Deterministic Spectral Residual Inference for Swarm Interaction Networks: A DSFB Framework for Structural Phase Stability

To the best of our knowledge, this is one of the first structured frameworks that integratesspectral residual drift, slew, and deterministic envelopes as structural forensic signals with finite-time detectability conditions and perturbation bounds, supporting trust-gated LT (t) and earlyanomaly certificates in swarm interaction networks.Swarm systems and active matter models are typically analyzed through either deterministicdynamical equations with assumed structure or stochastic statistical mechanics frameworksemphasizing ensemble behavior and noise-driven phase transitions. This paper introduces acomplementary perspective: deterministic structural inference of swarm interaction dynamicsthrough spectral residual monitoring.We model the swarm interaction network as a time-varying graph with Laplacian operatorL(t), whose spectral structure governs collective coordination, consensus, synchronization, andfragmentation behavior. Rather than estimating agent states probabilistically, the proposed ap-proach monitors residual dynamics of spectral quantities—particularly the algebraic connectivityλ2 (L)—through deterministic residual envelopes, drift and slew diagnostics, and trust-weightedinteraction interpretation inspired by the Deterministic Structural Forensic Bayesian (DSFB)paradigm.While the present exposition centers on algebraic connectivity as the primary scalar di-agnostic, the framework extends naturally to multi-eigenvalue residual stacks and mode-shapeconsistency measures, supporting richer structural monitoring across multiple interaction scales.The framework provides mechanisms for early detection of coordination degradation, connec-tivity collapse, and structural anomalies in multi-agent systems. By tracking deviations betweenpredicted and observed spectral dynamics, the method can anticipate collective instability priorto macroscopic swarm failure. We formalize spectral residual operators, envelope bounds, de-terministic spectral predictors, trust updates, and structural anomaly certificates. Theoreticalresults connect spectral residual drift to impending coordination loss, establish finite-time de-tectability conditions, and provide perturbation bounds relating spectral residual magnitude totopology change.Simulation scenarios based on Vicsek-style alignment systems and consensus dynamics demon-strate that spectral residual monitoring detects interaction degradation, adversarial agents, and1fragmentation precursors with consistent early-warning behavior. The results position determin-istic spectral residual inference as a structural diagnostic layer bridging swarm robotics, controltheory, and active matter physics. Rather than replacing existing control or statistical frame-works, the approach introduces a complementary structural inference perspective emphasizingcollective modal health and interaction topology integrity.