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Many real networks are embedded in space, where in some of them the linkslength decay as a power law distribution with distance. Indications that suchsystems can be characterized by the concept of dimension were found recently.Here, we present further support for this claim, based on extensive numericalsimulations for model networks embedded on lattices of dimensions $d_e=1$ and$d_e=2$.  We evaluate the dimension $d$ from the power law scaling of (a) the mass ofthe network with the Euclidean radius $r$ and (b) the probability of return tothe origin with the distance $r$ travelled by the random walker. Bothapproaches yield the same dimension. For networks with $\delta < d_e$, $d$ isinfinity, while for $\delta > 2d_e$, $d$ obtains the value of the embeddingdimension $d_e$. In the intermediate regime of interest $d_e \leq \delta < 2d_e$, our numerical results suggest that $d$ decreases continously from $d =\infty$ to $d_e$, with $d - d_e \sim (\delta - d_e)^{-1}$ for $\delta$ close to$d_e$. Finally, we discuss the scaling of the mass $M$ and the Euclideandistance $r$ with the topological distance $\ell$. Our results suggest that inthe intermediate regime $d_e \leq \delta < 2 d_e$, $M(\ell)$ and $r(\ell)$ donot increase with $\ell$ as a power law but with a stretched exponential,$M(\ell) \sim \exp [A \ell^{\delta' (2 - \delta')}]$ and $r(\ell) \sim \exp [B\ell^{\delta' (2 - \delta')}]$, where $\delta' = \delta/d_e$. The parameters$A$ and $B$ are related to $d$ by $d = A/B$, such that $M(\ell) \simr(\ell)^d$. For $\delta < d_e$, $M$ increases exponentially with $\ell$, asknown for $\delta=0$, while $r$ is constant and independent of $\ell$. For$\delta \geq 2d_e$, we find power law scaling, $M(\ell) \sim \ell^{d_\ell}$ and$r(\ell) \sim \ell^{1/d_{min}}$, with $d_\ell \cdot d_{min} = d$.

Complex networks embedded in space: Dimension and scaling relations between mass, topological distance and Euclidean distance