Nonuniversal Power Law Distribution of Intensities of the Self-Excited Hawkes Process: A Field-Theoretical Approach

The Hawkes self-excited point process provides an efficient representation of the bursty intermittent dynamics of many physical, biological, geological, and economic systems. By expressing the probability for the next event per unit time (called “intensity”), say of an earthquake, as a sum over all past events of (possibly) long-memory kernels, the Hawkes model is non-Markovian. By mapping the Hawkes model onto stochastic partial differential equations that are Markovian, we develop a field theoretical approach in terms of probability density functionals. Solving the steady-state equations, we predict a power law scaling of the probability density function of the intensities close to the critical point $n=1$ of the Hawkes process, with a nonuniversal exponent, function of the background intensity ${\nu }_0$ of the Hawkes intensity, the average timescale of the memory kernel and the branching ratio $n$. Our theoretical predictions are confirmed by numerical simulations.

Figure 1

Mapping from (a) a non-Markovian description of the total intensity $\widehat{\nu }(t)$, obeying the original Hawkes process (1), to (b) a Markovian description of the excess intensity $\{\widehat{z}(t,x){\}}_x$ on the auxiliary field variable $x\in {\mathit{R}}_{+}$, which obeys an SPDE (5). Notice the shocks (6) impact all excess intensity $\{\widehat{z}(t,x){\}}_x$ for all different variables $x$ simultaneously (orange plane at $t={\widehat{t}}_i$). One can observe the dependence $n(x)/x$ of the jump size given in Eq. (5) along the field variable $x$ (blue chain arrow). Along the time axis, the exponential decay $\sim e^{-t/x}$ is shown to be faster (slower) for smaller (larger) $x$ (red broken arrow). The sample trajectory was generated with the branching ratio density $n(x)=c_{\mathrm{ini}}(x-x_{\mathrm{ini}})+n_{\mathrm{ini}}$ if $x\in [x_{\mathrm{ini}},x_{\mathrm{fin}}]$ [$n(x)=0$ otherwise], with ${\nu }_0=0.05$, $x_{\mathrm{ini}}=0.5$, $x_{\mathrm{fin}}=10.0$, $c_{\mathrm{ini}}=0.0044$, $n_{\mathrm{ini}}=0.021$, $\mathrm{\Delta }x=(x_{\mathrm{fin}}-x_{\mathrm{ini}})/200$, and $\mathrm{\Delta }t=0.006$.

Figure 2

Numerical steady state PDFs of the Hawkes intensity $\widehat{\nu }$ for the double exponential case with $({\tau }_1,{\tau }_2)=(1,3)$, $(n_1,n_2)=(0.5,0.499)$ or $(n_1,n_2)=(0.5,0.49)$, near the critical point. (a) Background intensity ${\nu }_0=0.01$, leading to the power law exponent 0.96. (b) ${\nu }_0=0.1$, leading to the power law exponent 0.6. (c) ${\nu }_0=0.75$, leading to the negative (i.e., growing PDF) power law exponent $-2.0$. Here the sampling time interval and total sampling time are $dt=0.001$ and $T_{\mathrm{tot}}=10000$ from the initial condition $\widehat{z}(0)=0$. The initial 10% of the sample was discarded from the statistics for initialization.

Figure 3

Phase space description of the dynamical system following the LC equations. (a) The one-dimensional velocity field is defined by $V(s)≔ds/dl=-\mathcal{H}(s)$. For subcriticality ($n<1$, top), there are two fixed points at $s=0$ (attractor) and $s=\mu <0$ (repeller). The repeller merges with the attractor at criticality ($n=1$, bottom), which is a consistent picture of transcritical bifurcations. (b) The two-dimensional velocity field is defined by $\mathit{V}(\mathit{s})≔d\mathit{s}/dl$ with $\mathit{s}≔(s_1,s_2)$. By linearization $\mathit{V}(\mathit{s})\simeq -\mathit{H}\mathit{s}$, the eigenvectors ${\mathit{e}}_1$ and ${\mathit{e}}_2$ are introduced by $\mathit{H}{\mathit{e}}_i={\lambda }_i{\mathit{e}}_i$ with $0\le {\lambda }_1<{\lambda }_2$. For subcriticality ($n<1$, left), the origin $\mathit{s}=\mathbf{0}$ is the stable attractor. At criticality ($n=1$, right), the origin becomes marginal in terms of the linear stability (${\lambda }_1=0$): a repeller merges with the attractor along the ${\mathit{e}}_1$ (i.e., a transcritical bifurcation).