Estimating Parameters of Large CTMP from Single Trajectory with Application to Stochastic Network Epidemics Models

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.