Link persistence and conditional distances in multiplex networks

Recent progress towards unraveling the hidden geometric organization of real multiplexes revealed significant correlations across the hyperbolic node coordinates in different network layers, which facilitated applications like translayer link prediction and mutual navigation. But, are geometric correlations alone sufficient to explain the topological relation between the layers of real systems? Here, we provide the negative answer to this question. We show that connections in real systems tend to persist from one layer to another irrespective of their hyperbolic distances. This suggests that in addition to purely geometric aspects, the explicit link formation process in one layer impacts the topology of other layers. Based on this finding, we present a simple modification to the recently developed geometric multiplex model to account for this effect, and show that the extended model can reproduce the behavior observed in real systems. We also find that link persistence is significant in all considered multiplexes and can explain their layers' high edge overlap, which cannot be explained by coordinate correlations alone. Furthermore, by taking both link persistence and hyperbolic distance correlations into account, we can improve translayer link prediction. These findings guide the development of multiplex embedding methods, suggesting that such methods should account for both coordinate correlations and link persistence across layers.

Figure 1

Link persistence and hyperbolic distance correlations in the IPv4/IPv6 Internet, arXiv, Physicians, Drosophila, C. Elegans, and Human Brain multiplexes. The plots show the translayer connection probabilities $p_{\text{trans}}^{\text{all}}\left(x_1\right),p_{\text{trans}}^{\text{c}}\left(x_1\right),p_{\text{trans}}^{\text{d}}\left(x_1\right)$, the connection probabilities $p_2^{\text{all}}\left(x_2\right),p_2^{\text{c}}\left(x_2\right),p_2^{\text{d}}\left(x_2\right)$, and the conditional average hyperbolic distances $E^{\text{all}}\left[x_2\right|x_1],E^{\text{c}}\left[x_2\right|x_1],E^{\text{d}}\left[x_2\right|x_1]$. The $y$ axes in the connection probability plots are in logarithmic scale. The insets show $p_{\text{trans}}^{\text{c}}\left(x_1\right)$ in linear scale. The bins in the $x$ axes have size 1 except from the insets where larger bins have been used to reduce fluctuations. The horizontal dashed lines indicate the value of the estimated link-persistence probability $w$ (Sec. 6e). The vertical dashed lines indicate the hyperbolic disk radii $R_1,R_2$ of layers 1 and 2.

Figure 2

Link persistence in synthetic versions of the multiplexes in Fig. 1 constructed using the GMM-LP. The plots show the translayer connection probabilities between layers 1 and 2 and the connection probabilities in layer 2, as in Fig. 1. The first two columns are the results with the real node coordinates, while the next two columns are the results with the inferred node coordinates.

Figure 3

Hyperbolic distance correlations in synthetic versions of the multiplexes in Fig. 1 constructed using the GMM-LP. The plots show the conditional average hyperbolic distances as in Fig. 1. For each multiplex, the first plot shows the results with the real node coordinates while the second plot shows the results with the inferred node coordinates.

Figure 4

(a) Edge overlap $O$ in real systems vs synthetic counterparts with and without link persistence. (b) Edge overlap $O$ in real systems vs link-persistence probability $w$.

Figure 5

(a) Connection probability $p_2^{\text{all}}\left(x_2\right)$. (b) Distribution of persistent links over the hyperbolic distances in layer 2. (c) Degree-depended average clustering $\overline{c}\left(k\right)$. (d) Degree distribution $P\left(k\right)$. The results correspond to layer 2 of a two-layer synthetic multiplex with $N={10}^4$ nodes, ${\gamma }_1=2.8,{\gamma }_2=2.3,T_1=0.7,T_2=0.5,{\overline{k}}_1={\overline{k}}_2=8$, link-persistence probability $w=0.4$, and correlation strengths as shown in the legends. In (a) the green dotted line is the theoretical prediction for the uncorrelated case $[\eta \left(x_2\right)$ given by Eq. (20)], while the solid line is the connection probability in the ${\mathbb{H}}^2$ model [Eq. (13)]. In (a), (b) the vertical dashed lines indicate the hyperbolic disk radius $R_2$. In (c), (d) the solid lines are the corresponding results in a synthetic network constructed with the ${\mathbb{H}}^2$ model with the same parameters as layer 2.

Figure 6

Complementary cumulative distribution function (CCDF) of the degree product $k_1\times k_2$ of nodes connected by persistent and nonpersistent links. The results correspond to the persistent links in layer 2 of the multiplexes of Fig. 5, and to the links (nonpersistent) in a network constructed with the ${\mathbb{H}}^2$ model with the same parameters as layer 2.

Figure 7

Translayer connection probabilities $p_{\text{trans}}^{\text{all}}\left(x_1\right),p_{\text{trans}}^{\text{c}}\left(x_1\right),p_{\text{trans}}^{\text{d}}\left(x_1\right)$ at different correlation strengths. The results correspond to the synthetic multiplexes of Fig. 5. The $y$ axes in the plots are in logarithmic scale, while the insets show the same results in linear scale. The green dotted lines in the first plot ($\nu =g=0.02$) are the theoretical predictions for the uncorrelated case [Eqs. (30), (33), and (35)].

Figure 8

Translayer link prediction (layer 1 to layer 2) in the multiplexes of Fig. 1. Left column: area under the receiver operating characteristic curve (AUROC). Right column: area under the precision-recall curve (AUPR).

Figure 9

Conditional angular distance CDF. The results are from a two-layer synthetic multiplex with $N_1=5000$ and $N_2=3000$ nodes, and angular correlation strength $g=0.5$. All nodes in the smaller layer also exist in the larger. The conditional CDFs from left to right correspond to $\mathrm{\Delta }{\theta }_1=0.1,0.5,1.0,2.0,2.5,3.0$ radians. The red solid lines are the empirical distributions, while the dashed black lines are the corresponding theoretical predictions given by Eqs. (D4) and (D5) with $N=N_2$.

Figure 10

The results are from a two-layer synthetic multiplex with $N_1=5000$ and $N_2=3000$ nodes, ${\gamma }_1=2.1,{\gamma }_2=2.5,T_1=T_2=0.5$, and ${\overline{k}}_1={\overline{k}}_2=6$ ($R_1=23,R_2=16.8$). The correlation strengths are $\nu =0.5,g=0.7$, and all nodes in the smaller layer also exist in the larger. The plots show conditional hyperbolic distance CDFs and PDFs (histograms) for $r_1=18,r_1^{\prime }=20$, and $\mathrm{\Delta }{\theta }_1=0.1,0.5,1.5$ radians (from left to right, corresponding respectively to hyperbolic distances $x_1=32,35.2,37.2$). The $y$ axis in the inset of (a) is in logarithmic scale. The $x$ axis in (b) is binned into bins of size 0.1. The solid lines in the plots are the empirical distributions, while the dashed black lines are the corresponding theoretical predictions. The empirical distributions are computed over all node pairs with $r_1=18\pm 0.5,r_1^{\prime }=20\pm 0.5$ at the corresponding angular distance $\mathrm{\Delta }{\theta }_1\pm 0.05$. The empirical distributions in (b) are average distributions over 20 simulation runs.