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

Abstract

Many real networks are embedded in space, and often the distribution of the link lengths $r$ follows a power law, $p(r)\ensuremath{\sim}{r}^{\ensuremath{-}\ensuremath{\delta}}$. Indications that such systems can be characterized by the concept of dimension were found recently. Here, we present further support for this claim, based on extensive numerical simulations of model networks with a narrow degree distribution, embedded in lattices of dimensions ${d}_{e}=1$ and ${d}_{e}=2$. For networks with $\ensuremath{\delta}<{d}_{e}$, $d$ is infinity, while for $\ensuremath{\delta}>2{d}_{e}$, $d$ has the value of the embedding dimension ${d}_{e}$. In the intermediate regime of interest ${d}_{e}\ensuremath{\le}\ensuremath{\delta}<2{d}_{e}$, our numerical results suggest that $d$ decreases continuously from $d=\ensuremath{\infty}$ to ${d}_{e}$, with $d\ensuremath{-}{d}_{e}\ensuremath{\propto}(2\ensuremath{-}{\ensuremath{\delta}}^{\ensuremath{'}})/[{\ensuremath{\delta}}^{\ensuremath{'}}({\ensuremath{\delta}}^{\ensuremath{'}}\ensuremath{-}1)]$ and ${\ensuremath{\delta}}^{\ensuremath{'}}=\ensuremath{\delta}/{d}_{e}$. We also analyze how the mass $M$ and the Euclidean distance $r$ increase with the topological distance $\ensuremath{\ell}$ (minimum number of links between two sites in the network). Our results suggest that in the intermediate regime ${d}_{e}\ensuremath{\le}\ensuremath{\delta}<2{d}_{e}$, $M(\ensuremath{\ell})$ and $r(\ensuremath{\ell})$ increase with $\ensuremath{\ell}$ as a stretched exponential, $M(\ensuremath{\ell})\ensuremath{\sim}\mathrm{exp}[Ad{\ensuremath{\ell}}^{{\ensuremath{\delta}}^{\ensuremath{'}}(2\ensuremath{-}{\ensuremath{\delta}}^{\ensuremath{'}})}]$ and $r(\ensuremath{\ell})\ensuremath{\sim}\mathrm{exp}[A{\ensuremath{\ell}}^{{\ensuremath{\delta}}^{\ensuremath{'}}(2\ensuremath{-}{\ensuremath{\delta}}^{\ensuremath{'}})}]$, such that $M(\ensuremath{\ell})\ensuremath{\sim}r{(\ensuremath{\ell})}^{d}$. For $\ensuremath{\delta}<{d}_{e}$, $M$ increases exponentially with $\ensuremath{\ell}$ (as known for $\ensuremath{\delta}=0$), while $r$ is constant and independent of $\ensuremath{\ell}$. For $\ensuremath{\delta}\ensuremath{\ge}2{d}_{e}$, we find the expected power-law scaling, $M(\ensuremath{\ell})\ensuremath{\sim}{\ensuremath{\ell}}^{{d}_{\ensuremath{\ell}}}$ and $r(\ensuremath{\ell})\ensuremath{\sim}{\ensuremath{\ell}}^{1/{d}_{\mathrm{min}}}$, with ${d}_{\ensuremath{\ell}}{d}_{\mathrm{min}}=d$. In ${d}_{e}=1$, we find the expected result, ${d}_{\ensuremath{\ell}}={d}_{\mathrm{min}}=1$, while in ${d}_{e}=2$ we find surprisingly that although $d=2$, ${d}_{\ensuremath{\ell}}>2$ and ${d}_{\mathrm{min}}<1$, in contrast to regular lattices.