Cascade · synthesis

One small thing, enormous consequences — three ways the world does it.

THRESHOLD

0.3 0.4 0.5 0.6 0.7 0.8 0% 25% 50% 75% 100% density d → fraction burned threshold · d_c ≈ 0.59 ★ your run site-percolation threshold · square lattice ≈ 0.5927

drag the marker across the knee

Cross a line and the same spark changes everything.

Cliffs, not ramps.

An epidemic past R = 1.

CRITICALITY

1 10 100 1k 1000 100 10 1 avalanche size — cells toppled → 1 in N drops no typical size · τ ≈ 1.3 1 in 2 drops ≥ 1 cell ★ Bak–Tang–Wiesenfeld 1987 · 64² field · τ ≈ 1.1–1.3 (approx.)

drag the marker out along the tail

Systems drift to their own edge and stay there.

Nobody set the dial.

An idea going viral.

CASCADE

0 45 90 the grid’s 180 nodes, in no particular order % of the grid dark 6 of 180 nothing on the outside says which node 1 → 1.1% dark ★ Motter–Lai 2002 · small-world grid (Watts–Strogatz 1998) · 180 nodes

drag the marker along the grid

One failure can take the whole connected system.

Nothing looks load-bearing.

A grid failing on a hot day.

One edge you cross. One a system finds by itself. One you’d never have picked.