Absorbing phase transition in a unidirectionally coupled layered network

We study the contact process on layered networks in which each layer is unidirectionally coupled to the next layer. Each layer has elements sitting on i) Erd{\"o}s-R{\'e}yni network, ii) a $d$-dimensional lattice. The layer at the top which is not connected to any layer. The top layer undergoes absorbing transition in the directed percolation class for the corresponding topology. The critical point for absorbing transition is the same for all layers. For Erdos-Reyni network order parameter $\rho(t)$ decays as $t^{-\delta_l}$ at the critical point for $l'$th layer with $\delta_l \sim 2^{1-l}$. This can be explained with a hierarchy of differential equations in the mean-field approximation. The dynamic exponent $z$ is $0.5$ for all layers and the value of $\nu_{\parallel}$ tends to 2 for larger $l$. For a d-dimensional lattice, we observe stretched exponential decay of order parameter for all but top layer at the critical point.
Comment: 5 pages, 9 figures