Robustness of power-law networks: its assessment and optimization

Abstract

Many practical complex networks, such as the Internet, WWW and social networks, are discovered to follow power-law distribution in their degree sequences, i.e., the number of nodes with degree $i$ in these networks is proportional to $i^{-\beta }$ for some exponential factor $\beta > 0$. However, these networks also expose their vulnerabilities to a great number of threats such as adversarial attacks on the Internet, cyber-crimes on the WWW or malware propagations on social networks. Although power-law networks have been found robust under random attacks and vulnerable to intentional attacks via experimental observations, how to better understand their vulnerabilities from a theoretical point of view still remains open. In this paper, we study the vulnerability of power-law networks under random attacks and adversarial attacks using the in-depth probabilistic analysis on the theory of random power-law graph models. Our results indicate that power-law networks are able to tolerate random failures if their exponential factor $\beta$ is $<$2.9, and they are more robust against intentional attacks if $\beta$ is smaller. Furthermore, we reveal the best range $[1.8, 2.5]$ for the exponential factor $\beta$ by optimizing the complex networks in terms of both their vulnerabilities and costs. When $\beta < 1.8$, the network maintenance cost is very expensive, and when $\beta > 2.5$ the network robustness is unpredictable since it depends on the specific attacking strategy.