On the emergence of highly variable distributions in the autonomous system topology

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Abstract
Recent studies observe that vertex degree in the autonomous systems (AS) graph exhibits a highly variable distribution [14, 21]. The most prominent explanatory model for this phenomenon is the Barabasi-Albert (B-A) model [5, 2]. A central feature of the B-A model is preferential connectivity --- meaning that the likelihood a new node in a growing graph will connect to an existing node is proportional to the existing node's degree. In this paper we ask whether a more general explanation than the B-A model, and absent the assumption of preferential connectivity, is consistent with empirical data. We are motivated by two observations: first, AS degree and AS size are highly correlated [10]; and second, highly variable AS size can arise simply through exponential growth. We construct a model incorporating exponential growth in the size of the Internet and in the number of ASes, and show that it yields a size distribution exhibiting a power-law tail. In such a model, if an AS's link formation is roughly proportional to its size, then AS out-degree will also show high variability. Moreover, our approach is more flexible than previous work, since the choice of which AS to connect to does not impact high variability, thus can be freely specified. We instantiate such a model with empirically derived estimates of historical growth rates and show that the resulting degree distribution is in good agreement with that of real AS graphs.

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