F2-CommNet: Fourier–Fractional neural networks with Lyapunov stability guarantees for hallucination-resistant community detection

Community detection is a crucial task in network research, applicable to social systems, biology, cybersecurity, and knowledge graphs. Recent advancements in graph neural networks (GNNs) have exhibited significant representational capability; yet, they frequently experience instability and erroneous clustering, often referred to as "hallucinations." These artifacts stem from sensitivity to high-frequency eigenmodes, over-parameterization, and noise amplification, undermining the robustness of learned communities. To mitigate these constraints, we present F²-CommNet, a Fourier-Fractional neural framework that incorporates fractional-order dynamics, spectrum filtering, and Lyapunov-based stability analysis. The fractional operator implements long-memory dampening that mitigates oscillations, whereas Fourier spectral projections selectively attenuate eigenmodes susceptible to hallucination. Theoretical analysis delineates certain stability criteria under Lipschitz non-linearities and constrained disturbances, resulting in a demonstrable expansion of the Lyapunov margin. Experimental validation on synthetic and actual networks indicates that F²-CommNet reliably diminishes hallucination indices, enhances stability margins, and produces interpretable communities in comparison to integer-order GNN baselines. This study integrates fractional calculus, spectral graph theory, and neural network dynamics, providing a systematic method for hallucination-resistant community discovery.