The ubiquitous proliferation of online social networks has led to the widescale emergence of relational graphs expressing unique patterns in link formation and descriptive user node features. Matrix Factorization and Completion have become popular methods for Link Prediction due to the low rank nature of mutual node friendship information, and the availability of parallel computer architectures for rapid matrix processing. Current Link Prediction literature has demonstrated vast performance improvement through the utilization of sparsity in addition to the low rank matrix assumption. However, the majority of research has introduced sparsity through the limited L1 or Frobenius norms, instead of considering the more detailed distributions which led to the graph formation and relationship evolution. In particular, social networks have been found to express either Pareto, or more recently discovered, Log Normal distributions. Employing the convexity-inducing Lovasz Extension, we demonstrate how incorporating specific degree distribution information can lead to large scale improvements in Matrix Completion based Link prediction. We introduce Log-Normal Matrix Completion (LNMC), and solve the complex optimization problem by employing Alternating Direction Method of Multipliers. Using data from three popular social networks, our experiments yield up to 5% AUC increase over top-performing non-structured sparsity based methods.