Phase transition of biconnected components in scale-free networks

In information-transport and biological systems, there can be more than one pathway between two nodes, so that there is a backup in case one pathway is inactive. 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 the biconnectivity. Thus, here, we consider a critical behavior of the largest biconnected component as links are added and form a random scale-free network. The critical exponents $\beta_{\rm (BC)}$ and $\beta_{\rm (SC)}$ associated with the order parameter of the percolation transition of biconnected and single-connected components, respectively, are compared. We obtain that $\beta_{\rm (BC)}/\beta_{\rm (SC)}=\lambda-1$ for $2 < \lambda < 3$ and 2 for $\lambda > 3$, where $\lambda$ is the exponent of the degree distribution in scale-free networks. We also obtain the finite-size scaling behavior of the order parameter analytically and numerically.