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.
Highlights
Dynamical processes on networks are one of the main topics in network science (Barrat et al 2008; Castellano et al 2009; Newman 2003; 2010; Porter and Gleeson 2016)
We focus on dynamical processes that are described in terms of continuous-time Markov chains (CTMCs), other types of dynamical processes are possible (for example where the statespace is continuous (Deffuant et al 2000; Fortunato 2004; Hegselmann and Krause 2002), and/or the dynamics deterministic (Nakao and Mikhailov 2010; Rodrigues et al 2016; Ward and Grindrod 2014) or non-Markovian (Kiss et al 2015; Castro et al 2018))
In the “CTMC dynamics on networks” section we present a unified CTMC framework that captures a wide range of network dynamics, the typical summary statistics of interest and how network symmetries can be used to lump the state-space using a matrix-oriented approach
Summary
Dynamical processes on networks are one of the main topics in network science (Barrat et al 2008; Castellano et al 2009; Newman 2003; 2010; Porter and Gleeson 2016). In contrast to the use of approximate mean-field theories or simulation studies (Stoll et al 2012), recent approaches in mathematical epidemiology have focused on the exact analysis of infection spread dynamics occurring on small networks, for example to quantify the importance of nodes, in terms of outbreak size, vaccination and early infection in SIR epidemics (Holme 2017), and to compute SIS extinction times using computational algebra for all sufficiently small graphs (Holme and Tupikina 2018). By focusing on small networks, it is possible to include heterogeneous rates of infection and recovery in the context of particular applications, such as the spread of hospital-acquired infections in intensive care units (López-García 2016), and to analyse these systems in terms of a number of performance measures. The aim is usually to compute summary statistics related to the dynamical process (Economou et al 2015), instead of focusing on analysing the complete transient dynamics of the process, which are usually more complex to study (Keeling and Ross 2009)
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