A repair strategy to the attacked random and scale-free networks
  With a simple attack and repair evolution model, we investigate the stability and structural changes 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 I_c has a power-law dependence on the repair probability p_re. We also analyze the effects of the repair strategy to the attack tolerance of the networks. We observe that there is a threshold, (k_max)_c, for the maximum degree. The maximum degree k_max at time s will be no smaller than (k_max)_c. We give further information on the evolution of the networks by comparing the changes of the topological parameters, such as degree distribution P(k), average degree <k>, shortest path length L, clustering coefficient C, assortativity r, under the initial and stationary states. 
L.P. Chi, C.B. Yang and X. Cai