Given a strongly connected digraph $G$, consider its adjacency matrix $A \in \mathbb{R}^n$ and out-degree diagonal matrix $D$. Consider now the combinatorial Laplacian $L_c=D-A$ and the random-walk Laplacian $L_r=D^{-1}L_c = I-P$ where $P$ is the row-stochastic normalized adjacency matrix.
It is well known that the two continuous-time dynamics $\dot{x}_c=-L_cx_c$ and $\dot{x}_r=-L_rx_r$ converge to a consensus. More specifically, given an initial condition $x_0$, we have that $$ x(\infty)_c=\mathbf{1}\gamma_c^{'} x_0, \quad \quad \quad x(\infty)_r=\mathbf{1}\gamma_r^{'} x_0 $$
Now, it is known that $\gamma_r$ is the eigenvector centrality of $G$ (it is the solution of $\gamma_r= P' \gamma_r$) so my question is: is $\gamma_c$ some other known centrality measure? Is there any interpretation of this vector?