On the friendship paradox and inversity: A network property with applications to privacy-sensitive network interventions.

We provide the mathematical and empirical foundations of the friendship paradox in networks, often stated as "Your friends have more friends than you." We prove a set of network properties on friends of friends and characterize the concepts of ego-based and alter-based means. We propose a network property called inversity that quantifies the imbalance in degrees across edges and prove that the sign of inversity determines the ordering between ego-based or alter-based means for any network, with implications for interventions.

Nuclear localization of heparanase 2 (Hpa2) attenuates breast carcinoma growth and metastasis.

Unlike the intense research effort devoted to exploring the significance of heparanase in cancer, very little attention was given to Hpa2, a close homolog of heparanase. Here, we explored the role of Hpa2 in breast cancer. Unexpectedly, we found that patients endowed with high levels of Hpa2 exhibited a higher incidence of tumor metastasis and survived less than patients with low levels of Hpa2.

Holographic codes from hyperinvariant tensor networks.

Holographic quantum-error correcting codes are models of bulk/boundary dualities such as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, where a higher-dimensional bulk geometry is associated with the code's logical degrees of freedom. Previous discrete holographic codes based on tensor networks have reproduced the general code properties expected from continuum AdS/CFT, such as complementary recovery. However, the boundary states of such tensor networks typically do not exhibit the expected correlation functions of CFT boundary states.