Multiplexity amplifies geometry in networks

TL;DR

This work focuses on mutual clustering--a measure of the amount of triangles that are present in all layers among the same triplets of nodes--and finds that this clustering is abnormally high in many real-world networks, even when clustering in each individual layer is weak.

Abstract

Many real-world network are multilayer, with nontrivial correlations across layers. Here we show that these correlations amplify geometry in networks. We focus on mutual clustering--a measure of the amount of triangles that are present in all layers among the same triplets of nodes--and find that this clustering is abnormally high in many real-world networks, even when clustering in each individual layer is weak. We explain this unexpected phenomenon using a simple multiplex network model with latent geometry: links that are most congruent with this geometry are the ones that persist across layers, amplifying the cross-layer triangle overlap. This result reveals a different dimension in which multilayer networks are radically distinct from their constituent layers.

Figures (3)

• Figure 1: (a) The average mutual degree of the MG versus the mean of the two average degrees of the single layer networks for several real multiplexes. One multiplex can result in several data points as multiplexes with more than two layers are deconstructed into all possible pairwise MCCs. The style of the data points is the same for all MCCs corresponding to the same multiplex, and we only label each type once. For example, MCCs related to the Malaria multiplex are represented by gray hexagons. The black dashed line indicates the diagonal $\langle \widetilde{k}\rangle = (\langle k^{(1)}\rangle+\langle k^{(2)}\rangle)/2$. (b) The analogous data for the average local mutual clustering coefficient. The details of the networks shown in this figure can be found in Appendix \ref{['app:realmultiplexes']}. (c) An arXiv pairwise MCC and its mutual graph. In the bottom two layers, representing the pairwise MCC, nodes are colored by the local clustering coefficient $c_i$. In the top layer, representing the MG, the nodes are colored by the difference between the local mutual clustering coefficient and the mean of the clustering of the corresponding nodes in the constituent layers $\Delta c_i = \widetilde{c}_i - (c^{(1)}_i+c^{(2)}_i)/2$. In all layers, nodes with $k_i\leq 1$ are not shown as they do not contribute to the average local clustering coefficient.
• Figure 2: (a) The scaling exponent $\sigma_c$ of the mutual clustering coefficient $\langle \widetilde{c}\rangle \sim N^{\sigma_c}$ as a function of the geometric couplings $\beta_1$ and $\beta_2$ of the constituent graphs for fully correlated layers and homogeneous degree distribution. The black dashed lines define five non-overlapping regions of the parameter space, based on how the mutual clustering coefficient relates to its single layer counterparts. The parametrization of these lines can be found in the SI supp (b) The scaling exponents $\sigma_c$ as a function of $\beta=\beta_1=\beta_2$ for various power-law exponents $\gamma$. Results were obtained by numerical integration of Eq. S42 in the SI supp for $N\in[10^6,10^9]$. The inset shows the analytic result (see the SI supp for details) for $\sigma_c$ as a function of $\gamma$ when $\beta=0$. (c) The mutual clustering coefficient as a function of $\beta=\beta_1=\beta_2$ for various correlation strengths $g\in[0,1]$. In all cases the individual layers were generated with $N=32000$, $\gamma=50$, $\langle k\rangle = 20$ and $\nu=1$. (d) The mutual clustering coefficient as a function of the degree distribution exponent $\gamma=\gamma_1=\gamma_2$. Various correlations strengths $\nu\in[0,1]$ are shown. The individual layers use $N=32000$, $\beta=0$, $\langle k\rangle =20$ and $g=1$. (e-h)The relation between the geometric couplings of the MG and that of its constituent graphs, for homogeneous (e,g) and heterogeneous degree distributions (f,h). The black, dotted line corresponds to the effective coupling $\widetilde{\beta}=1$. Perfect correlations are studied in panels (e,f) whereas weaker correlations are explored in (g,h). For all realizations $N=1500$ and $\langle k^{(1)}\rangle=\langle k^{(2)}\rangle=50$.
• Figure 3: (a) The relation between the geometric couplings of the mutual graph and constituent single layer networks for several real multiplexes. White color corresponds to the transition point $\widetilde{\beta}=1$. (b) The effective coupling $\widetilde{\beta}$ is plotted against the smallest of the geometric couplings of the underlying layers. (c) The relationship between the average angular distance of connected nodes in the mutual graph and the mean of the average angular distances in the two corresponding single-layer MCCs.