Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping

Abstract

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.

Rare events and first passage time statistics from the energy landscape

“…4,5,38,39 We therefore examined whether the underlying database could be coarse-grained via a suitable regrouping scheme to extend the range of applicability. A number of lumping schemes have been discussed in the literature, [49][50][51][52][53][54][55][56][57]101 and community detection algorithms [102][103][104][105][106][107][108][109][110] could also be employed to partition a transition network. However, care must be taken that the resulting groups of nodes accurately reflect the metastable macrostates; otherwise the lumped transition network may not appropriately represent the dynamical properties of the original network.…”

“…Nonetheless, when eigendecomposition fails kPS is the method of choice for obtaining the FPT distribution, and hence insightful mechanistic information, for dynamical systems featuring rare events. 67 We investigated the extent to which regrouping [48][49][50][51][52][53][54][55][56][57] the states in a kinetic transition network is helpful for improving the numerical performance of eigendecomposition. Dimensionality reduction of kinetic transition networks via cluster is attractive because eigendecomposition has cubic scaling with system size, 38,39 and the dynamics of the lumped network may be more readily interpreted in terms of the key features of the underlying energy landscape.…”

“…Lumping states in the original network together is an attractive approach for reducing dimensionality. [48][49][50][51][52][53][54][55][56][57] We consider a lumping scheme that recursively regroups local minima separated by low-lying barriers below a free energy threshold. [58][59][60] If rapidly interconverting states can establish local relative equilibrium on the time scale of the rate-determining process, the intergroup rates in the coarse-grained network can be simply determined by the local equilibrium approximation (LEA) 60 while preserving the rare event of interest.…”

Optimal dimensionality reduction of Markov chains using graph transformation

“….}. 1,[61][62][63][64][65][66][67][68][69][70][71][72][73][74] To ensure that a reduced Markovian model is a valid approximation to the original system, the community structure C must appropriately characterize the metastable macrostates. Determination of C is particularly challenging in the metastable regime, where spectral methods [71][72][73]138 are numerically unstable (Sec.…”

“…Reducing the dimensionality of a Markov chain, while accurately preserving global dynamical properties of the original model, is an active area of research. 1,[61][62][63][64][65][66][67][68][69][70][71][72][73][74] Coarse-graining facilitates computational analyses that are prohibitively expensive for the original model, such as sampling of the A ← B transition path ensemble by kinetic Monte Carlo (kMC) methods. [75][76][77][78][79][80][81] Moreover, the reduced Markov chain only preserves the slowest dynamical processes and so is less ill-conditioned than the original representation.…”