Graph dynamical systems (GDS) model dynamic processes on a (static) graph. Stochastic GDS has been used for network-based epidemics models such as the contact process and the reversible contact process. In this paper, we consider stochastic GDS that are also continuous-time Markov processes (CTMP), whose transition rates are linear functions of some dynamics parameters $\theta$ of interest (i.e., healing, exogeneous, and endogeneous infection rates). Our goal is to estimate $\theta$ from a single, finite-time, continuously observed trajectory of the CTMP. Parameter estimation of CTMP is challenging when the state space is large; for GDS, the number of Markov states are \emph{exponential} in the number of nodes of the graph. We showed that holding classes (i.e., Markov states with the same holding time distribution) give efficient partitions of the state space of GDS. We derived an upperbound on the number of holding classes for the contact process, which is polynomial in the number of nodes. We utilized holding classes to solve a smaller system of linear equations to find $\theta$. Experimental results show that finding reasonable results can be achieved even for short trajectories, particularly for the contact process. In fact, trajectory length does not significantly affect estimation error.

翻译：图动力系统（GDS）对（静态）图上的动态过程进行建模。随机GDS已被用于基于网络的流行病模型，如接触过程和可逆接触过程。本文考虑一类同时也是连续时间马尔可夫过程（CTMP）的随机GDS，其转移率是关于某些感兴趣动力学参数$\theta$（即治愈率、外生感染率和内生感染率）的线性函数。我们的目标是从CTMP的单条有限时间连续观测轨迹中估计$\theta$。当状态空间较大时，CTMP的参数估计具有挑战性；对于GDS，马尔可夫状态的数量与图节点数呈指数关系。我们证明，保持类（即具有相同保持时间分布的马尔可夫状态）能对GDS的状态空间进行有效划分。我们推导了接触过程保持类数量的上界，该上界关于节点数为多项式阶。利用保持类，我们求解一个更小的线性方程组以求得$\theta$。实验结果表明，即使对于短轨迹（尤其是接触过程），也能获得合理的结果。实际上，轨迹长度对估计误差影响不显著。