Geometric Nodal Degree Distributions Arise in Barabási-Albert Graphs!

In random graphs, node degrees are random variables that can have different statistical properties in comparison to the (empirical) degree distribution of the entire network. In the present work, we analyze the degree distribution of individual nodes in Barabási-Albert graphs, and observe them to be geometrically distributed with parameter dependent on the node's age. This result directly shows an “old-get-rich” phenomenon in the BA model, i.e., older nodes are more likely to have a higher degree than more recent nodes. We also conclude that old nodes tend to have a flatter degree distribution, while later nodes have a more pronounced exponential tail. The result provides an approach for estimating nodal degree distribution in a finite BA graph. Finally, we reconcile with the well known result that the degree distribution of the entire network is power law (heavy-tailed) in spite of the individual nodes being geometrically distributed (light-tailed). In fact, we show that an infinitesimally small fraction of early nodes in the network make the degree distribution power-law. This suggests that in heterogeneous models, e.g., network growth models, the degree distribution of the node can be very different in comparison to the network-wide empirical distribution, possibly of a different class altogether.