Limited path percolation in complex networks

We study the stability of network communication after removal of a fraction q=1-p of links under the assumption that communication is effective only if the shortest path between nodes i and j after removal is shorter than aℓij(a≥1) where ℓij is the shortest path before removal. For a large class of networks, we find analytically and numerically a new percolation transition at p˜c=(κ0-1)(1-a)/a, where κ0≡⟨k2⟩/⟨k⟩ and k is the node degree. Above p˜c, order N nodes can communicate within the limited path length aℓij, while below p˜c, Nδ (δ<1) nodes can communicate. We expect our results to influence network design, routing algorithms, and immunization strategies, where short paths are most relevant.