Random graphs and real networks with weak geometric coupling (2312.07416v1)

Abstract: Geometry can be used to explain many properties commonly observed in real networks. It is therefore often assumed that real networks, especially those with high average local clustering, live in an underlying hidden geometric space. However, it has been shown that finite size effects can also induce substantial clustering, even when the coupling to this space is weak or non existent. In this paper, we study the weakly geometric regime, where clustering is absent in the thermodynamic limit but present in finite systems. Extending Mercator, a network embedding tool based on the Popularity$\times$Similarity $\mathbb{S}^1$/$\mathbb{H}^2$ static geometric network model, we show that, even when the coupling to the geometric space is weak, geometric information can be recovered from the connectivity alone. The fact that several real networks are best described in this quasi-geometric regime suggests that the transition between non-geometric and geometric networks is not a sharp one.