Figure 1

(Color online) $\left({\mathrm{a}}_1\right)$, $\left({\mathrm{b}}_1\right)$, and $\left({\mathrm{c}}_1\right)$: graphical representations of the models (a), (b), and (c), respectively. The three networks have 1000 vertices; their number of edges is $L=1881$ $\left({\mathrm{a}}_1\right)$, $L=1982$ $\left({\mathrm{a}}_2\right)$, and $L=1982$ $\left({\mathrm{a}}_3\right)$. Note that all edges of network $\left({\mathrm{a}}_1\right)$ are short-range connections, while in network $\left({\mathrm{c}}_1\right)$ edges connecting distant vertices do exist. $\left({\mathrm{a}}_2\right)$, $\left({\mathrm{b}}_2\right)$, and $\left({\mathrm{c}}_2\right)$: Degree distribution of the networks represented in $\left({\mathrm{a}}_1\right)$, $\left({\mathrm{b}}_1\right)$, and $\left({\mathrm{c}}_1\right)$, respectively. Models (b) and (c) are scale-free. The slope of the straight lines are $-2.24$ $\left({\mathrm{b}}_2\right)$ and $-2.95$ $\left({\mathrm{c}}_2\right)$. $\left({\mathrm{a}}_3\right)$, $\left({\mathrm{b}}_3\right)$, and $\left({\mathrm{c}}_3\right)$: Local clustering coefficient $C\left(k\right)$ of the networks represented in $\left({\mathrm{a}}_1\right)$, $\left({\mathrm{b}}_1\right)$, and $\left({\mathrm{c}}_1\right)$, respectively. The behavior of $C\left(k\right)$ follows power laws for the three models. The slope of the straight lines are $-0.40$ $\left({\mathrm{a}}_3\right)$, $-0.46$ $\left({\mathrm{b}}_3\right)$, and $-0.58$ $\left({\mathrm{b}}_3\right)$. Notice the double logarithmic scales in all graphs.Reuse & Permissions

Figure 2

(Color online) $\overline{\mathcal{N}}\left(\mathcal{L}\right)$ as a function of the $\mathcal{L}$. From bottom to top, models a, b, and c. The straight line correspond to a network of dimension 2. The results are used to estimate the dimension of the three spatial networks considered (see text for more details). Notice the double logarithmic scales of the picture.Reuse & Permissions

Figure 3

(Color online) Layer average degree ${⟨k⟩}_{\mathcal{L}}$ as a function of shell number $\mathcal{L}$. The curve with the highest peak corresponds to model c, the intermediate one and the flat one to models b and a, respectively.Reuse & Permissions

Figure 4

(Color online) Analysis of the degree-degree correlations of the three models considered. Nearest neighbors’ average function ${\overline{k}}_{nn}\left(j\right)$ against $j$. From bottom to top, models a, b, and c. The corresponding straight lines are drawn as a guide of eyes. Note that models b and c exhibit dissortative mixing, while model a is slightly assortative.Reuse & Permissions

Figure 5

(Color online) (a) Graphical representation of a small network ($N=1000$ and $L=1985$) corresponding to model (d). Note that vertices concentrate in certain areas of the space, and that the long-range connections of the network link vertices of large degree. (b) Degree distribution of model (d). The slope of the straight line is $-2.42$. (c) Degree-dependent clustering coefficient $C\left(k\right)$ of the model. The slope of the straight line is $-0.97$. (d) Nearest-neighbor average function ${\overline{k}}_{nn}$ as function of degree $j$. The slope of the straight line is $-0.52$. Note that $P\left(k\right)$, $C\left(k\right)$, and ${\overline{k}}_{nn}$ fall off as power law functions. (e) Average degree ${⟨k⟩}_{\mathcal{L}}$ as a function of shell number $\mathcal{L}$. From ${⟨k⟩}_{\mathcal{L}=1}$ the average degree interestingly decays following a power law (inset of the picture). The straight line plotted in the inset is drawn as a guide for eyes.Reuse & Permissions