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

We derive an explicit analytical formula for edge-level bifurcation thresholds in complex networks. Unlike classical spectral approaches that yield a single global transition point, our result assigns a critical coupling to each edge as a function of its local triangle count. We show that: Ccrit (e) = 2 (A − β) / tri (e) · g'' (0) where tri (e) denotes the number of triangles containing edge e and g is any even stabilization function with positive curvature at the origin. The threshold is strictly decreasing in the local triangle count, and edges with tri (e) = 0 are never stabilizable at any finite coupling. We validate the analytical prediction across five synthetic graph families (Erdős–Rényi, Barabási–Albert, Watts–Strogatz, stochastic block model, random regular), achieving 96–100% prediction accuracy and large effect sizes (Cohen’s d > 2. 5). This work introduces a heterogeneous, motif-dependent stability landscape for complex networks and transforms the classical question of global phase transition into an edge-level structural stability problem.