About Weighted Random Sampling in Preferential Attachment Models

The Barabasi-Albert model is a very popular model for creating random scale-free graphs. Despite its widespread use, there is a subtle ambiguity in the definition of the model and, consequently, the dependent models and applications. This ambiguity is a result of the model's tight relation with the field of unequal probability random sampling, which dictates the exact process of edge creation after a newborn node has been added. In this work, we identify the ambiguity, assess its impact and propose a more precise definition of the Barabasi-Albert model that is expressed as an application of unequal probability random sampling.