Structural Properties of Scale-Free Networks

Structural Properties of Scale-FreeNetworksReuven Cohen, Shlomo Havlin, and Daniel ben-AvrahamWILEY-VCH Verlag Berlin GmbHJanuary 20034 Structural Properties of Scale-Free NetworksReuven Cohen and Shlomo HavlinMinerva Center and Department of Physics,Bar-Ilan University, Ramat-Gan, IsraelDaniel ben-AvrahamPhysics Department and Center for Statistical Physics (CISP),Clarkson University, Potsdam NY 13699-5820, USAAbstractMany networks have been reported recently to follow a scale-free degree distribution in whichthe fraction of sites having connections follows a power law: . In this chapterwe study the structural properties of such networks. We show that the average distance be-tween sites in scale-free networks is much smaller than that in regular random networks, andbears an interesting dependence on the degree exponent . We study percolation in scale-freenetworks and show that in the regime the networks are resilient to random break-down and the percolation transition occurs only in the limit of extreme dilution. On the otherhand, attack of the most highly connected nodes easily disrupts the nets. We compute the per-colation critical exponents and find that percolation in scale-free networks is non-universal,i.e. depends on and different from the mean-field behavior in dimensions . Finally, wesuggest a novel and efficient method for immunization against the spread of diseases in socialnetworks, or ...