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  .
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 .
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Barabasi-Albert
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very popular model for creating random
popular scheme for creating
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but has been previously shown to have ambiguities in its definition. In this paper we discuss a new
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model and, consequently, the dependent models and applications. This ambiguity is a result of the model's tight relation with the field of
BA model by identifying the tight relation between the preferential attachment process and
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, 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
. 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
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Barabasi-Albert model that is expressed as an application of
model that addresses this ambiguity. Using the connection with
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, 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