1,483 research outputs found
Simulating non-Markovian stochastic processes
We present a simple and general framework to simulate statistically correct
realizations of a system of non-Markovian discrete stochastic processes. We
give the exact analytical solution and a practical an efficient algorithm alike
the Gillespie algorithm for Markovian processes, with the difference that now
the occurrence rates of the events depend on the time elapsed since the event
last took place. We use our non-Markovian generalized Gillespie stochastic
simulation methodology to investigate the effects of non-exponential
inter-event time distributions in the susceptible-infected-susceptible model of
epidemic spreading. Strikingly, our results unveil the drastic effects that
very subtle differences in the modeling of non-Markovian processes have on the
global behavior of complex systems, with important implications for their
understanding and prediction. We also assess our generalized Gillespie
algorithm on a system of biochemical reactions with time delays. As compared to
other existing methods, we find that the generalized Gillespie algorithm is the
most general as it can be implemented very easily in cases, like for delays
coupled to the evolution of the system, where other algorithms do not work or
need adapted versions, less efficient in computational terms.Comment: Improvement of the algorithm, new results, and a major reorganization
of the paper thanks to our coauthors L. Lafuerza and R. Tora
Temporal Gillespie algorithm: Fast simulation of contagion processes on time-varying networks
Stochastic simulations are one of the cornerstones of the analysis of
dynamical processes on complex networks, and are often the only accessible way
to explore their behavior. The development of fast algorithms is paramount to
allow large-scale simulations. The Gillespie algorithm can be used for fast
simulation of stochastic processes, and variants of it have been applied to
simulate dynamical processes on static networks. However, its adaptation to
temporal networks remains non-trivial. We here present a temporal Gillespie
algorithm that solves this problem. Our method is applicable to general Poisson
(constant-rate) processes on temporal networks, stochastically exact, and up to
multiple orders of magnitude faster than traditional simulation schemes based
on rejection sampling. We also show how it can be extended to simulate
non-Markovian processes. The algorithm is easily applicable in practice, and as
an illustration we detail how to simulate both Poissonian and non-Markovian
models of epidemic spreading. Namely, we provide pseudocode and its
implementation in C++ for simulating the paradigmatic
Susceptible-Infected-Susceptible and Susceptible-Infected-Recovered models and
a Susceptible-Infected-Recovered model with non-constant recovery rates. For
empirical networks, the temporal Gillespie algorithm is here typically from 10
to 100 times faster than rejection sampling.Comment: Minor changes and updates to reference
A gillespie algorithm for non-markovian stochastic processes
The Gillespie algorithm provides statistically exact methods for simulating stochastic dynamics modeled 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
Practical and scalable simulations of non-Markovian stochastic processes
Discrete stochastic processes are widespread in natural systems with many
applications across physics, biochemistry, epidemiology, sociology, and
finance. While analytic solutions often cannot be derived, existing simulation
frameworks can generate stochastic trajectories compatible with the dynamical
laws underlying the random phenomena. However, most simulation algorithms
assume the system dynamics are memoryless (Markovian assumption), under which
assumption, future occurrences only depend on the present state of the system.
Mathematically, the Markovian assumption models inter-event times as
exponentially distributed variables, which enables the exact simulation of
stochastic trajectories using the seminal Gillespie algorithm. Unfortunately,
the majority of stochastic systems exhibit properties of memory, an inherently
non-Markovian attribute. Non-Markovian systems are notoriously difficult to
investigate analytically, and existing numerical methods are computationally
costly or only applicable under strong simplifying assumptions, often not
compatible with empirical observations.To address these challenges, we have
developed the Rejection-based Gillespie algorithm for non-Markovian Reactions
(REGIR), a general and scalable framework to simulate non-Markovian stochastic
systems with arbitrary inter-event time distributions. REGIR can achieve
arbitrary user-defined accuracy while maintaining the same asymptotic
computational complexity as the Gillespie algorithm. We illustrate REGIR's
modeling capabilities in three important biochemical systems, namely microbial
growth dynamics, stem cell differentiation, and RNA transcription. In all three
cases, REGIR efficiently models the underlying stochastic processes and
demonstrates its utility to accurately investigate complex non-Markovian
systems. The algorithm is implemented as a python library REGIR
Stochastic Wilson-Cowan models of neuronal network dynamics with memory and delay
We consider a simple Markovian class of the stochastic Wilson-Cowan type
models of neuronal network dynamics, which incorporates stochastic delay caused
by the existence of a refractory period of neurons. From the point of view of
the dynamics of the individual elements, we are dealing with a network of
non-Markovian stochastic two-state oscillators with memory which are coupled
globally in a mean-field fashion. This interrelation of a higher-dimensional
Markovian and lower-dimensional non-Markovian dynamics is discussed in its
relevance to the general problem of the network dynamics of complex elements
possessing memory. The simplest model of this class is provided by a
three-state Markovian neuron with one refractory state, which causes firing
delay with an exponentially decaying memory within the two-state reduced model.
This basic model is used to study critical avalanche dynamics (the noise
sustained criticality) in a balanced feedforward network consisting of the
excitatory and inhibitory neurons. Such avalanches emerge due to the network
size dependent noise (mesoscopic noise). Numerical simulations reveal an
intermediate power law in the distribution of avalanche sizes with the critical
exponent around -1.16. We show that this power law is robust upon a variation
of the refractory time over several orders of magnitude. However, the avalanche
time distribution is biexponential. It does not reflect any genuine power law
dependence
Stochastic processes with distributed delays: chemical Langevin equation and linear-noise approximation
We develop a systematic approach to the linear-noise approximation for
stochastic reaction systems with distributed delays. Unlike most existing work
our formalism does not rely on a master equation, instead it is based upon a
dynamical generating functional describing the probability measure over all
possible paths of the dynamics. We derive general expressions for the chemical
Langevin equation for a broad class of non-Markovian systems with distributed
delay. Exemplars of a model of gene regulation with delayed auto-inhibition and
a model of epidemic spread with delayed recovery provide evidence of the
applicability of our results.Comment: 21 pages, 7 figure
Optimized Gillespie algorithms for the simulation of Markovian epidemic processes on large and heterogeneous networks
Numerical simulation of continuous-time Markovian processes is an essential
and widely applied tool in the investigation of epidemic spreading on complex
networks. Due to the high heterogeneity of the connectivity structure through
which epidemics is transmitted, efficient and accurate implementations of
generic epidemic processes are not trivial and deviations from statistically
exact prescriptions can lead to uncontrolled biases. Based on the Gillespie
algorithm (GA), in which only steps that change the state are considered, we
develop numerical recipes and describe their computer implementations for
statistically exact and computationally efficient simulations of generic
Markovian epidemic processes aiming at highly heterogeneous and large networks.
The central point of the recipes investigated here is to include phantom
processes, that do not change the states but do count for time increments. We
compare the efficiencies for the susceptible-infected-susceptible, contact
process and susceptible-infected-recovered models, that are particular cases of
a generic model considered here. We numerically confirm that the simulation
outcomes of the optimized algorithms are statistically indistinguishable from
the original GA and can be several orders of magnitude more efficient.Comment: 12 pages, 9 figure
Markovian Dynamics on Complex Reaction Networks
Complex networks, comprised of individual elements that interact with each
other through reaction channels, are ubiquitous across many scientific and
engineering disciplines. Examples include biochemical, pharmacokinetic,
epidemiological, ecological, social, neural, and multi-agent networks. A common
approach to modeling such networks is by a master equation that governs the
dynamic evolution of the joint probability mass function of the underling
population process and naturally leads to Markovian dynamics for such process.
Due however to the nonlinear nature of most reactions, the computation and
analysis of the resulting stochastic population dynamics is a difficult task.
This review article provides a coherent and comprehensive coverage of recently
developed approaches and methods to tackle this problem. After reviewing a
general framework for modeling Markovian reaction networks and giving specific
examples, the authors present numerical and computational techniques capable of
evaluating or approximating the solution of the master equation, discuss a
recently developed approach for studying the stationary behavior of Markovian
reaction networks using a potential energy landscape perspective, and provide
an introduction to the emerging theory of thermodynamic analysis of such
networks. Three representative problems of opinion formation, transcription
regulation, and neural network dynamics are used as illustrative examples.Comment: 52 pages, 11 figures, for freely available MATLAB software, see
http://www.cis.jhu.edu/~goutsias/CSS%20lab/software.htm
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