GVD-FID $\downarrow$	Eva $\alpha$ $\uparrow$	Eva $\beta$ $\uparrow$	Eva F1 $\uparrow$	CAS AUC $\uparrow$	CAS F1 $\uparrow$
cVAE	BNCI2014_001	1.374 $\pm$ 0.020	0.621 $\pm$ 0.090	0.002 $\pm$ 0.002	0.004 $\pm$ 0.003	0.588 $\pm$ 0.013	0.555 $\pm$ 0.015
	BNCI2014_002	1.154 $\pm$ 0.008	0.486 $\pm$ 0.045	0.006 $\pm$ 0.004	0.012 $\pm$ 0.008	0.550 $\pm$ 0.022	0.491 $\pm$ 0.066
	BNCI2015_001	1.885 $\pm$ 0.052	0.023 $\pm$ 0.005	0.005 $\pm$ 0.001	0.007 $\pm$ 0.001	0.584 $\pm$ 0.002	0.546 $\pm$ 0.011
	Shin2017A	1.617 $\pm$ 0.055	0.668 $\pm$ 0.140	0.002 $\pm$ 0.002	0.003 $\pm$ 0.003	0.514 $\pm$ 0.013	0.470 $\pm$ 0.043
	Zhou2016	1.049 $\pm$ 0.003	0.240 $\pm$ 0.043	0.018 $\pm$ 0.006	0.034 $\pm$ 0.011	0.920 $\pm$ 0.007	0.799 $\pm$ 0.026
	5-dataset avg.	1.416	0.407	0.006	0.012	0.631	0.572

Appendix

Table 11: Full per-dataset generative performance (continued).

Method	Dataset	Temp. corr. $\uparrow$	Lag-ACF $\uparrow$	Energy $\to 1$	Dyn. frac. $\to 1$	Adjacent $\to 1$
GVD-CFM	BNCI2014_001	0.920 $\pm$ 0.006	0.999 $\pm$ 0.000	0.995 $\pm$ 0.023	1.024 $\pm$ 0.052	1.070 $\pm$ 0.013
	BNCI2014_002	0.838 $\pm$ 0.003	0.999 $\pm$ 0.000	0.948 $\pm$ 0.018	0.938 $\pm$ 0.036	0.977 $\pm$ 0.017
	BNCI2015_001	0.958 $\pm$ 0.001	0.999 $\pm$ 0.000	0.938 $\pm$ 0.016	0.976 $\pm$ 0.007	0.976 $\pm$ 0.015
	Shin2017A	0.958 $\pm$ 0.002	0.994 $\pm$ 0.001	1.068 $\pm$ 0.006	1.161 $\pm$ 0.011	1.317 $\pm$ 0.005
	Zhou2016	0.877 $\pm$ 0.017	0.992 $\pm$ 0.005	0.815 $\pm$ 0.039	0.835 $\pm$ 0.011	0.871 $\pm$ 0.027
	5-dataset avg.	0.910	0.997	0.953	0.987	1.042

Appendix

Table 12: Full per-dataset complementary diagnostics corresponding to Table 2 : temporal and dynamic fidelity. Entries are mean

\pm

standard deviation over three generator seeds. For ratio metrics, ranking is by proximity to 1.

Method	Dataset	Temp. corr. $\uparrow$	Lag-ACF $\uparrow$	Energy $\to 1$	Dyn. frac. $\to 1$	Adjacent $\to 1$
cVAE	BNCI2014_001	0.399 $\pm$ 0.031	0.948 $\pm$ 0.017	0.986 $\pm$ 0.014	1.381 $\pm$ 0.031	1.039 $\pm$ 0.023
	BNCI2014_002	0.512 $\pm$ 0.060	0.976 $\pm$ 0.006	0.956 $\pm$ 0.011	0.850 $\pm$ 0.014	1.013 $\pm$ 0.008
	BNCI2015_001	0.798 $\pm$ 0.019	0.990 $\pm$ 0.003	1.023 $\pm$ 0.014	0.520 $\pm$ 0.001	1.038 $\pm$ 0.013
	Shin2017A	0.216 $\pm$ 0.033	0.913 $\pm$ 0.026	0.651 $\pm$ 0.018	1.290 $\pm$ 0.060	0.886 $\pm$ 0.023
	Zhou2016	0.710 $\pm$ 0.018	0.958 $\pm$ 0.004	0.876 $\pm$ 0.005	0.862 $\pm$ 0.006	0.955 $\pm$ 0.006
	5-dataset avg.	0.527	0.957	0.898	0.981	0.986

Appendix

Table 13: Full per-dataset temporal and dynamic diagnostics (continued).

Method	Dataset	Precision $\uparrow$	Diversity $\to 1$	Coverage $\uparrow$
GVD-CFM	BNCI2014_001	0.967 $\pm$ 0.012	0.945 $\pm$ 0.002	0.175 $\pm$ 0.008
	BNCI2014_002	0.899 $\pm$ 0.026	0.952 $\pm$ 0.012	0.211 $\pm$ 0.003
	BNCI2015_001	0.859 $\pm$ 0.037	0.956 $\pm$ 0.007	0.203 $\pm$ 0.006
	Shin2017A	0.032 $\pm$ 0.036	1.168 $\pm$ 0.005	0.222 $\pm$ 0.017
	Zhou2016	1.000 $\pm$ 0.000	0.914 $\pm$ 0.022	0.317 $\pm$ 0.012
	5-dataset avg.	0.751	0.987	0.226

Appendix

Table 14: Full per-dataset novelty, diversity, and coverage diagnostics corresponding to Table 2 . Entries are mean

\pm

standard deviation over three generator seeds. Diversity has ideal value 1.

Method	Dataset	Precision $\uparrow$	Diversity $\to 1$	Coverage $\uparrow$
cVAE	BNCI2014_001	0.799 $\pm$ 0.090	0.709 $\pm$ 0.002	0.027 $\pm$ 0.001
	BNCI2014_002	0.615 $\pm$ 0.016	0.890 $\pm$ 0.015	0.029 $\pm$ 0.002
	BNCI2015_001	0.101 $\pm$ 0.031	1.037 $\pm$ 0.012	0.022 $\pm$ 0.003
	Shin2017A	0.875 $\pm$ 0.122	0.503 $\pm$ 0.034	0.008 $\pm$ 0.000
	Zhou2016	0.572 $\pm$ 0.024	0.946 $\pm$ 0.009	0.085 $\pm$ 0.005
	5-dataset avg.	0.593	0.817	0.034

Appendix

Table 15: Full per-dataset novelty, diversity, and coverage diagnostics (continued).

Dataset	Variant	Rel. GVD-FID $\downarrow$	Eva F1 $\uparrow$	CAS AUC $\uparrow$	CAS F1 $\uparrow$
BNCI2014_001	No DCT	1.030 $\pm$ 0.017	0.641 $\pm$ 0.090	0.773 $\pm$ 0.027	0.691 $\pm$ 0.023
	DCT, spectral only	1.016 $\pm$ 0.015	0.825 $\pm$ 0.021	0.784 $\pm$ 0.006	0.695 $\pm$ 0.005
	GVD-CFM (DCT + temporal branch)	1.006 $\pm$ 0.014	0.864 $\pm$ 0.027	0.789 $\pm$ 0.015	0.707 $\pm$ 0.007
BNCI2014_002	No DCT	1.036 $\pm$ 0.009	0.628 $\pm$ 0.090	0.757 $\pm$ 0.009	0.685 $\pm$ 0.009
	DCT, spectral only	0.988 $\pm$ 0.012	0.723 $\pm$ 0.042	0.789 $\pm$ 0.006	0.716 $\pm$ 0.006
	GVD-CFM (DCT + temporal branch)	0.977 $\pm$ 0.016	0.740 $\pm$ 0.022	0.780 $\pm$ 0.015	0.709 $\pm$ 0.005

Appendix

Table 16: Full per-dataset DCT and temporal-branch ablation corresponding to Table 3 : generative fidelity and utility. No DCT and DCT, spectral only are trained without the temporal branch; the last row of each block is the full GVD-CFM. Entries are mean

\pm

standard deviation over three generator seeds. Best values are bold ; second-best values are underlined italics .

Dataset	Variant	Temp. corr. $\uparrow$	Lag-ACF $\uparrow$	Energy $\to 1$	Dyn. frac. $\to 1$	Adjacent $\to 1$
BNCI2014_001	No DCT	0.065 $\pm$ 0.022	0.465 $\pm$ 0.158	0.992 $\pm$ 0.027	1.005 $\pm$ 0.059	1.231 $\pm$ 0.034
	DCT, spectral only	0.939 $\pm$ 0.003	0.999 $\pm$ 0.000	1.013 $\pm$ 0.026	1.025 $\pm$ 0.031	1.052 $\pm$ 0.024
	GVD-CFM (DCT + temporal branch)	0.920 $\pm$ 0.006	0.999 $\pm$ 0.000	0.995 $\pm$ 0.023	1.024 $\pm$ 0.052	1.070 $\pm$ 0.013
BNCI2014_002	No DCT	0.140 $\pm$ 0.018	0.649 $\pm$ 0.177	0.974 $\pm$ 0.019	0.980 $\pm$ 0.013	1.193 $\pm$ 0.024
	DCT, spectral only	0.842 $\pm$ 0.003	0.999 $\pm$ 0.001	0.984 $\pm$ 0.032	0.963 $\pm$ 0.002	0.990 $\pm$ 0.024
	GVD-CFM (DCT + temporal branch)	0.838 $\pm$ 0.003	0.999 $\pm$ 0.000	0.948 $\pm$ 0.018	0.938 $\pm$ 0.036	0.977 $\pm$ 0.017

Appendix

Table 17: Full per-dataset DCT and temporal-branch ablation: temporal and dynamic fidelity. Entries are mean

\pm

standard deviation over three generator seeds. For ratio metrics, ranking is by proximity to 1. Best values are bold ; second-best values are underlined italics .

Comparison	Metric	BNCI2014_001	BNCI2014_002	BNCI2015_001	Shin2017A	Zhou2016
No DCT $\rightarrow$ DCT, spectral only	Rel. GVD-FID	-0.014 (-1.1)	-0.048 ( -5.5 )	-0.017 ( -4.7 )	-0.247 ( -3.0 )	-0.007 ( -3.4 )
	Eva F1	+0.184 ( 3.4 )	+0.095 (1.7)	-0.100 ( -2.7 )	+0.173 ( 5.7 )	-0.053 (-1.8)
	CAS AUC	+0.011 (0.7)	+0.032 ( 5.1 )	+0.057 ( 27.4 )	+0.003 (0.3)	-0.003 (-0.6)
	CAS F1	+0.004 (0.3)	+0.031 ( 5.0 )	+0.052 ( 7.7 )	+0.004 (0.3)	-0.007 (-1.6)
	Temp. corr.	+0.874 ( 68.2 )	+0.702 ( 66.6 )	+0.022 ( 12.7 )	+0.945 ( 80.9 )	+0.030 ( 4.8 )
	Lag-ACF	+0.534 ( 5.9 )	+0.350 ( 3.4 )	+0.015 ( 13.0 )	+0.743 ( 5.1 )	+0.013 ( 5.3 )

Appendix

Table 18: Paired per-dataset comparison for the DCT and temporal-branch ablation. Each entry is the change in the dataset mean (second variant minus first) followed by the Welch

t

-statistic computed from three generator seeds per variant,

t=\Delta/\sqrt{(\sigma_{1}^{2}+\sigma_{2}^{2})/3}

. Entries with

|t|\geq 2

are bold ; a dash marks zero seed variance. For Rel. GVD-FID a negative change is an improvement; for Energy the ideal value is 1.

Dataset	Variant	Rel. GVD-FID $\downarrow$	Eva F1 $\uparrow$	CAS AUC $\uparrow$	CAS F1 $\uparrow$	Temp. corr. $\uparrow$	Lag-ACF $\uparrow$	Energy $\to 1$
BNCI2014_001	GVD-CFM (stable support)	1.006 $\pm$ 0.014	0.864 $\pm$ 0.027	0.789 $\pm$ 0.015	0.707 $\pm$ 0.007	0.920 $\pm$ 0.006	0.999 $\pm$ 0.000	0.995 $\pm$ 0.023
	No support + ridge	0.917 $\pm$ 0.006	0.292 $\pm$ 0.032	0.667 $\pm$ 0.019	0.614 $\pm$ 0.017	0.977 $\pm$ 0.001	1.000 $\pm$ 0.000	0.826 $\pm$ 0.007
BNCI2014_002	GVD-CFM (stable support)	0.977 $\pm$ 0.016	0.740 $\pm$ 0.022	0.780 $\pm$ 0.015	0.709 $\pm$ 0.005	0.838 $\pm$ 0.003	0.999 $\pm$ 0.000	0.948 $\pm$ 0.018
	No support + ridge	0.909 $\pm$ 0.001	0.481 $\pm$ 0.024	0.736 $\pm$ 0.009	0.674 $\pm$ 0.010	0.929 $\pm$ 0.001	0.999 $\pm$ 0.000	0.850 $\pm$ 0.010
BNCI2015_001	GVD-CFM (stable support)	0.978 $\pm$ 0.005	0.775 $\pm$ 0.073	0.746 $\pm$ 0.011	0.681 $\pm$ 0.002	0.958 $\pm$ 0.001	0.999 $\pm$ 0.000	0.938 $\pm$ 0.016
	No support + ridge	0.921 $\pm$ 0.005	0.372 $\pm$ 0.050	0.698 $\pm$ 0.012	0.638 $\pm$ 0.006	0.980 $\pm$ 0.000	0.999 $\pm$ 0.000	0.874 $\pm$ 0.006

Appendix

Table 19: Full per-dataset stable-support ablation summarized in Section 6.3 . GVD-CFM uses the canonical ridge-free stable support; the no-support control uses the ridge required to obtain SPD matrices and the spectral-only velocity network. Entries are mean

\pm

standard deviation over three generator seeds. Best values are bold ; second-best values are underlined italics .

Dataset	Basis	Rel. GVD-FID $\downarrow$	Eva F1 $\uparrow$	CAS AUC $\uparrow$	CAS F1 $\uparrow$
BNCI2014_001	No DCT	1.030 $\pm$ 0.017	0.641 $\pm$ 0.090	0.773 $\pm$ 0.027	0.691 $\pm$ 0.023
	Random orthogonal	1.071 $\pm$ 0.003	0.461 $\pm$ 0.045	0.780 $\pm$ 0.021	0.695 $\pm$ 0.011
	GVD-CFM (DCT)	1.006 $\pm$ 0.014	0.864 $\pm$ 0.027	0.789 $\pm$ 0.015	0.707 $\pm$ 0.007
	KLT	1.023 $\pm$ 0.006	0.802 $\pm$ 0.008	0.787 $\pm$ 0.012	0.697 $\pm$ 0.010
BNCI2014_002	No DCT	1.036 $\pm$ 0.009	0.628 $\pm$ 0.090	0.757 $\pm$ 0.009	0.685 $\pm$ 0.009
	Random orthogonal	1.048 $\pm$ 0.015	0.578 $\pm$ 0.050	0.783 $\pm$ 0.006	0.700 $\pm$ 0.015

Appendix

Table 20: Full temporal-basis ablation across the five main EEG datasets: generative fidelity and downstream utility. Each per-dataset entry is mean

\pm

standard deviation over three generator seeds. All four variants retain all

B=100

temporal coordinates. Random orthogonal uses a fixed random orthogonal basis, KLT the training-set empirical temporal Karhunen–Loève basis, and GVD-CFM (DCT) the fixed orthonormal DCT-II basis with the full network including the temporal branch; the other variants use the spectral-only velocity network. Best values are bold ; second-best values are underlined italics .

Dynamic Functional Connectivity

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TVGL-CFM:Generating and Forecasting Time-Varying Trajectories of Dynamic Networks with Conditional Flow Matching
Jul 18, 2026Om Roy, Yashar Moshfeghi, Keith Malcolm SmithTime Series Generation Dynamic Graph Learning
University of Strathclyde
GRFBrain: Graph-Structured Rectified Flows for EEG Dynamic Modeling
Sep 29, 2026Haohui Jia, Zheng Chen, Jathurshan Pradeepkumar +4Electroencephalography Functional Connectivity
Faculty of Information Science and Technology, Hokkaido University, Japan · SANKEN, The University of Osaka, Japan · Department of Computer Science, University of Illinois Urbana-Champaign, USA +1
Structural-Functional Brain Connectivity Generation via Multimodal Hypergraph-based Flow Matching
Oct 2, 2026Chyong Yi Poh, Hwa Hui Tew, Junn Yong Loo +4Multimodal Generation Functional Connectivity
School of Information Technology, Monash University Malaysia · School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore · Department of Radiology, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

Jul 18, 2026J/K move · Enter open · S save

TVGL-CFM:Generating and Forecasting Time-Varying Trajectories of Dynamic Networks with Conditional Flow Matching

Om Roy, Yashar Moshfeghi, Keith Malcolm Smith

University of Strathclyde

Time Series Generation Dynamic Graph Learning Conditional Generative Modeling

Open paper AskPDF arXiv

Many complex systems such as brain networks, financial markets, and gene-regulatory circuits are described not by a fixed graph but by one that changes over time. A standard way to summarise such structure at each instant is the sparse precision (inverse-covariance) matrix, and the time-varying graphical lasso (TVGL) turns a multivariate signal into a smooth chain of these matrices. We introduce TVGL-CFM, a single model that learns the distribution of such chains and can both generate new, realistic time-varying network trajectories for a given class and forecast how an observed trajectory will continue. Each precision matrix lives on a curved space of positive-definite matrices, but a log-Euclidean chart flattens an entire trajectory into an ordinary vector space, so a simple conditional flow-matching model can be trained and sampled there while every decoded matrix is guaranteed to be a valid precision matrix. For forecasting we start the flow not from noise but from a rough extrapolation of the recent history, so the model only has to learn a small correction. Across EEG motor-imagery, chaotic systems, and gene-expression data, TVGL-CFM generates trajectories that keep the class-discriminative structure of real data, and it forecasts future connectivity more accurately than raw-signal baselines. Generating the structured precision trajectory directly is therefore more faithful than generating raw signals and estimating connectivity afterwards.

Abstract

Dynamic Functional Connectivity

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TVGL-CFM:Generating and Forecasting Time-Varying Trajectories of Dynamic Networks with Conditional Flow Matching

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Structural-Functional Brain Connectivity Generation via Multimodal Hypergraph-based Flow Matching