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Register a new userNon-universal power law distribution of intensities of the self-excited Hawkes process: a field-theoretical approach
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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 (PDF) of the intensities close to the critical point $n=1$ of the Hawkes process, with a non-universal exponent, function of the background intensity $ u_0$ of the Hawkes intensity, the average time scale of the memory kernel and the branching ratio $n$. Our theoretical predictions are confirmed by numerical simulations.
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A field theoretical framework is developed for the Hawkes self-excited point process with arbitrary memory kernels by embedding the original non-Markovian one-dimensional dynamics onto a Markovian infinite-dimensional one. The corresponding Langevin dynamics of the field variables is given by stochastic partial differential equations that are Markovian. This is in contrast to the Hawkes process, which is non-Markovian (in general) by construction as a result of its (long) memory kernel. We derive the exact solutions of the Lagrange-Charpit equations for the hyperbolic master equations in the Laplace representation in the steady state, close to the critical point of the Hawkes process. The critical condition of the original Hawkes process is found to correspond to a transcritical bifurcation in the Lagrange-Charpit equations. We predict a power law scaling of the PDF of the intensities in an intermediate asymptotics regime, which crosses over to an asymptotic exponential function beyond a characteristic intensity that diverges as the critical condition is approached. We also discuss the formal relationship between quantum field theories and our formulation. Our field theoretical framework provides a way to tackle complex generalisation of the Hawkes process, such as nonlinear Hawkes processes previously proposed to describe the multifractal properties of earthquake seismicity and of financial volatility.
The origin(s) of the ubiquity of Zipfs law (an inverse power law form for the probability density function (PDF) with exponent $1+1$) is still a matter of fascination and investigation in many scientific fields from linguistic, social, economic, computer sciences to essentially all natural sciences. In parallel, self-excited dynamics is a prevalent characteristic of many systems, from seismicity, financial volatility and financial defaults, to sociology, consumer behaviors, computer sciences, The Internet, neuronal discharges and spike trains in biological neuron networks, gene expression and even criminology. Motivated by financial and seismic modelling, we bring the two threads together by introducing a general class of nonlinear self-excited point processes with fast-accelerating intensities as a function of tension. Solving the corresponding master equations, we find that a wide class of such nonlinear Hawkes processes have the PDF of their intensities described by Zipfs law on the condition that (i) the intensity is a fast-accelerating function of tension and (ii) the distribution of the point fertilities is symmetric. This unearths a novel mechanism for Zipfs law, providing a new understanding of its ubiquity.
Both theoretical and applied economics have a great deal to say about many aspects of the firm, but the literature on the extinctions, or demises, of firms is very sparse. We use a publicly available data base covering some 6 million firms in the US and show that the underlying statistical distribution which characterises the frequency of firm demises - the disappearances of firms as autonomous entities - is closely approximated by a power law. The exponent of the power law is, intriguingly, close to that reported in the literature on the extinction of biological species.
To describe and analyze the dynamics of Self-Organized Criticality (SOC) systems, a four-state continuous-time Markov model is proposed in this paper. Different to computer simulation or numeric experimental approaches commonly employed for explaining the power law in SOC, in this paper, based on this Markov model, using E.T.Jayness Maximum Entropy method, we have derived a mathematical proof on the power law distribution for the size of these events. Both this Makov model and the mathematical proof on power law present a new angle on the universality of power law distributions, they also show that the scale free property exists not necessary only in SOC system, but in a class of dynamical systems which can be modelled by the proposed Markov model.
We introduce a persistent random walk model with finite velocity and self-reinforcing directionality, which explains how exponentially distributed runs self-organize into truncated Levy walks observed in active intracellular transport by Chen et. al. [textit{Nat. mat.}, 2015]. We derive the non-homogeneous in space and time, hyperbolic PDE for the probability density function (PDF) of particle position. This PDF exhibits a bimodal density (aggregation phenomena) in the superdiffusive regime, which is not observed in classical linear hyperbolic and Levy walk models. We find the exact solutions for the first and second moments and criteria for the transition to superdiffusion.
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