Stability in the evolution of random networks

TL;DR

This paper investigates how random networks evolve under attack and reconstruction, introducing a new stability measure and revealing that networks reach a stationary state with a normal degree distribution and increased clustering.

Contribution

It presents a novel stability metric, invulnerability I(s), and demonstrates the network's evolution to a stationary state with distinct degree distribution and clustering properties.

Findings

Stationary invulnerability I_c depends on initial average degree <k> with a power-law slope of about -1.485.

Degree distribution in the stationary state is normal, not Poisson.

Clustering coefficient increases significantly in the stationary state.

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>.