Interdependent Lattice Networks in High Dimensions

We study the mutual percolation of two interdependent lattice networks ranging from two to seven dimensions, denoted as $D$. We impose that the length of interdependent links connecting nodes in the two lattices be less than or equal to a certain value, $r$. For each value of $D$ and $r$, we find the mutual percolation threshold, $p_c[D,r]$ below which the system completely collapses through a cascade of failures following an initial destruction of a fraction $(1-p)$ of the nodes in one of the lattices. We find that for each dimension, $D<6$, there is a value of $r=r_I>1$ such that for $r\geq r_I$ the cascading failures occur as a discontinuous first order transition, while for $r<r_I$ the system undergoes a continuous second order transition, as in the classical percolation theory. Remarkably, for $D=6$, $r_I=1$ which is the same as in random regular (RR) graphs with the same degree (coordination number) of nodes. We also find that in all dimensions, the interdependent lattices reach maximal vulnerability (maximal $p_c[D,r]$) at a distance $r=r_{max}>r_I$, and for $r>r_{max}$ the vulnerability starts to decrease as $r\to\infty$. However the decrease becomes less significant as $D$ increases and $p_c[D,r_{max}]-p_c[D,\infty]$ decreases exponentially with $D$. We also investigate the dependence of $p_c[D,r]$ on the system size as well as how the nature of the transition changes as the number of lattice sites, $N\to\infty$.