mathematics//probability//stochastic process//branching process

A branching process is a stochastic process in which each event gives rise to a random number of new events, independently of the others, and it is the simplest model of how a local failure turns into a cascade: a tripped line loading its neighbours, a crashed service whose clients retry and overload the next, a neutron splitting a nucleus. The one number that matters is the **branching ratio** \(R\), the mean number of new events each event causes.

A branching process is a stochastic process in which each event gives rise to a random number of new events, independently of the others, and it is the simplest model of how a local failure turns into a cascade: a tripped line loading its neighbours, a crashed service whose clients retry and overload the next, a neutron splitting a nucleus. The one number that matters is the branching ratio RRR, the mean number of new events each event causes.

Counting the first event, a cascade has on average 1+R+R2+…1+R+R^2+\dots1+R+R2+… events, which for R<1R<1R<1 sums to

E[size]=11−R.\mathbb E[\text{size}]=\frac{1}{1-R}.E[size]=1−R1​.

At R=0.5R=0.5R=0.5 a failure brings on average one more; at 0.90.90.9, ten in all; at 0.990.990.99, a hundred. The growth is gentle and then violent as RRR approaches one, the same 1/(1−x)1/(1-x)1/(1−x) shape as the waits of an M/M/1 queue. At R=1R=1R=1 every cascade still ends, but cascade sizes spread over every scale; above one, a cascade has a positive probability of never stopping, which in a finite system means it stops only when it runs out of things to break.

The size of a cascade is set by RRR, and RRR is set by the coupling and the margins. Neighbours that run close to their limits fail when they inherit load, so RRR rises; breakers that isolate a fault, load shedding, firebreaks, spare capacity and clients that back off instead of retrying all lower it. A large blackout is explained by the ratio of the network far more than by the failure that started it.

In a network the components differ, and each failure loads specific neighbours by specific amounts. Linearizing that redistribution gives a matrix, and the role of RRR is taken by its spectral radius: above one, failures multiply on average, the stability question of a linear system asked of a grid.

The model assumes independent offspring and an inexhaustible supply of things to fail. Real cascades saturate, and their branching ratio changes as they spread (a grid that has lost half its lines carries the rest closer to their limits), so the model gives the threshold and the shape, while the sizes of real events come from simulation (cascading failure).

The same mathematics carries other names: the reproduction number of an epidemic, the multiplication factor of a nuclear reactor (critical at exactly one, the regime a reactor is held in on purpose), the self-excitation of a Hawkes process of events that trigger events.