Probability mass of nodes over directed graphs

Question

Consider a Markov Chain such that the transition matrix $A$ formed is doubly stochastic and the associated undirected graph $G=(V,E)$ is strongly connected, irreducible and aperiodic. $V$ represents the vertex set and $E$ represents the edge set. The probability of a state in this graph can be easily calculated by looking at the degree of nodes. Suppose the probability of each node was $\alpha$. Now, if I construct a directed graph $G'$=$(V',E')$ by augmenting $G$ (adding extra nodes and edges) in such a way that the degree of any node $i \in V$ doesn't change. Also suppose the transition matrix $A^*$ associated with $G'$ is row stochastic strongly connected, aperiodic, positive recurrent and irreducible Does this imply the probability mass of all the nodes which were originally present, i.e. $i \in V$ will still be equal? I understand that it will be less than $\alpha$ in this case because the mass will be distributed in over augmented graph but will the original nodes (i.e, $i \in V$) still have the same probability mass? If yes, why?