Failure of interdependent lattice networks in high dimensions

We study the mutual percolation of two interdependent lattice networks for dimensions ranging from 2 to 7. We impose that the length of interdependency 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 selected at random 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. In all dimensions, the interdependent lattices reach maximal vulnerability (maximal $p_c[D,r]$) at a distance $r=r_{max}>r_I$, before beginning to decrease as $r\to\infty$. The decrease becomes less significant as $D$ increases and $p_c[D,r_{max}]-p_c[D,\infty]$ decreases exponentially with $D$.