@gartner-88; @sznitman-91; @benaim-leboudec-08]. Suppose that the initial conditions $\{\mu_N(0)\}_{N \geq 1}$ converge weakly to a deterministic measure $\nu \in M_1(\mathcal{Z})$. Then for any fixed $T > 0$, the empirical measure process $(\mu_N(t), 0 \leq t \leq T)$ converges in $D([0,T],M_1(\mathcal{Z}))$ to the solution to the ODE $$\begin{aligned}
\dot{\mu}(t) = \Lambda_{\mu(t)}^* \mu(t), \, 0 \leq t \leq T, \, \mu(0) = \nu,
\label{eqn:MVE}\end{aligned}$$ where, for any $\xi \in M_1(\mathcal{Z})$, $\Lambda_{\xi}$ denotes the $|\mathcal{Z}|\times |\mathcal{Z}|$ rate matrix when the empirical measure is $\xi$ (i.e. $\Lambda_\xi(z,z^\prime)= \lambda_{z,z^\prime}(\xi)$ for $(z,z^\prime) \in \mathcal{E}$, $\Lambda_{\xi}(z,z^\prime) = 0$ when $(z,z^\prime) \notin \mathcal{E}$, and $\Lambda_{\xi}(z,z) = -\sum_{z^\prime \neq z} \lambda_{z,z^\prime}(\xi)$ for $z \in \mathcal{Z}$), $\Lambda^*_\xi$ denotes its transpose, and $D([0,T],M_1(\mathcal{Z}))$ denotes the space of $M_1(\mathcal{Z})$-valued càdlàg functions on $[0,T]$ equipped with the Skorohod-$J_1$ topology (we assume that all paths are left continuous at $T$). The above ODE is referred to as the McKean-Vlasov equation. The assumption \[assm:a2\] on the Lipschitz continuity of the transition functions $\lambda_{z,z^\prime}$ implies that the above ODE is well-posed. Therefore, for large values of $N$, the above mean-field convergence result suggests that one can approximate the trajectory of the empirical measure process $\mu_N$ using the McKean-Vlasov equation, and hence, one can study properties of 