I consider a graph that changes randomly over (discrete) time denoted by $(G_t)_{t=0}^{\infty}$ where I call $G_0=(V_0, E_0)$, $V_0$ being the vertex and $E_0$ the edge set my initial condition where $\mathrm{card}(V_0)=n$.
Assuming that the dynamics preserve the vertex set and the number of edges, i.e., $V_t=V_0$ and $\mathrm{card}(E_t)=\mathrm{card}(E_0)=m$ for all $t$, and that $G_0$ is a $\mathcal{G}(n,m)$ graph can I say that $G_t$ at any time $t$ is also a $\mathcal{G}(n,m)$ graph and therefore apply results from theory of $\mathcal{G}(n,m)$ graphs? In particular, I would like to employ results on the largest connected component.
The dynamics itself consists of deleting and rewiring edges.
Right now I am torn between the fact that my dynamics give me a "trajectory" in $\mathcal{G}(n,m)$ and every $G_t$ could therefore be seen as a $\mathcal{G}(n,m)$ graph but $G_t$ is not just some random graph drawn from $\mathcal{G}(n,m)$ but the result of the dynamics started at $G_0$. I am not sure which one of the two is the right view on $G_t$.
Thanks in advance for your help!