  Random network models generated using sparse exchangeable graphs have
provided a mechanism to study a wide variety of complex real-life networks. In
particular, these models help with investigating power-law properties of degree
distributions, number of edges, and other relevant network metrics which
support the scale-free structure of networks. Previous work on such graphs
imposes a marginal assumption of univariate regular variation (e.g., power-law
tail) on the bivariate generating graphex function. In this paper, we study
sparse exchangeable graphs generated by graphex functions which are
multivariate regularly varying. We also focus on a different metric for our
study: the distribution of the number of common vertices (connections) shared
by a pair of vertices. The number being high for a fixed pair is an indicator
of the original pair of vertices being connected. We find that the distribution
of number of common connections are regularly varying as well, where the tail
indices of regular variation are governed by the type of graphex function used.
Our results are verified on simulated graphs by estimating the relevant tail
index parameters.
