  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.
