9,691 research outputs found
Inference of kinetic Ising model on sparse graphs
Based on dynamical cavity method, we propose an approach to the inference of
kinetic Ising model, which asks to reconstruct couplings and external fields
from given time-dependent output of original system. Our approach gives an
exact result on tree graphs and a good approximation on sparse graphs, it can
be seen as an extension of Belief Propagation inference of static Ising model
to kinetic Ising model. While existing mean field methods to the kinetic Ising
inference e.g., na\" ive mean-field, TAP equation and simply mean-field, use
approximations which calculate magnetizations and correlations at time from
statistics of data at time , dynamical cavity method can use statistics of
data at times earlier than to capture more correlations at different time
steps. Extensive numerical experiments show that our inference method is
superior to existing mean-field approaches on diluted networks.Comment: 9 pages, 3 figures, comments are welcom
Moment Closure - A Brief Review
Moment closure methods appear in myriad scientific disciplines in the
modelling of complex systems. The goal is to achieve a closed form of a large,
usually even infinite, set of coupled differential (or difference) equations.
Each equation describes the evolution of one "moment", a suitable
coarse-grained quantity computable from the full state space. If the system is
too large for analytical and/or numerical methods, then one aims to reduce it
by finding a moment closure relation expressing "higher-order moments" in terms
of "lower-order moments". In this brief review, we focus on highlighting how
moment closure methods occur in different contexts. We also conjecture via a
geometric explanation why it has been difficult to rigorously justify many
moment closure approximations although they work very well in practice.Comment: short survey paper (max 20 pages) for a broad audience in
mathematics, physics, chemistry and quantitative biolog
Model reduction of biochemical reactions networks by tropical analysis methods
We discuss a method of approximate model reduction for networks of
biochemical reactions. This method can be applied to networks with polynomial
or rational reaction rates and whose parameters are given by their orders of
magnitude. In order to obtain reduced models we solve the problem of tropical
equilibration that is a system of equations in max-plus algebra. In the case of
networks with nonlinear fast cycles we have to solve the problem of tropical
equilibration at least twice, once for the initial system and a second time for
an extended system obtained by adding to the initial system the differential
equations satisfied by the conservation laws of the fast subsystem. The two
steps can be reiterated until the fast subsystem has no conservation laws
different from the ones of the full model. Our method can be used for formal
model reduction in computational systems biology
Variational approximations for stochastic dynamics on graphs
We investigate different mean-field-like approximations for stochastic
dynamics on graphs, within the framework of a cluster-variational approach. In
analogy with its equilibrium counterpart, this approach allows one to give a
unified view of various (previously known) approximation schemes, and suggests
quite a systematic way to improve the level of accuracy. We compare the
different approximations with Monte Carlo simulations on a reversible
(susceptible-infected-susceptible) discrete-time epidemic-spreading model on
random graphs.Comment: 29 pages, 5 figures. Minor revisions. IOP-style
A matrix product algorithm for stochastic dynamics on networks, applied to non-equilibrium Glauber dynamics
We introduce and apply a novel efficient method for the precise simulation of
stochastic dynamical processes on locally tree-like graphs. Networks with
cycles are treated in the framework of the cavity method. Such models
correspond, for example, to spin-glass systems, Boolean networks, neural
networks, or other technological, biological, and social networks. Building
upon ideas from quantum many-body theory, the new approach is based on a matrix
product approximation of the so-called edge messages -- conditional
probabilities of vertex variable trajectories. Computation costs and accuracy
can be tuned by controlling the matrix dimensions of the matrix product edge
messages (MPEM) in truncations. In contrast to Monte Carlo simulations, the
algorithm has a better error scaling and works for both, single instances as
well as the thermodynamic limit. We employ it to examine prototypical
non-equilibrium Glauber dynamics in the kinetic Ising model. Because of the
absence of cancellation effects, observables with small expectation values can
be evaluated accurately, allowing for the study of decay processes and temporal
correlations.Comment: 5 pages, 3 figures; minor improvements, published versio
Asymptotology of Chemical Reaction Networks
The concept of the limiting step is extended to the asymptotology of
multiscale reaction networks. Complete theory for linear networks with well
separated reaction rate constants is developed. We present algorithms for
explicit approximations of eigenvalues and eigenvectors of kinetic matrix.
Accuracy of estimates is proven. Performance of the algorithms is demonstrated
on simple examples. Application of algorithms to nonlinear systems is
discussed.Comment: 23 pages, 8 figures, 84 refs, Corrected Journal Versio
The Michaelis-Menten-Stueckelberg Theorem
We study chemical reactions with complex mechanisms under two assumptions:
(i) intermediates are present in small amounts (this is the quasi-steady-state
hypothesis or QSS) and (ii) they are in equilibrium relations with substrates
(this is the quasiequilibrium hypothesis or QE). Under these assumptions, we
prove the generalized mass action law together with the basic relations between
kinetic factors, which are sufficient for the positivity of the entropy
production but hold even without microreversibility, when the detailed balance
is not applicable. Even though QE and QSS produce useful approximations by
themselves, only the combination of these assumptions can render the
possibility beyond the "rarefied gas" limit or the "molecular chaos"
hypotheses. We do not use any a priori form of the kinetic law for the chemical
reactions and describe their equilibria by thermodynamic relations. The
transformations of the intermediate compounds can be described by the Markov
kinetics because of their low density ({\em low density of elementary events}).
This combination of assumptions was introduced by Michaelis and Menten in 1913.
In 1952, Stueckelberg used the same assumptions for the gas kinetics and
produced the remarkable semi-detailed balance relations between collision rates
in the Boltzmann equation that are weaker than the detailed balance conditions
but are still sufficient for the Boltzmann -theorem to be valid. Our results
are obtained within the Michaelis-Menten-Stueckelbeg conceptual framework.Comment: 54 pages, the final version; correction of a misprint in Attachment
Dynamic message-passing approach for kinetic spin models with reversible dynamics
A method to approximately close the dynamic cavity equations for synchronous
reversible dynamics on a locally tree-like topology is presented. The method
builds on a graph expansion to eliminate loops from the normalizations of
each step in the dynamics, and an assumption that a set of auxilary
probability distributions on histories of pairs of spins mainly have
dependencies that are local in time. The closure is then effectuated by
projecting these probability distributions on -step Markov processes. The
method is shown in detail on the level of ordinary Markov processes (),
and outlined for higher-order approximations (). Numerical validations of
the technique are provided for the reconstruction of the transient and
equilibrium dynamics of the kinetic Ising model on a random graph with
arbitrary connectivity symmetry.Comment: 6 pages, 4 figure
- …