Branching Process & Criticality

Galton-Watson branching process: each individual independently produces k offspring with P(k). The mean offspring m = E[k] determines fate. m < 1 (subcritical): extinction is certain. m = 1 (critical): extinction probability = 1 but with power-law survival times; avalanche sizes ~ x⁻³/². m > 1 (supercritical): survival probability = 1 - q where q is the smallest fixed point of G(s) = E[s^k]. Left panel: a single branching tree (color = generation). Right panel: log-log distribution of total population sizes across many runs — the power law at criticality is the signature of self-organized criticality in neural avalanches, earthquakes (Gutenberg-Richter), and forest fires.