Phase Boundary of a Stochastic Watts-Threshold SIS Model on Random Networks

Complex contagion models, in which adoption requires reinforcement from multiple neighbors, have been extensively studied in the monotone (no-recovery) setting, but the phase diagram of threshold models with SIS-like recovery on networks remains unmapped. We study a stochastic Watts-threshold SIS model on Erdos-Renyi and Barabasi-Albert networks and reconstruct its extinction-persistence phase boundary in the joint parameter space of transmission rate $β$, adoption threshold $θ$, and infectious duration $d$. Using adaptive Delaunay-based sampling and weighted logistic regression on over 180,000 Monte Carlo trials, we find that: (i) the boundary is well described by a six-parameter interaction model whose structure is invariant across both topologies; (ii) the transition is sharp, with the 10-90\% extinction-probability band spanning only $Δθ\approx 0.005$-$0.008$; and (iii) the adoption threshold is the dominant parameter governing epidemic feasibility, with transmission rate and infectious duration playing secondary and asymmetric roles. The characterization provides a quantitative reference for the complex-contagion analogue of the classical SIS epidemic threshold.