Spatial network surrogates for disentangling complex system structure from spatial embedding of nodes

Networks with nodes embedded in a metric space have gained increasing interest in recent years. The effects of spatial embedding on the networks' structural characteristics, however, are rarely taken into account when studying their macroscopic properties. Here, we propose a hierarchy of null models to generate random surrogates from a given spatially embedded network that can preserve certain global and local statistics associated with the nodes' embedding in a metric space. Comparing the original network's and the resulting surrogates' global characteristics allows one to quantify to what extent these characteristics are already predetermined by the spatial embedding of the nodes and links. We apply our framework to various real-world spatial networks and show that the proposed models capture macroscopic properties of the networks under study much better than standard random network models that do not account for the nodes' spatial embedding. Depending on the actual performance of the proposed null models, the networks are categorized into different classes. Since many real-world complex networks are in fact spatial networks, the proposed approach is relevant for disentangling the underlying complex system structure from spatial embedding of nodes in many fields, ranging from social systems over infrastructure and neurophysiology to climatology.

Figure 1

Sketch of the rewiring process that generates randomized surrogates of a given original network by applying (a) GeoModel I and (b) GeoModel II. Nodes $i$, $j$, $k$, and $l$ are drawn at random, such that $i$ is linked to $j$, and $k$ is linked to $l$ (solid lines), but no links are present between $i$ and $k$, and $j$ and $l$ (dashed lines). According to the chosen network model, the distances $d_{--}$ between the nodes are evaluated. If the nodes form an approximate (a) kite or (b) diamond with the connections between them present as depicted, previous links are replaced by links connecting $i$ with $k$ and $j$ with $l$.

Figure 2

Evolution of the (a) global clustering coefficient $\mathcal{C}$ and (b) average path length $\mathcal{L}$ with the number of rewirings for an ensemble of $n=100$ surrogates generated from the interstate network by applying GeoModel I and using different tolerances ${\epsilon }_I$. Solid lines indicate the respective value of $\mathcal{C}$ and $\mathcal{L}$ of the interstate network itself. (c) Hamming distance $\mathcal{H}$ between the surrogate networks and the original network. Scatter symbols denote the mean value and error bars indicate one standard deviation of each measure.

Figure 3

(a) Distribution of KS statistics $\kappa$ measuring the maximum distance in the global cumulative link-length distribution $P\left(l\right)$ between the interstate network and each of the $n=100$ network surrogates obtained by applying GeoModel I under different tolerances $\epsilon$ and $20M$ rewirings. (b) Average KS statistics ${\kappa }_{\mathrm{mean}}$ and Hamming distance $\mathcal{H}$ after $20M$ successful rewirings depending on the choice of tolerance $\epsilon$. Error bars denote one standard deviation for the Hamming distance and the 5th and 95th percentile of the distribution of KS statistics. The solid line indicates the critical value ${\kappa }_{\mathrm{crit}}$ below which the surrogates' and the original network's link-length distributions are considered statistically indistinguishable under a confidence level of $\alpha =0.95$.

Figure 4

Evolution of (a) global clustering coefficient $\mathcal{C}$ and (b) average path length $\mathcal{L}$ with the number of rewirings averaged over ensembles of $n=100$ surrogates generated from the interstate network by applying the different random network models (dashed lines). For GeoModel I and GeoModel II, the tolerances are set to ${\epsilon }_I=0.17$ and ${\epsilon }_{II}=0.24$, respectively. Scatter symbols denote the mean value of each measure. Error bars indicate one standard deviation and are shown if their size exceeds that of the symbol. Solid lines indicate the value of $\mathcal{C}$ and $\mathcal{L}$ in the original network.

Figure 5

Same as Fig. 4, but for the airline network and tolerances of ${\epsilon }_I=0.04$ and ${\epsilon }_{II}=0.06$. Error bars are not shown as they do not exceed the size of the symbols.

Figure 6

Average relative deviation of global clustering coefficient $\mathrm{\Delta }\mathcal{C}$ and average path length $\mathrm{\Delta }\mathcal{L}$ from the respective original values computed over an ensemble of $n=100$ surrogate networks after $20M$ successful rewirings by applying (a) random rewiring, (b) random link switching, (c) GeoModel I, and (d) GeoModel II. The tolerances ${\epsilon }_I$ and ${\epsilon }_{II}$ used for each network and random network model are given in Table 1. Error bars denote the standard deviation in $\mathrm{\Delta }\mathcal{C}$ and $\mathrm{\Delta }\mathcal{L}$ and are shown if their size exceeds that of the corresponding symbol.

Figure 7

(a) Cumulative intrinsic probability distribution $P_{\mathrm{int}}\left(l^{\prime }\right)$ for the different networks under study. (b) KS statistics indicating the maximum distance between each cumulative probability distribution $P_{\mathrm{int}}\left(l^{\prime }\right)$ and that of a random uniform distribution for each network under study (sorted in ascending order).