About Weighted Random Sampling in Preferential Attachment Models

The Barab\'asi-Albert model is a popular scheme for creating scale-free graphs but has been previously shown to have ambiguities in its definition. In this paper we discuss a new ambiguity in the definition of the BA model by identifying the tight relation between the preferential attachment process and unequal probability random sampling. While the probability that each individual vertex is selected is set to be proportional to their degree, the model does not specify the joint probabilities that any tuple of $m$ vertices is selected together for $m>1$. We demonstrate the consequences using analytical, experimental, and empirical analyses and propose a concise definition of the model that addresses this ambiguity. Using the connection with unequal probability random sampling, we also highlight a confusion about the process via which nodes are selected on each time step, for which -- despite being implicitly indicated in the original paper -- current literature appears fragmented.