there is an excellent degree of connectivity between high-level nodes. However, conversely, for low-level nodes it may be very difficult to get connected to a nearby high-level node. This might pose a substantial obstruction to the build-up of a hierarchical network. Due to the drawbacks of the existing networks models, we propose two alternatives. According to our discussion, it would be desirable to consider a variant of the ultra-small scale-free geometric network, where two sites $x,y\in\Z^d$ are connected by an edge if $|x-y|\le \max\{R_x,R_y\}$ (instead of $|x-y|\le \min\{R_x,R_y\}$). In other words, in order to connect $x$ and $y$ by an edge it is no longer necessary that both $x$ lies within the radius of influence of $y$ *and* $y$ lies within the radius of influence of $x$. It suffices that $x$ lies within the radius of influence of $y$ *or* $y$ lies within the radius of influence of $x$. We consider a spatial variant of this network model, called *scale-free Gilbert graph*, where the vertices are given by a homogeneous Poisson point process on the $d$-dimensional torus with side length $n$, for some $d,n\ge2$. See Figure \[sfbmFig\] for an illustration of this graph in dimension $d=2$.

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As $n\to\infty$, 