Hawkes Branching Renewal Criticality

The Hawkes process is a self-exciting point process where each event increases the likelihood of subsequent events. It can be viewed as a branching process with an exogenous immigration baseline rate μ and an expected descendant cluster size governed by the branching ratio η = α / β. It is essential to differentiate between the dynamics of a single exogenous seed at time 0 (with no further immigrants in its descendant cluster) and a continuous immigration process.

Worked Example

Let us compute the expected properties of a single seed cluster and the stationary mean of the continuous immigration process given Immigration μ = 1, Kernel decay β = 1, Branching ratio η = 0.5, Time t = 3, and Generations G = 5.

The kernel amplitude is α = η × β = 0.5 × 1 = 0.5.
The expected post-seed descendant intensity at time t is m(t) = α exp((α - β) t) = 0.5 exp((0.5 - 1) × 3) = 0.5 exp(-1.5) ≈ 0.111565.
The expected cluster count by time t including the seed is M(t) = 1 + (α)/(α - β) (exp((α - β) t) - 1) = 1 + 0.5/(-0.5) (exp(-1.5) - 1) = 1 - (0.22313 - 1) ≈ 1.77687.
The expected total mass through generation G is \sum_(g=0)^5 η^g = 1 + 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 = 1.96875.
The stationary immigration mean is (μ)/(1 - η) = 1/(1 - 0.5) = 2.0.

Experiment

Using the interactive module Hawkes Branching Renewal Criticality:

Select a value for the Immigration baseline rate.
Select the Kernel decay parameter to control how fast the excitation decays.
Adjust the Branching ratio to see its effect on subcritical, critical, and supercritical regimes.
Select the evaluation Time for transient observations.
Set the Generations to compute the mass through generation G.

Limitations

The presented metrics are purely expectations and do not represent deterministic event counts or a specific random simulation of the point process. Furthermore, when analyzing supercritical behaviors (η ≥ 1), the finite-time expected values for a seed cluster remain finite and valid, whereas the continuous positive immigration process lacks a finite stationary mean. Do not add the stationary baseline to a transient seed curve in a supercritical or critical model where stationarity is undefined. If the baseline is strictly zero, the stationary mean is simply an empty zero-rate baseline, although a separately seeded cluster can still grow.

References

For the mathematical framework of Hawkes processes as branching processes, refer to:

Hawkes, A. G., & Oakes, D. (1974). A Cluster Process Representation of a Self-Exciting Process. Journal of Applied Probability, 11(3), 493-503.

Hawkes Branching Renewal Criticality