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.