to the graph $G(X^{(n)})$, see Theorem \[asyGrowProp\]. To be more precise, defining $$D^{\p,(\alpha)}_{\out,n}=\sum_{(\xi,r)\in X^{(n)}}|\xi|^\alpha1_{((o,R^*),(\xi,r))\text{ is an edge in }G^\p(\{(o,R^*)\}\cup X^{(n)})},$$ where $\alpha\ge 0$ and $n\ge1$ we have the following result.

\[redElThm\] Assume that $s=d$. Then, the expected out-degree $\E D^{\p,(0)}_{\out,n}$ is asymptotically bounded from above by a constant, i.e., $\E D^{\p,(0)}_{\out,n}\in O(1)$. , if $\alpha>0$, then $\E D^{\p,(\alpha)}_{\out,n}\in O(n^{\alpha-\delta})$ for some $\delta>0$.

In Section \[multiSec\], we show that removing redundancies does not destroy the property of connectivity. However, this thinning operation *does* influence the quality of connectivity, in the sense that chemical distances will increase. Indeed, instead of moving from $x\in X^{(n)}$ to $y\in X^{(n)}$ directly along an edge in the graph $G(X^{(n)})$, introducing a multilayer topology via $G^\p(X^{(n)})$ might force us to move through a potentially large number of layers, before we can get from $x$ to $y$. Still, chemical distances increase at most by a logarithmic factor in the size of the sampling window.

\[intMedProp\] There exists $c_1>0$ whp any $x,y\in X^{(n)}$ that are adjacent in $G(X^{(n)})$ can be connected by a path in $G^\p(X^{(n)})$ consisting of at most $c_1\log n$ hops.

Proofs of Theorems \[outEdgeThm\], \[inEdgeThm\] and  \[asyGrowProp\] {#sfSec}
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In the present section, we investigate the 