Stability of complex networks under the evolution of attack and repair

With a simple attack and repair evolution model, we investigate and compare the stability of the Erdos-Renyi random graphs (RG) and Barabasi-Albert scale-free (SF) networks. We introduce a new quantity, invulnerability I(s), to describe the stability of the system. We find that both RG and SF networks can evolve to a stationary state. The stationary value Ic has a power-law dependence on the average degree <k>_rg for RG networks; and an exponential relationship with the repair probability p_sf for SF networks. We also discuss the topological changes of RG and SF networks between the initial and stationary states. We observe that the networks in the stationary state have smaller average degree <k> but larger clustering coefficient C and stronger assortativity r.