Structural properties of scale-free networks

Many networks have been reported recently to follow a scale-free degree distribution in which the fraction of sites having k connections follows a power law: P(k) = ck^-γ. In this chapter we study the structural properties of such networks. We show that the average distance between sites in scale-free networks is much smaller than that in regular random networks, and bears an interesting dependence on the degree exponent γ. We study percolation in scale-free networks and show that in the regime 2 < γ < 3 the networks are resilient to random breakdown and the percolation transition occurs only in the limit of extreme dilution. On the other hand, attack of the most highly connected nodes easily disrupts the nets. We compute the percolation critical exponents and find that percolation in scale-free networks is non-universal, i.e. depends on γ and different from the mean-field behavior in dimensions d > 6. Finally, we suggest a novel and efficient method for immunization against the spread of diseases in social networks, or the spread of viruses and worms in computer networks.