Whole Sampling Generation of Scale-Free Graphs

This paper presents the development of a new class of algorithms that accurately implement the preferential attachment mechanism of the Barab\'asi-Albert (BA) model to generate scale-free graphs. Contrary to existing approximate preferential attachment schemes, our methods are exact in terms of the proportionality of the vertex selection probabilities to their degree and run in linear time with respect to the order of the generated graph. Our algorithms are based on a principle of random sampling which is called whole sampling and is a new perspective for the study of preferential attachment. We show that they obey the definition of the original BA model that generates scale-free graphs and discuss their higher-order properties. Finally, we extend our analytical presentation with computer experiments that focus on the degree distribution and several measures surrounding the local clustering coefficient.