Nonlinear consensus on networks: equilibria, effective resistance and trees of motifs

The paper proves an effective-resistance instability criterion: at an equilibrium x*, with Jacobian J = L+ − L−, if there exist nodes i,j with r−_{ij} > r+_{ij}, then J has a positive eigenvalue on 〈1〉⊥ and x* is unstable (Theorem 3.12). This builds on the decomposition J = L+ − L− (equation (3.7)) and a Rayleigh-quotient comparison via Lemma 3.11, together with the standard linear-instability implication (Proposition 3.2) . The candidate solution proves exactly the same instability condition but by a different route: it uses the Dirichlet/minimum-energy characterisation of effective resistance (equivalent to Lemma 3.7) to construct an explicit test vector v that achieves v^T L− v = 1/r−_{ij} and ensures v^T L+ v ≥ 1/r+_{ij}, hence v^T J v > 0, and then projects to 〈1〉⊥. This is a clean alternative to the paper’s Rayleigh-quotient-maximisation proof and is mathematically sound. Minor omissions (handling the r−_{ij} = ∞ edge case explicitly and noting the need to restrict to 〈1〉⊥) are easily patched and do not affect correctness. Overall: both are correct, with different proofs. Key definitions and steps align with the paper’s framework (Definition 3.6, Lemma 3.7, equation (3.7), Theorem 3.12) .