Local motif counts predict edge-level bifurcation thresholds in complex networks

We introduce a class of edge-state models in which each edge carries a scalar variable evolving under a local potential composed of an instability drive anda triangle-mediated stabilization term. Within this framework, we derive an explicit formula for the critical coupling at which each edge’s trivial fixedpoint loses local stability: Ccrit (e) = 2 (A − β) / tri (e) · g″ (0) where tri (e) is the number of triangles containing edge e and g is any even, globally confining stabilization function with g′′ (0) > 0. The threshold isstrictly decreasing in tri (e) ; edges participating in no triangle have no finite stability threshold within this model. We validate the formula on five synthetic graph families (Erd˝os–R´enyi, Barab´asi–Albert, Watts–Strogatz, stochastic block model, random regular) with N = 200 nodes, achieving 96–100% prediction accuracy. Tests on random regular graphs confirm that the triangle effect is not a proxy for degree. As an independent empirical test, we analyse the C. elegans neural connectome (297 neurons, 2148 edges): triangle-free edges have 1. 88× higherbetweenness centrality (p = 5 × 10−19), contain all 15 network bridges, and targeted removal of triangle-free edges disconnects the network after removing a single edge versus 124 ± 101 for random removal. On Kuramoto oscillator dynamics, triangle count does not significantly predict synchronization order (ρ = 0. 057, p = 0. 18), clarifying that the framework captures structural fragility rather than dynamical priority. Permutation tests yield p 2. 5).