Limited Percolation on Complex Networks

We study the stability of network communication under removal of $q=1-p$ links when communication between nodes is possible only through a subset of the paths connecting them. We find a new percolation transition $\tilde{p}$ below which only a fractal fraction of nodes $N^{\gamma}$ can communicate, where $\gamma$ is a function of the accepted communication paths. Above $\tilde{p}$, order $N$ nodes can communicate. The results may be useful for the design of communication networks and immunization strategies.