Hyperbolic geometric graph

A hyperbolic geometric graph (HGG) or hyperbolic geometric network (HGN) is special spatial network where nodes are (1) sprinkled according to a probability distribution onto a hyperbolic space of constant negative curvature and (2) an edge between two nodes is present if they are close according to a function of the metric, a HGG generalizes a random geometric graph (RGG) whose embedding space is Euclidean.

Mathematical formulation

Mathematically, a HGG is a graph $G(V,E)$ with a vertex set V (cardinality $N=|V|$) and a edge set E constructed by considering the nodes as points placed onto a 2-dimensional hyperbolic space $\mathbb{H}^2_\zeta$ of constant negative Gaussian curvature, $-\zeta^2$ and cut-off radius $R$, i.e. the radius of the Poincaré disk which can be visualized using a hyperboloid model. Each point $i$ has hyperbolic polar coordinates $(r_i,\theta_i)$ with $0\leq r_i\leq R$ and $0\leq \theta_i < 2\pi$.

The hyperbolic law of cosines allows to measure the distance $d_{ij}$ between two points $i$ and $j$,

$\cosh(\zeta d_{ij})=\cosh(\zeta r_i)\cosh(\zeta r_j)$$-\sinh(\zeta r_i)\sinh(\zeta r_j)\cos\bigg(\underbrace{\pi\!-\!\bigg|\pi-|\theta_i\!-\!\theta_j|\bigg|}_{\Delta}\bigg).$

The angle $\Delta$ is the (smallest) angle between the two position vectors.

In the simplest case, an edge $(i,j)$ is established iff (if and only if) two nodes are within a certain neighborhood radius $r$, $d_{ij}\leq r$, this corresponds to an influence threshold.

Connectivity decay function

In general, a link will be established with a probability depending on the distance $d_{ij}$. A connectivity decay function $\gamma(s): \mathbb{R}^+\to[0,1]$ represents the probability of assigning an edge to a pair of nodes at distance $s$. In this framework, the simple case of hard-code neighborhood like in random geometric graphs is referred to as truncation decay function.

Findings

For $\zeta=1$ (Gaussian curvature $K=-1$), HGGs form an ensemble of networks for which is possible to express the degree distribution analytically as closed form for the limiting case of large number of nodes. This is worth mentioning since this is not true for many ensemble of graphs.

Applications

HGGs have been suggested as promising model for social networks where the hyperbolicity appears through a competition between similarity and popularity of an individual.