Vulnerability of networks: Fractional percolation on random graphs

We present a theoretical framework for understanding nonbinary, nonindependent percolation on networks with general degree distributions. The model incorporates a partially functional (PF) state of nodes so that both intensity and extensity of error are characterized. Two connected nodes in a PF state cannot sustain the load and therefore break their link. We give exact solutions for the percolation threshold, the fraction of giant cluster, and the mean size of small clusters. The robustness-fragility transition point for scale-free networks with a degree distribution p(k) proportional to k(-alpha) is identified to be alpha = 3. The analysis reveals that scale-free networks are vulnerable to targeted attack at hubs: a more complete picture of their Achilles' heel turns out to be not only the hubs themselves but also the edges linking them together.