Stability in the evolution of random networks

Abstract

With a simple model, we study the evolution of random networks under attack and reconstruction. We introduce a new quality, invulnerability I(s), to describe the stability of the system. We find that the network can evolve to a stationary state. The stationary value I_c has a power-law dependence on the initial average degree <k>, with the slope is about -1.485. In the stationary state, the degree distribution is a normal distribution, rather than a typical Poisson distribution for general random graphs. The clustering coefficient in the stationary state is much larger than that in the initial state. The stability of the network depends only on the initial average degree <k>, which increases rapidly with the decrease of <k>.