Fiedler Algebraic Connectivity (λ₂) as a Structural Health Index for Legged Robot Joint Networks: Empirical Validation Across 1,000 Graph Topologies and Real-Physics dm_control Simulation

We investigate Fiedler algebraic connectivity (λ₂) as a structural health index (SHI) for legged robot joint networks. Across 1,000 randomly generated graph topologies (5 families), λ₂ achieves the highest Spearman correlation with network collapse resistance (ρ=0.912, p<10⁻³⁰⁰), outperforming mean degree (ρ=0.856), betweenness centrality (ρ=0.822), and clustering coefficient (ρ=0.735). In legged robot subgraphs (n=161), ρ=0.865. In real-physics dm_control quadruped walk simulations (100 episodes, MuJoCo 3.3.7), we find that λ₂ — when constructed from instantaneous actuator forces — does not precede velocity, power, or stability signals as an early warning metric. We provide an honest mechanistic explanation and conclude that λ₂ primary value is as a topology-level diagnostic of joint network integrity (structural health index), not as a real-time kinematic early warning system. Part of the AGSA (Algebraic Geometric Signal Analysis) framework.