Multilayer weighted social network model

Recent empirical studies using large-scale data sets have validated the Granovetter hypothesis on the structure of the society in that there are strongly wired communities connected by weak ties. However, as interaction between individuals takes place in diverse contexts, these communities turn out to be overlapping. This implies that the society has a multilayered structure, where the layers represent the different contexts. To model this structure we begin with a single-layer weighted social network (WSN) model showing the Granovetterian structure. We find that when merging such WSN models, a sufficient amount of interlayer correlation is needed to maintain the relationship between topology and link weights, while these correlations destroy the enhancement in the community overlap due to multiple layers. To resolve this, we devise a geographic multilayer WSN model, where the indirect interlayer correlations due to the geographic constraints of individuals enhance the overlaps between the communities and, at the same time, the Granovetterian structure is preserved.

Figure 1

Link percolation analysis for $L=1$ (left) and $L=2$ (right). The upper figures show the relative size of the largest connected component, $R_{\mathrm{LCC}}$, as a function of the fraction of the removed links $f$. The lower figures show the susceptibility $\chi$. Red solid (green dashed) lines correspond to the case when links are removed in ascending (descending) order of the link weights. The error bars show standard errors.

Figure 2

Snapshots of the copy-and-shuffle model with different $p$ shuffling parameter values and $N=300$. Red (blue) links are in the first (second) layer, and green links are in both layers.

Figure 3

Percolation thresholds for various shuffle fraction values $p$ for the copy-and-shuffle model. The green upper and red lower lines denote the critical points $f_c^d$ and $f_c^a$, respectively. The critical points are determined by the peak of the susceptibility. The points are calculated for 50 independent runs. The blue dashed line is calculated using Eq. (3).

Figure 4

Characteristic quantities for the copy-and-shuffle multilayer WSN model. The difference $\Delta f_c$ between the percolation thresholds decreases with the shuffling probability $p$, while $\overline{c}/{\overline{c}}_0$, the ratio of the average number of communities a node belongs to at parameter value $p$ and $p=0$, decreases. There is no regime, where $\Delta f_c$ is significantly larger than zero and $\overline{c}/{\overline{c}}_0$ is considerably larger than one. The results are averaged by 50 independent samples and the errors are smaller than the symbol size.

Figure 5

Link percolation analysis for (a) $\alpha =0$ and (b) $\alpha =6$. The upper figures show the relative size of the largest connected component, $R_{\mathrm{LCC}}$, as a function of the fraction of the removed links $f$. The lower figures show the susceptibility $\chi$. Note that the scale of the horizontal axis is different from Fig. 1. Red solid (green dashed) lines correspond to the case when links are removed in ascending (descending) order of the link weight. The results are obtained by 50 independent samples. The error bars show standard errors.

Figure 6

Sample of double-layer networks of the geographic model for $N=300$. Links only in the first and the second layers are shown in blue and red, respectively, and links shared by both layers are depicted in green.

Figure 7

Percolation thresholds $f_c^a$ and $f_c^d$ for the geographic model as a function of $\alpha$. The percolation thresholds are determined as the point where maximum susceptibility is observed and then averaged over 50 independent samples. (Inset) Average degrees divided by $L$ as a function of $\alpha$ for $L=1$ and $L=2$. The values for $L=2$ are smaller than those for $L=1$ when $\alpha$ is sufficiently large because there are links appearing in both layers.

Figure 8

This figure is a similar plot in Fig. 4 for the geographic multilayer WSN model. Here $\Delta f_c$ and $\overline{c}/{\overline{c}}_0$ are shown as a function of $\alpha$. Note that for this model ${\overline{c}}_0$ also depends on $\alpha$. For $\alpha \ge 6$ we have $\Delta f_c$ significantly larger than 0 and $\overline{c}/{\overline{c}}_0$ close to 2. The results are obtained by 50 independent samples.

Figure 9

These figures show the same quantities as Fig. 4 for several values of $p_{\Delta }$ (a), $p_r$ (b), and $p_d$ (c). The results are obtained by simulation of $N=10000$ and averaged over 20 independent samples.