72 research outputs found
A General Model of Dynamics on Networks with Graph Automorphism Lumping
In this paper we introduce a general Markov chain model of dynamical processes on networks. In this model, nodes in the network can adopt a finite number of states and transitions can occur that involve multiple nodes changing state at once. The rules that govern transitions only depend on measures related to the state and structure of the network and not on the particular nodes involved. We prove that symmetries of the network can be used to lump equivalent states in state-space. We illustrate how several examples of well-known dynamical processes on networks correspond to particular cases of our general model. This work connects a wide range of models specified in terms of node-based dynamical rules to their exact continuous-time Markov chain formulation
Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping
We propose a unified framework to represent a wide range of continuous-time discrete-state Markov processes on networks, and show how many network dynamics models in the literature can be represented in this unified framework. We show how a particular sub-set of these models, referred to here as single-vertex-transition (SVT) processes, lead to the analysis of quasi-birth-and-death (QBD) processes in the theory of continuous-time Markov chains. We illustrate how to analyse a number of summary statistics for these processes, such as absorption probabilities and first-passage times. We extend the graph-automorphism lumping approach [Kiss, Miller, Simon, Mathematics of Epidemics on Networks, 2017; Simon, Taylor, Kiss, J. Math. Bio. 62(4), 2011], by providing a matrix-oriented representation of this technique, and show how it can be applied to a very wide range of dynamical processes on networks. This approach can be used not only to solve the master equation of the system, but also to analyse the summary statistics of interest. We also show the interplay between the graph-automorphism lumping approach and the QBD structures when dealing with SVT processes. Finally, we illustrate our theoretical results with examples from the areas of opinion dynamics and mathematical epidemiology
Lifted Probabilistic Inference: An MCMC Perspective
The general consensus seems to be that lifted
inference is concerned with exploiting model
symmetries and grouping indistinguishable
objects at inference time. Since first-order
probabilistic formalisms are essentially tem-
plate languages providing a more compact
representation of a corresponding ground
model, lifted inference tends to work especially well in these models. We show that the
notion of indistinguishability manifests itself
on several dferent levels {the level of constants, the level of ground atoms (variables),
the level of formulas (features), and the level
of assignments (possible worlds). We discuss
existing work in the MCMC literature on ex-
ploiting symmetries on the level of variable
assignments and relate it to novel results in
lifted MCMC
Oscillating epidemics in a dynamic network model: stochastic and mean-field analysis
An adaptive network model using SIS epidemic propagation with link-type-dependent link activation and deletion is considered. Bifurcation analysis of the pairwise ODE approximation and the network-based stochastic simulation is carried out, showing that three typical behaviours may occur; namely, oscillations can be observed besides disease-free or endemic steady states. The oscillatory behaviour in the stochastic simulations is studied using Fourier analysis, as well as through analysing the exact master equations of the stochastic model. By going beyond simply comparing simulation results to mean-field models, our approach yields deeper insights into the observed phenomena and help better understand and map out the limitations of mean-field models
SIS epidemic propagation on hypergraphs
Mathematical modeling of epidemic propagation on networks is extended to
hypergraphs in order to account for both the community structure and the
nonlinear dependence of the infection pressure on the number of infected
neighbours. The exact master equations of the propagation process are derived
for an arbitrary hypergraph given by its incidence matrix. Based on these,
moment closure approximation and mean-field models are introduced and compared
to individual-based stochastic simulations. The simulation algorithm, developed
for networks, is extended to hypergraphs. The effects of hypergraph structure
and the model parameters are investigated via individual-based simulation
results
Perspectives on Multi-Level Dynamics
As Physics did in previous centuries, there is currently a common dream of
extracting generic laws of nature in economics, sociology, neuroscience, by
focalising the description of phenomena to a minimal set of variables and
parameters, linked together by causal equations of evolution whose structure
may reveal hidden principles. This requires a huge reduction of dimensionality
(number of degrees of freedom) and a change in the level of description. Beyond
the mere necessity of developing accurate techniques affording this reduction,
there is the question of the correspondence between the initial system and the
reduced one. In this paper, we offer a perspective towards a common framework
for discussing and understanding multi-level systems exhibiting structures at
various spatial and temporal levels. We propose a common foundation and
illustrate it with examples from different fields. We also point out the
difficulties in constructing such a general setting and its limitations
Exact maximal reduction of stochastic reaction networks by species lumping
Motivation: Stochastic reaction networks are a widespread model to describe
biological systems where the presence of noise is relevant, such as in cell
regulatory processes. Unfortu-nately, in all but simplest models the resulting
discrete state-space representation hinders analytical tractability and makes
numerical simulations expensive. Reduction methods can lower complexity by
computing model projections that preserve dynamics of interest to the user.
Results: We present an exact lumping method for stochastic reaction networks
with mass-action kinetics. It hinges on an equivalence relation between the
species, resulting in a reduced network where the dynamics of each
macro-species is stochastically equivalent to the sum of the original species
in each equivalence class, for any choice of the initial state of the system.
Furthermore, by an appropriate encoding of kinetic parameters as additional
species, the method can establish equivalences that do not depend on specific
values of the parameters. The method is supported by an efficient algorithm to
compute the largest species equivalence, thus the maximal lumping. The
effectiveness and scalability of our lumping technique, as well as the physical
interpretability of resulting reductions, is demonstrated in several models of
signaling pathways and epidemic processes on complex networks. Availability:
The algorithms for species equivalence have been implemented in the software
tool ERODE, freely available for download from https://www.erode.eu
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