Phase transition in the biconnectivity of scale-free networks

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In information-transport and biological systems, sometimes there is more than one pathway between two nodes, so that there is a backup in case one pathway becomes defective. The size of such biconnected nodes can be an important measure of the robustness of a system. The giant biconnected components of diverse real-world networks suggest the importance of scale-free topology in biconnectivity. Thus, here, we consider the critical behavior of the largest biconnected component (BC) as links are added and form a random scale-free network. The critical exponents β((BC)) and β((SC)) associated with the order parameter of the percolation transition of the biconnected component and the single-connected component (SC), respectively, are compared. We obtain a ratio β((BC))/β((SC))=λ-1 for 2<λ<3 and 2 for λ>3, where λ is the exponent of the degree distribution in scale-free networks. We also determine the finite-size scaling behavior of the order parameter analytically and numerically.