to a network with a giant connected component and power-law degree distributions. However, this model does not offer an inherently defined hierarchy of nodes. , if we think of a high-level node as having the purpose of providing access to all low-level nodes in some region, then this would suggest that the occurrence of edges should be spatially positively correlated. However, if $x,y\in\Z^d$ are arbitrary sites in $\Z^d$ and $y^\p$ is close to $y$, then putting an edge between $x$ and $y$ does not influence at all the probability of seeing an edge between $x$ and $y^\p$.

A second option could be to consider the ultra-small scale-free geometric networks on $\Z^d$ introduced in [@ultraSmall]. Two sites $x,y\in\Z^d$ are connected by an edge in this graph if $|x-y|\le \min\{R_x,R_y\}$, where $\{R_z\}_{z\in\Z^d}$ denotes a family of iid heavy-tailed random variables. For any site $z\in\Z^d$ the value $R_z$ can be thought of as the radius of influence of $z$, ultra-small scale-free geometric networks offer a natural possibility for defining the network hierarchy. Similar remarks apply to scale-free percolation [@sfPerc]. Recently, also a Poisson-based continuum analogue of this model has been investigated, see [@wuth1; @wuth2]. From a modeling point of view, this means that 