[Paper Review] A Gillespie algorithm for non-Markovian stochastic processes: Laplace transform approach

Naoki Masuda, Luis E. C. Rocha|arXiv (Cornell University)|Jan 7, 2016

Complex Network Analysis Techniques3 citations

TL;DR

This paper proposes a novel Gillespie algorithm for non-Markovian stochastic processes using Laplace transforms to model renewal processes with completely monotone survival functions, enabling efficient simulation of long-tailed inter-event time distributions. The method outperforms existing approaches in speed and accurately captures positive correlation in inter-event times, with minimal quantitative impact on epidemic dynamics even at empirically observed correlation levels.

ABSTRACT

The Gillespie algorithm provides statistically exact methods to simulate stochastic dynamics modelled as interacting sequences of discrete events including systems of biochemical reactions or earthquakes, networks of queuing processes or spiking neurons, and epidemic and opinion formation processes on social networks. Empirically, inter-event times of various human activities, in particular human communication, and some natural phenomena are often distributed according to long-tailed distributions. The Gillespie algorithm and its extant variants either assume the Poisson process, which produces exponentially distributed inter-event times, not long-tailed distributions, assume particular functional forms for time courses of the event rate, or works for non-Poissonian renewal processes including the case of long-tailed distributions of inter-event times but at a high computational cost. In the present study, we propose an innovative Gillespie algorithm for renewal processes on the basis of the Laplace transform. It uses the fact that a class of point processes is represented as a mixture of Poisson processes with different event rates. The method allows renewal processes whose survival function of inter-event times is completely monotone functions and works faster than a recently proposed Gillespie algorithm for general renewal processes. We also propose a method to generate sequences of event times with a given distribution of inter-event times and a tunable amount of positive correlation between inter-event times. We demonstrate our algorithm with exact simulations of epidemic processes on networks. We find that positive correlation in inter-event times modulates dynamics but in a quantitatively minor way with the amount of positive correlation comparable with empirical data.

Motivation & Objective

• To develop an efficient and statistically exact algorithm for simulating non-Markovian stochastic processes with long-tailed inter-event time distributions.
• To overcome the limitations of existing Gillespie algorithms that assume exponential inter-event times or require high computational costs for general renewal processes.
• To enable simulation of event sequences with tunable positive correlation between inter-event times while preserving the target marginal distribution.
• To demonstrate the method’s effectiveness in modeling epidemic processes on networks with realistic human interaction patterns.

Proposed method

• The method represents a class of point processes as a mixture of Poisson processes with stochastically varying event rates, leveraging the Laplace transform of the inter-event time distribution.
• It uses the property that completely monotone survival functions correspond to mixtures of exponential distributions, enabling exact sampling via inverse transform sampling on the mixture parameter.
• The algorithm samples event times by first drawing a rate from the mixing distribution and then generating an exponential event time with that rate.
• It avoids the high computational cost of prior methods by exploiting the analytical structure of completely monotone functions through Laplace transform inversion.
• The approach allows for the generation of correlated inter-event times by introducing dependence between successive rates in the mixture process.
• The method is validated through exact simulations of epidemic spreading on networks under various inter-event time distributions and correlation structures.

Experimental results

Research questions

• RQ1 Can a Gillespie algorithm be designed to efficiently simulate non-Markovian processes with long-tailed inter-event time distributions without relying on Poisson assumptions?
• RQ2 How does positive correlation between inter-event times affect the dynamics of epidemic processes on networks?
• RQ3 Can the proposed method generate event sequences with a given inter-event time distribution and tunable correlation while maintaining statistical exactness?
• RQ4 How does the performance of the new algorithm compare to existing methods in terms of computational efficiency for general renewal processes?
• RQ5 To what extent does correlation in human interaction patterns alter epidemic spreading dynamics in network models?

Key findings

• The proposed algorithm achieves faster simulation speeds than existing Gillespie algorithms for general renewal processes, particularly when the survival function of inter-event times is completely monotone.
• The method enables exact simulation of event sequences with a specified inter-event time distribution and controllable positive correlation between successive intervals.
• Positive correlation in inter-event times was found to modulate epidemic dynamics only quantitatively, even at correlation levels comparable to empirical data.
• The model successfully captures realistic human communication patterns through the combination of long-tailed inter-event time distributions and tunable correlation.
• The Laplace transform-based approach provides a general and computationally efficient framework for simulating non-Poissonian renewal processes in complex systems.