Analytical prediction of the mixed-to-pure simple phase boundary for z=3

Derive an analytical expression for the phase boundary that separates the mixed contagion regime from the pure simple contagion regime when the complex contagion threshold corresponds to z=3 in the mixed simple/complex contagion model on homogeneous activity-driven networks, where susceptible nodes adopt via simple contagion with probability p (per-contact infection probability β) or via complex contagion requiring at least z infected neighbors among m concurrent contacts.

Background

The paper studies a mixed contagion model on homogeneous activity-driven networks in which, at each activation, a susceptible node adopts either via simple contagion with probability p (per-contact infection probability β) or via complex contagion if at least z neighbors are infected among m contacts. For z=2, the authors obtain an analytical critical line p_c(β) separating regimes (simple-dominated vs. complex-dominated).

When examining higher thresholds, specifically z=3, they show that Method 1 captures the pure complex/mixed boundary, but they explicitly state the lack of an analytical prediction for the boundary separating the mixed regime from the pure simple regime, relying instead on numerical simulations. This leaves open the task of deriving that analytical boundary for z=3.