非马尔可夫随机过程的Gillespie算法
A Gillespie algorithm for non-Markovian stochastic processes
摘要：
Gillespie算法提供了用于模拟随机动态的统计精确方法，所述随机动态被建模为离散事件的交互序列，包括生化反应或地震发生的系统、排队过程或尖峰神经元的网络以及社交网络上的流行病和意见形成过程。经验上，各种现象的事件间时间服从长尾分布。Gillespie算法及其变体或者假设泊松过程（即，指数分布的事件间时间），使用事件率的时间过程的特定函数，或用于非泊松更新过程，包括事件间时间的长尾分布的情况，但计算成本高。在本研究中，我们提出了一个创新的更新过程的拉普拉斯变换的基础上的Gillespie算法。该算法利用一类点过程表示为具有不同事件率的泊松过程的混合这一事实。该方法适用于事件间隔时间生存函数完全单调的多变量更新过程。它是一个精确的算法，比最近提出的Gillespie算法一般更新过程，这是精确的，只有在无限多个过程的限制。我们还提出了一种方法来生成事件时间序列与事件间时间的正相关性可调量。我们证明了我们的算法与网络上的流行病过程的精确模拟，发现在事件间时间的正相关性的现实量只有轻微影响的流行病动态。
英文摘要：
The Gillespie algorithm provides statistically exact methods for simulating stochastic dynamics modelled as interacting sequences of discrete events including systems of biochemical reactions or earthquake occurrences, networks of queuing processes or spiking neurons, and epidemic and opinion formation processes on social networks. Empirically, the inter-event times of various phenomena obey long-tailed distributions. The Gillespie algorithm and its variants either assume Poisson processes (i.e., exponentially distributed inter-event times), use particular functions for time courses of the event rate, or work 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. The algorithm makes use of the fact that a class of point processes is represented as a mixture of Poisson processes with different event rates. The method is applicable to multivariate renewal processes whose survival function of inter-event times is completely monotone. It is an exact algorithm and works faster than a recently proposed Gillespie algorithm for general renewal processes, which is exact only in the limit of infinitely many processes. We also propose a method to generate sequences of event times with a tunable amount of positive correlation between inter-event times. We demonstrate our algorithm with exact simulations of epidemic processes on networks, finding that a realistic amount of positive correlation in inter-event times only slightly affects the epidemic dynamics.
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