The proposed F²-CommNet framework integrates fractional-order neural dynamics, Fourier spectral filtering, and Lyapunov stability control into a unified graph-based learning system for dynamic community detection. The model enhances interpretability and robustness by embedding memory-dependent evolution and spectral regularization into the community learning process. The framework is illustrated in Figure 1.

Step 1: Graph Input.

Given a sequence of graph snapshots {G_t = (V, E_t)} with node features Xt∈ℝ|V|×d and Laplacian matrices L_t = D_t−A_t, the model initializes node representations for temporal propagation.

Step 2: Fractional Dynamics.

Node embeddings evolve under Caputo fractional-order differential equations:

where the fractional derivative DCTα introduces long-memory smoothing and non-local temporal effects into neural propagation.

Step 3: Fourier Spectral Filtering.

Each snapshot is decomposed into Laplacian eigenmodes:

where unstable high-frequency modes are attenuated by a spectral kernel ϕ(λ_k), thereby reducing noise amplification.

Step 4: Stability Monitoring.

To establish the Lyapunov-based stability bound, we assume a symmetric positive definite matrix P≻0 such that the Lyapunov functional V(x) = x^⊤Px is well-defined and radially unbounded. This standard assumption in fractional-order stability theory enables the derivation of provable error bounds under the proposed dynamics.

A Lyapunov margin ρ is estimated as

while the hallucination index η_k = λ_kF−c_k is monitored for each eigenmode to assess spectral stability.

Step 5: Community Partitioning.

The stabilized embeddings are clustered into communities {C_t} by maximizing the standard Newman–Girvan modularity on each snapshot. The term ”stability-aware” denotes that modularity optimization is conducted on embeddings that have been previously refined via fractional and Lyapunov-based stabilization modules, rather than on unprocessed features or adjacency matrices.

We generalize graph neural evolution by introducing a Caputo fractional derivative of order α ∈ (0, 1):

where x_i(t) denotes the state of node i, c_i>0 is a leakage coefficient, w_ij represents connection weights, f(·) is a non-linear activation, and u_i(t) is an external forcing term.

The Caputo derivative is defined as

where Γ(·) denotes the Gamma function. This expression reveals that the derivative depends on the entire historical trajectory x(τ) for τ ≤ t, embedding long-memory effects within the dynamics. Compared with the integer-order case (α = 1), fractional dynamics dampen oscillations and enlarge the convergence basin, improving robustness to perturbations.

To practically compute the fractional derivative DCTα, we adopt the truncated Grünwald–Letnikov (GL) approximation:

where H denotes the memory horizon that limits the fractional historical dependence. This discretization yields an efficient and numerically stable implementation suitable for large-scale dynamic graphs.

From a modeling perspective, the window H defines the effective memory horizon of the Caputo operator: larger H retains longer-range temporal dependence, but also increases computational cost. In practice, we require H≪T, where T is the sequence length, and we find that choosing H/T in a small range (around 5–15% of T) preserves the desired long-memory behavior while keeping the overall complexity near-linear. The dataset-specific choices of H are summarized in Section 4.3.

For the collective evolution of all node features, we express the fractional-order neural dynamics in matrix form:

where X_t is the node feature matrix, C is a leakage coefficient matrix, W is a weight matrix for inter-node connections, and U_t represents external forcing. This matrix form guides the temporal propagation of node representations within our F²-CommNet model.

Let L = UΛU^⊤ denote the Laplacian decomposition, where Λ = diag(λ₁, …, λ_n) contains eigenvalues and U = [u₁, …, u_n] the corresponding eigenvectors. Spectral projection and reconstruction are expressed as

For each eigenmode u_k, the hallucination index is defined by

where F is the forcing gain and c_k is the leakage term. Modes with η_k>0 are deemed unstable, while η_k < 0 indicates spectral stability. Because high-frequency eigenmodes (large λ_k) amplify noise, F²-CommNet employs adaptive Fourier filtering:

where ϕ(λ_k) is a decay kernel that suppresses unstable modes and preserves low-frequency structure.

Integrating the spectral projection, adaptive filtering (where ϕ(λ_k) is often parameterized as g_θ(Λ)), and reconstruction, the comprehensive Fourier spectral filtering operation for the feature matrix Xt(α) is expressed as:

This operation effectively purifies the node embeddings by removing noise-amplifying high-frequency components.

Intuitively, the decay kernel ϕ(λ_k) controls the degree of suppression applied to each spectral mode. High-frequency components associated with large eigenvalues λ_k tend to exhibit oscillatory or unstable behavior in dynamic graphs. A stronger decay factor, therefore, effectively damps these fluctuations, reducing the likelihood of hallucinated communities while preserving low-frequency structural information.

We summarize the key ideas behind the stability analysis and present the main results in a concise form for improved readability.

Error dynamics and hallucinations. Let x^*(t) denote an ideal (hallucination-free) community trajectory and define the deviation

The fractional-order error dynamics can be written as

where Δu(t) models perturbations such as noise or modeling mismatch. Intuitively, a persistent non-zero e(t) corresponds to hallucinated community states.

Lyapunov function and Mittag–Leffler bound. We consider the quadratic Lyapunov function

with P symmetric positive definite. If there exists P≻0 and a margin ρ>0 such that

then the error norm admits the fractional-order bound

where E_α(·) is the Mittag–Leffler function and ||Δu(t)|| ≤ ū. The Mittag–Leffler term generalizes the exponential decay in integer-order systems, capturing the memory-dependent, non-local convergence behavior of fractional dynamics.

Equation 16 implies that the error is ultimately bounded and cannot diverge if ρ>0 and ū is finite, thereby preventing long-term hallucinations.

Spectral stability margin and fractional effect. For interpretability, we summarize the stability condition using a spectral margin

A larger ρ indicates a larger region of attraction and stronger robustness to perturbations. In fractional-order dynamics (α < 1), the effective forcing term is attenuated by a factor depending on Γ(1−α), which increases the margin ρ compared to the integer-order case. As a result, fewer Laplacian eigenmodes violate the stability condition, and the hallucination indices η_k tend to become negative, which matches the empirical reduction in unstable high-frequency modes.

Summary. Rather than providing full proofs, we emphasize the practical implications: (i) the Lyapunov condition (Equation 15) and margin (Equation 17) quantify robustness against hallucinations; (ii) fractional dynamics enlarge this stability region; and (iii) Fourier spectral filtering further pushes η_k to the stable regime. These theoretical insights are empirically validated by the stability and hallucination metrics reported in Section 4.

The stability analysis adheres to conventional fractional-order Lyapunov theory and necessitates two technical prerequisites: (i) the non-linear activation f(·) exhibits Lipschitz continuity, and (ii) the external disturbances are constrained. In practical GNN training, these assumptions are approximately valid. Initially, while ReLU is not globally Lipschitz at the origin, it is piecewise 1-Lipschitz virtually universally, and contemporary optimizers (such as Adam with minimal step sizes) maintain the iterates inside compact domains where the local Lipschitz constant remains finite. This relaxation is widely adopted in stability analyses of neural and graph-based models (Pascanu et al., 2013; Scaman and Virmaux, 2018). Second, the disturbances induced by stochastic training noise and spectral approximation errors remain bounded due to gradient clipping, finite-step discretization, and the bounded magnitude of node embeddings. Finally, the Lyapunov matrix P is computed numerically using the classical Bartels–Stewart solver (Golub and Van Loan, 2013), which ensures P≻0 in all experiments. Together, these considerations justify the applicability of the theoretical assumptions in real training scenarios while preserving the rigor of the stability guarantees.

To practically enforce the Lyapunov stability conditions and ensure bounded error dynamics, our framework actively refines the node embeddings. The next-step embeddings X_t+1 are determined by solving an optimization problem that balances fidelity to the spectrally filtered embeddings X^t with a regularization term directly tied to the system's stability:

where λ>0 is a hyperparameter balancing the two terms. Here, ρ(Z) represents the spectral stability margin associated with the candidate embedding Z. By minimizing ρ⁻¹(Z), the optimization actively guides the model toward configurations that maximize ρ, thereby enhancing system stability and mitigating the formation of hallucination-prone structures. This term ensures that the information energy in the embeddings remains bounded, preventing persistent instability or convergence to spurious equilibria.

The stability-aware refinement and clustering stage is closely interconnected: modularity is optimized on embeddings X_t+1 that have been specifically regularized by the Lyapunov margin ρ(Z). The hallucination indices {η_k} serve as diagnostic metrics to assess if the resultant communities engage unstable spectral modes. In this context, stability considerations indirectly influence the final partition via embedding refinement, but the clustering target continues to adhere to the conventional modularity metric.

We summarize the complete training and inference workflow of F²-CommNet in Algorithm 1 and Figure 1, which integrates fractional dynamics, Fourier spectral filtering, Lyapunov stability monitoring, and stability-aware modularity optimization.

We analyze the computational complexity of F²-CommNet in both training and inference phases by decomposing its workflow into the major steps of Algorithm 1. Let n = |V| be the number of nodes, d the embedding dimension, H the effective memory horizon for fractional dynamics, and r≪n the number of leading eigenpairs retained for spectral decomposition.

F²-CommNet amalgamates fractional-order neural dynamics, Fourier spectrum filtering, and Lyapunov-guided refinement into a cohesive stability-aware framework for dynamic community discovery. Fractional dynamics facilitate long-memory smoothing, whereas spectral filtering mitigates high-frequency modes susceptible to hallucinations. The resultant stability margin ρ and hallucination index η_k offer comprehensible robustness assurances, while the entire pipeline exhibits near-linear scalability, facilitating implementation on extensive dynamic graphs.

To evaluate the effectiveness and robustness of F²-CommNet, we conduct experiments on a diverse set of real-world and synthetic dynamic networks. All datasets are preprocessed into temporal snapshots {G_t} with consistent node sets and evolving edge relations. Statistics are summarized in Table 2.

Enron Email Network (EN) (Kojaku et al., 2024): A communication dataset with n = 36, 692 nodes and 367, 662 edges, where nodes are employees and edges represent time-stamped email exchanges. Communities correspond to functional groups within the company.

DBLP Co-authorship (DBLP) (Diboune et al., 2024): A co-authorship graph with n = 317, 080 authors and 1, 049, 866 edges. Snapshots are constructed yearly, reflecting the evolution of research communities.

Cora Citation Network (Cora-TS) (Hu et al., 2020): A citation graph adapted into temporal slices, with n = 19, 793 articles and 126, 842 citations. Node attributes are bag-of-words features; communities reflect scientific subfields.

Reddit Hyperlink Network (Reddit) (Kaiser et al., 2023): A large-scale temporal network with n = 55, 863 nodes and 858, 490 edges, where nodes are subreddits and edges represent hyperlinks shared by users. Community structure aligns with topical categories.

UCI Messages (UCI) (Prokop et al., 2024): A dynamic communication dataset with n = 1, 899 nodes and 59, 835 edges, representing private message exchanges on an online forum. Snapshots are segmented weekly to capture evolving social groups.

Human Protein-Protein Interaction (PPI) (Oughtred et al., 2021): A biological network with n = 3, 852 proteins and 76, 584 interactions. Communities correspond to functional protein complexes, with dynamics reflecting newly discovered interactions.

Synthetic Dynamic SBM (Syn-SBM) (Peixoto, 2019): A synthetic dynamic stochastic block model with n = 10, 000 nodes and 4 evolving communities. To study stability and hallucination resistance under noise, we inject temporal perturbations by randomly rewiring a proportion p of edges per snapshot. We consider three noise levels p ∈ {0.02, 0.05, 0.10} (low, moderate, high). Unless otherwise specified, the main experiments use p = 0.05, while Section 4.12 evaluates robustness across all noise settings.

We evaluate F²-CommNet against a diverse set of baselines spanning static, spectral, temporal, and stability-aware approaches:

Static GNNs: Graph Convolutional Network (GCN) (Kipf and Welling, 2017), Graph Attention Network (GAT) (Velickovic et al., 2018).

Spectral methods: Spectral Clustering (SC) (Shah, 2022).

Temporal GNNs: Temporal Graph Network (TGN) (Rossi et al., 2020), Dynamic Graph Convolutional Network (DyGCN) (Manessi et al., 2020).

Stability-enhanced methods: EvolveGCN (Pareja et al., 2020).

Proposed: F²-CommNet.

Table 3 summarizes the taxonomy of baseline methods considered in our experiments. We divide existing approaches into four main categories: (i) Static GNNs such as GCN and GAT, which capture spectral properties but lack temporal modeling and stability control; (ii) Spectral methods such as Spectral Clustering, which operate purely in the eigen-space of the Laplacian without temporal adaptation; (iii) Temporal GNNs, including TGN and DyGCN, which extend GNNs with dynamic node updates but still lack explicit hallucination suppression; and (iv) Stability-enhanced methods such as EvolveGCN, which introduce mechanisms to handle evolving graphs but without formal stability guarantees.

The proposed F²-CommNet unifies these perspectives by simultaneously supporting temporal modeling, spectral filtering, attention-based aggregation, Lyapunov-guided stability monitoring, and hallucination control. As shown in Table 3, it is the only method that explicitly checks all five properties, highlighting its principled design and broader coverage compared with existing baselines.

We also evaluated Transformer-style dynamic graph designs as possible baselines. Nonetheless, current graph Transformers are predominantly engineered for supervised temporal prediction tasks, including link prediction or node forecasting with time-stamped labels, and generally depend on quadratic self-attention methods. The two qualities render them unsuitable for our context, where (i) the aim is unsupervised structural community discovery instead of predictive accuracy, and (ii) extensive dynamic graphs necessitate near-linear scalability rather than attention-based O(n2) complexity. Modifying these transformer-based models for our label-free clustering task necessitates essential alterations to their architectures and training aims, resulting in indirect and sometimes inequitable comparisons. Consequently, adhering to known methodologies in dynamic community detection, we utilize GCN, GAT, SC, TGN, DyGCN, and EvolveGCN as the most representative and directly comparable benchmarks.

We also evaluated whether recent stability-enhanced GNNs, such as SO-GCN (Chen et al., 2025c), LDC-GAT (Chen et al., 2025b), and other constraint-based stability models could be included as baselines. However, these techniques are predominantly intended for semi-supervised node classification on static graphs and depend on label-driven losses, Jacobian-based regularization, or Lyapunov-style constraints, which are not applicable to our unsupervised dynamic community identification context. These limits also impose significant processing expense, rendering such models unworkable for the million-edge temporal graphs utilized in our research. Consequently, in alignment with existing practices in dynamic community detection, we see these stability-oriented systems as conceptually complementary rather than directly comparable baselines.

All experiments are implemented in PyTorch Geometric and executed on a single NVIDIA RTX 3090 GPU with 24GB memory. The Adam optimizer is used with learning rate 10⁻³, weight decay 10⁻⁵, and embedding dimension d = 64. The batch size is fixed at 128, and each model is trained for 200 epochs. Early stopping with patience 20 epochs is applied to prevent overfitting. Spectral filtering uses r = 50 Lanczos-approximated eigenmodes.

For the fractional dynamics, the truncation window H is treated as a dataset-dependent hyperparameter. For each dataset, we conduct a small validation sweep over a candidate set (e.g., H ∈ {5, 10, 20}) and select the smallest value that yields stable training and strong validation modularity. The final choices are as follows: Enron Email (EN) and UCI Messages use H = 5; Human PPI, Cora-TS, and Reddit Hyperlink use H = 10; DBLP Co-authorship and Synthetic SBM use H = 20. These values remain fixed for all reported experiments to ensure full reproducibility.

Unless otherwise stated, all reported numbers are averaged over 10 independent runs with random seeds {0, 1, …, 9}. For each dataset, method, and metric, we report the sample mean μ and the corresponding 95% confidence interval μ±δ, where δ=t0.975,9σ/10 and σ is the sample standard deviation. This unified statistical protocol ensures a fair and robust comparison across all experiments.

We evaluate F²-CommNet on the two largest datasets in our benchmark suite: Reddit and DBLP. To clarify methodological differences, the baselines are grouped into: (i) static GNNs (GCN, GAT), which do not model temporal evolution, and (ii) dynamic GNNs (DyGCN, EvolveGCN), which adapt to evolving graph structures. All results follow the unified statistical protocol described in Section 4.3, and are reported as mean ± 95% confidence intervals over 10 runs. As shown in Table 5, F²-CommNet achieves the highest ARI on both benchmarks, improving upon static baselines by 20–25% and upon the strongest dynamic baseline (EvolveGCN) by 10–15%. Moreover, the confidence intervals of F²-CommNet are significantly narrower, indicating reduced sensitivity to initialization and greater robustness on large-scale dynamic graphs.

From the comprehensive results in Table 6, several consistent patterns emerge across all seven benchmark datasets (Cora, Citeseer, PubMed, Reddit, Enron, DBLP, BioGRID).

(i) Stability improvement. F²-CommNet consistently achieves the highest stability margin ρ, with average gains of more than 2 × compared to GCN, GAT, and spectral clustering, and at least 30% relative improvement over the strongest temporal baselines such as TGN, DyGCN, and EvolveGCN. This confirms the effectiveness of fractional dynamics and Lyapunov-guided monitoring in enforcing robust equilibrium during dynamic community evolution.

(ii) Hallucination suppression. The hallucination index η_max is drastically reduced by F²-CommNet, reaching values as low as 0.20–0.29 across all datasets, compared with 0.30–0.52 for competing methods. Notably, on Reddit and BioGRID the reduction exceeds 40%, showing that Fourier spectral filtering effectively suppresses unstable high-frequency modes responsible for noisy communities.

(iii) Clustering quality enhancement. The stability and robustness improvements translate directly into superior clustering outcomes. F²-CommNet obtains the best Adjusted Rand Index (ARI), Normalized Mutual Information (NMI), and modularity Q in every case, with gains of 5–10% over GCN/GAT and 3–6% over temporal models like TGN and EvolveGCN. For example, on Cora the ARI improves from 0.75 (EvolveGCN) to 0.80, and on Reddit the NMI improves from 0.77 (EvolveGCN) to 0.85.

Overall These findings demonstrate that F²-CommNet achieves a balanced and principled advancement in stability, hallucination suppression, and clustering quality, providing a robust and generalizable framework for dynamic community detection across diverse domains.

Table 7 summarizes the metric-wise wins of F²-CommNet across seven benchmark datasets. We count victories over five evaluation criteria: stability margin ρ, hallucination index η_max, Adjusted Rand Index (ARI), Normalized Mutual Information (NMI), and modularity Q. As shown, F²-CommNet consistently dominates: it secures the best ρ on all six datasets where stability is well-defined, reduces η_max to the lowest levels on all datasets, and achieves the highest ARI, NMI, and Q in nearly all cases. In total, the model wins 32 out of 35 possible comparisons, demonstrating its robustness across diverse graph domains.

This result highlights that the integration of fractional dynamics, spectral filtering, and stability-aware regularization not only stabilizes training but also directly translates into superior clustering quality. The strong performance across heterogeneous datasets such as citation networks (Cora, Citeseer, PubMed), social networks (Reddit, DBLP), and biological graphs (BioGRID) confirm the generalizability of F²-CommNet.

Key findings. (i) F²-CommNet enlarges ρ by more than 3 × compared to GCN/GAT. (ii) The hallucination index η_max is reduced to nearly zero. (iii) These stability gains translate into better clustering quality.

Figure 2 shows training curves of modularity and stability margin ρ, confirming that F²-CommNet converges faster and to more stable solutions.

Qualitative results in Figure 3 visualize learned communities, showing that F²-CommNet yields cleaner and more compact clusters.

We evaluate five variants:

To evaluate the generality of each architectural component, we compare the five variants across three typical datasets: citation (Cora), social (Reddit), and biological networks (BioGRID).

Table 8 summarizes the cross-dataset ablation results. We focus on the stability margin ρ and the hallucination index η_max, as they directly reflect the stabilization effect of each architectural component.

In all three datasets, each module consistently enhances both the stability margin and the suppression of hallucinations. Fractional damping yields the highest individual benefit, although Lyapunov stability enhances the precision of the confidence intervals. The complete F²-CommNet attains superior performance across all domains, indicating that the stabilizing processes generalize beyond an individual dataset.

We analyze the sensitivity of F²-CommNet to fractional order α, leakage c_i, embedding dimension d, and window size w.

Training dynamics and sensitivity analysis.

Figure 4 provides a joint view of training behaviors and parameter sensitivity. In Figure 4, we compare the modularity Q and stability margin ρ across seven representative methods. Classical baselines such as GCN and Spectral clustering show slower convergence and weaker stability, while more advanced temporal models (DyGCN and EvolveGCN) demonstrate improved robustness. Our proposed F²-CommNet consistently achieves higher Q and larger ρ, validating both community quality and stability guarantees. We further analyze the role of the fractional order α. We observe that α ∈ [0.7, 0.9] yields the most balanced performance: smaller α enlarges the stability margin but slows down convergence due to excessive memory effects, whereas larger α accelerates convergence but weakens robustness, reflected by an increase in η_max. These results empirically support the theoretical trade-off derived in Equation 22 and highlight the importance of selecting moderate fractional orders in practice.

We further investigate how key architectural and fractional-order parameters influence model stability and clustering performance. Table 8 summarizes the ablation study, showing that each fractional component contributes positively to the stability margin ρ and clustering quality Q. The progressive inclusion of Fourier projection, fractional damping, and Lyapunov stability terms leads to a monotonic improvement, with the full F²-CommNet achieving the highest average performance across all metrics. This indicates that the combination of fractional dynamics and Lyapunov-based correction yields a synergistic stabilization effect rather than a simple additive gain.

Table 9 examines the impact of the fractional order α on both stability and hallucination suppression. As α decreases from 1.0 to 0.5, the stability margin ρ gradually increases while the hallucination index η_max decreases, reflecting a stronger damping of unstable eigenmodes. This behavior confirms that the fractional operator serves as a spectral regulator—suppressing noisy high-frequency responses while preserving coherent community structures. Notably, α≈0.7 provides a desirable trade-off between responsiveness and smoothness, consistent with the optimal setting adopted in our experiments.

Table 10 presents the detailed eigenmode analysis of the hallucination indices η_k under α = 1.0 and α = 0.7. Compared with the integer-order case, the fractional configuration compresses the dynamic range of η_k values, effectively reducing extreme oscillations at higher Laplacian frequencies (k>6). This spectral contraction explains the observed increase in temporal consistency across snapshots and validates the fractional damping mechanism described in Equation 22.

Effect of leakage c_i We additionally analyzed the leakage coefficient c_i, which regulates the inherent damping intensity in the fractional dynamics. Augmenting c_i enhances the Lyapunov stability margin ρ and expedites the attenuation of disturbances; nevertheless, excessive leaking may excessively dampen node activations, resulting in unduly smoothed embeddings and diminished community contrast. Conversely, minimal values of c_i diminish the effective damping, rendering the system more susceptible to noise, which subsequently elevates the hallucination index η_max and results in fragmented communities. In our experiments, we probed a moderate range of c_i values and observed that performance (in terms of ρ, η_max, ARI, and Q) remains stable within a band of c_i ∈ [0.2, 0.4]. We therefore fix c_i in this range for all reported results, which provides a robust trade-off between stability and representation strength.

Effect of embedding dimension d Performance improves with increasing feature dimension up to d = 128, beyond which overfitting emerges, suggesting that excessively large latent spaces capture noise rather than informative structural variations.

Effect of window size w A larger temporal window captures longer dependencies but increases computational overhead. Empirically, w = 64 offers a satisfactory balance between temporal expressiveness and efficiency, providing stable training and consistent community alignment across dynamic snapshots.

We further analyze the suppression of high-frequency Laplacian eigenmodes. Table 10 compares hallucination indices η_k = λ_kF−c_k under integer-order (α = 1.0) vs. fractional-order (α = 0.7). The results confirm that fractional dynamics suppress unstable high-frequency modes, consistent with the theoretical model. The theoretical derivation in Equation 22 suggests that fractional damping reduces the effective forcing term λ_kF, thereby shifting certain mid-frequency modes into the stable region.

Fractional-order dynamics thus provide a natural spectral regularization mechanism. Unlike integer-order propagation, which tends to amplify noise residing in higher Laplacian eigenmodes, the fractional operator introduces a smooth decay governed by α, effectively attenuating oscillatory perturbations and stabilizing graph filters. This behavior leads to smaller hallucination indices η_k and smoother temporal transitions across successive snapshots. Empirically, the suppression effect becomes more evident as α decreases, demonstrating that fractional damping not only mitigates over-smoothing but also prevents spectral drift caused by transient noise. Consequently, the fractional component can be interpreted as an adaptive low-pass filter that preserves informative structures while restraining unstable eigenmodes. This motivates the following analysis of spectral hallucination and stability margins.

We next study error trajectories under different noise intensities, based on the error dynamics formulation (Equations 11–15). As shown in Table 11, fractional dynamics consistently achieve tighter error bounds limsup_{t → ∞}||e(t)||, in line with the boundedness theorem (Equation 21).

Finally, we validate Lyapunov convergence by monitoring V(t) = e^⊤Pe. Table 12 demonstrates that α < 1 accelerates the decay of V(t), achieving faster stability, consistent with the sufficient conditions in Equations 20, 21.

To complement the main experiments, we further evaluate the robustness of F²-CommNet under controlled noise conditions using the synthetic dynamic SBM described in Section 4.1. As noted previously, the dataset includes three perturbation levels p ∈ {0.02, 0.05, 0.10}, corresponding to low, moderate, and high noise. This section examines the model's stability and hallucination suppression behavior across these noise regimes.

To validate the theoretical framework of F²-CommNet, we perform a hierarchy of simulations, ranging from toy graphs to synthetic networks and real-world benchmarks. This staged design illustrates how fractional dynamics, Fourier spectral filtering, and Lyapunov-based analysis jointly contribute to stability enhancement and hallucination suppression.

The Laplacian eigenvalues of the 10-vertex synthetic graph are

revealing a rich spectral structure. The smallest eigenvalue λ₁ = 0 corresponds to the trivial constant mode, mid-range modes (e.g., λ₃, λ₄) encode coarse community partitions, while the largest eigenvalues (λ₉, λ₁₀) correspond to highly oscillatory modes that dominate hallucination channels. As shown in Section 3.8, decreasing the fractional order α suppresses such unstable modes, enlarging the stability margin ρ and reducing the hallucination index η_max.

Experiment 1: Baseline Integer Dynamics.

Integer-order dynamics (α = 1.0) follow classical exponential decay. As illustrated in Figure 5, integer-order dynamics (α = 1.0) demonstrate exponential decay. However, high-frequency eigenmodes remain unstable, amplifying oscillations and destabilizing node trajectories. Although partial suppression occurs in low-frequency modes, the lack of robustness in high-frequency channels highlights the inherent limitations of classical integer-order updates, motivating the introduction of fractional damping.

Experiment 2: Fractional Damping.

When governed by fractional order α = 0.8, the system exhibits long-memory smoothing. As illustrated in Figure 6, the Mittag–Leffler decay suppresses oscillations and enforces stable convergence, even under moderate perturbations. Compared with integer-order dynamics, fractional damping converges more slowly at first but achieves greater long-term robustness. This matches the theoretical claim that fractional updates redistribute dissipation across time, thereby suppressing hallucination-prone modes.

Experiment 3: Parameter Sweep.

We sweep α ∈ [0.5, 1.0] to quantify robustness. As shown in Table 9 and Figure 7, smaller α consistently enlarges ρ and reduces η_max, though convergence slows for α ≤ 0.6. The range α ∈ (0.7, 0.9) offers the best trade-off between speed and stability, matching the theoretical condition in Equation 22.

Experiment 4: Perturbation Analysis.

We next test robustness under explicit edge perturbations (Δw₁₄ = 0.5, Δw₂₅ = 0.8, Δw₃₆ = 1.0). Integer-order dynamics amplify noise via unstable high-frequency modes, while fractional-order dynamics confine oscillations to bounded trajectories (Figure 8). Table 13 quantifies this effect, demonstrating fractional damping reduces η_max and enlarges ρ, consistent with the Lyapunov boundedness theorem (Equation 21).

Experiment 5: Spectral Hallucination Indices (Sections 3.10, 3.11).

Finally, we evaluate hallucination indices at the spectral level. Table 14 shows that fractional damping (α = 0.8) selectively stabilizes mid-frequency modes, shifting Mode 3 from unstable to stable. High-frequency modes remain unstable but with reduced growth, consistent with bounded dynamics observed in Experiment 4. Figure 9 confirms Lyapunov decay V(t) is monotone under fractional updates, validating the theoretical stability guarantees.

To complement the quantitative evaluation, we provide qualitative analyses on two representative real-world datasets, illustrating how F²-CommNet suppresses hallucination-prone structures and stabilizes community embeddings.

Cora: t-SNE embedding visualization. Figure 10 compares the node embeddings produced by GCN, EvolveGCN, and F²-CommNet using t-SNE. GCN and EvolveGCN exhibit scattered and overlapping clusters, indicating unstable high-frequency modes that distort community boundaries. In contrast, F²-CommNet produces compact and well-separated clusters with significantly fewer noisy points, visually confirming the suppression of hallucination artifacts predicted by the spectral analysis.

Reddit: Training stability curves. Figure 11 reports the evolution of the stability margin ρ(t) and hallucination index η_max(t) during training on Reddit. GCN and TGN exhibit strong oscillations and intermittent spikes in η_max, revealing the presence of unstable spectral modes. EvolveGCN partially mitigates this behavior but still suffers from fluctuations. F²-CommNet maintains consistently higher ρ(t) and substantially lower η_max(t) throughout training, demonstrating robust suppression of hallucination-prone eigenmodes on large-scale social networks.

Summary. Across citation and social networks, F²-CommNet consistently generates more stable, coherent, and hallucination-resistant embeddings. These qualitative results align with the theoretical predictions of the fractional damping mechanism.