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.

中文翻译：

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基于图自同构集的网络上连续时间离散状态马尔可夫过程摘要统计的精确分析

我们提出了一个统一的框架来代表网络上的各种连续时间离散状态马尔可夫过程，并展示了在这个统一的框架中可以代表多少文献中的网络动力学模型。我们展示了这些模型的特定子集（在此称为单顶点过渡（SVT）过程）如何导致连续时间马尔可夫理论中的准生死（QBD）过程分析链。我们说明了如何分析这些过程的大量汇总统计信息，例如吸收概率和首次通过时间。我们扩展了图-自同构集法[Kiss，Miller，Simon，网络流行病的数学，2017年；西蒙，泰勒，吻，J。数学。生物。62（4），2011]，通过提供此技术的面向矩阵的表示，并说明如何将其应用于网络上的各种动态过程。这种方法不仅可以用于求解系统的主方程，还可以分析感兴趣的摘要统计信息。我们还显示了在处理SVT流程时图自同构集总方法和QBD结构之间的相互作用。最后，我们用观点动态和数学流行病学领域的例子来说明我们的理论结果。