Bifurcation-Based Predictive Modelling of Socio-Political Dynamics

5) First-passage approximations & analytical shortcuts

If running full ensembles is costly, approximate MFPT using linearized dynamics around current state:

For the linearized SDE near current HH, use Ornstein-Uhlenbeck approximations to estimate MFPT to threshold HcH_c (Kramers' formula analog). This yields closed-form approximations for small noise and near-linear drift.
Use these approximations for rapid nowcasts; reserve ensembles for daily/weekly updates.

6) Visualization & products (operational)

Build a small dashboard with:

1. Emergence probability curve --- plot pem(t)p_{\text{em}}(t) for horizon 0--90 days with 95% credible bands.

2. MFPT & hazard panel --- show MFPT and instantaneous hazard rate h(t).

3. Actor probability heatmap --- table/heatmap of pi(t)p_i(t) over time (rows: actors, columns: 7/30/90 day horizons).

4. Ensemble fan charts --- H(t) and P(t) percentile bands to show dispersion.

5. Early-warning triggers --- flags when pem(30d)>p_{\text{em}}(30d) > threshold (e.g., 30%) or when any pi(30d)>0.3p_i(30d) > 0.3.

6. Drill-down for signals --- show which components (social/media/organizational/fund) most contributed to recent rise in HH.