Measuring and modeling correlations in multiplex networks

The interactions among the elementary components of many complex systems can be qualitatively different. Such systems are therefore naturally described in terms of multiplex or multilayer networks, i.e., networks where each layer stands for a different type of interaction between the same set of nodes. There is today a growing interest in understanding when and why a description in terms of a multiplex network is necessary and more informative than a single-layer projection. Here we contribute to this debate by presenting a comprehensive study of correlations in multiplex networks. Correlations in node properties, especially degree-degree correlations, have been thoroughly studied in single-layer networks. Here we extend this idea to investigate and characterize correlations between the different layers of a multiplex network. Such correlations are intrinsically multiplex, and we first study them empirically by constructing and analyzing several multiplex networks from the real world. In particular, we introduce various measures to characterize correlations in the activity of the nodes and in their degree at the different layers and between activities and degrees. We show that real-world networks exhibit indeed nontrivial multiplex correlations. For instance, we find cases where two layers of the same multiplex network are positively correlated in terms of node degrees, while other two layers are negatively correlated. We then focus on constructing synthetic multiplex networks, proposing a series of models to reproduce the correlations observed empirically and/or to assess their relevance.

Figure 1

In a multiplex representation, different types of relationships correspond to the distinct layers of a multilayer network. For instance, in the case of the neural system of C. elegans two neurons can communicate either by means of electrical signals, which are propagated through synapses and neuronal dendrites, or by means of the diffusion of ions and small molecules, which travel through intercellular channels called gap junctions. The two types of communication are encoded in the two layers of a multiplex network.

Figure 2

The fraction $N_L$ of nodes which appear in the top $L$ positions according to degree in both layers (Phys and Gen) of the BIOGRID network (squares) scales approximately as a power law $N_L\sim L^{0.56}$ (solid line, $r^2=0.96$). In particular, fewer than 20 nodes appear in both rankings up to $L\simeq 300$, meaning that there is almost no correlation between the degrees of the a node at the two layers and that it is very unlikely that a node is a hub on both Gen and Phys.

Figure 3

Distributions of node activity for (a) the six multiplex networks of continental airlines and for (b) APS and IMDb. In all airline networks, $P\left(B_i\right)$ can be fitted by power laws with exponents ranging from $1.8$ to $2.3$ (the exponents, together with the corresponding $p$-values in parentheses, are reported in the legend). This means that the typical number of layers in which a node is active is subject to unbounded fluctuations. The plots in panel (a) have been vertically displaced to enhance readability.

Figure 4

(a) The Zipf's plot of the node-activity vectors is a power law, both for APS and for IMDb. Also the rank distribution $P\left({\mathit{b}}_i\right|B_i)$ restricted to nodes having a given value of node activity $B_i$, for (b) APS and (c) IMDb, are power laws with exponential cutoff. The exponents of the power laws range between $0.5$ (dot-dashed blue line) and $1.0$ (dashed black line).

Figure 5

Distribution of layer activity for the continental airline networks, APS, and IMDb. In the six multiplex of continental airlines, which consist of $O\left({10}^2\right)$ layers, $P\left(N^{\alpha }\right)$ has a clear power-law shape. A somehow heterogeneous behavior is also observed for IMDb, although the number of layers is not large enough to allow a meaningful fit. The plots were vertically displaced to enhance readability.

Figure 6

The pairwise multiplexity has a power-law behavior in (a) airline networks, while it is exponential in (b) APS and IMDb. In panel (c) we report a graph of the first 20 airlines in Europe by number of covered airports. Each node of this graph represents a layer of the original multiplex network, while the weight of the edge connecting two nodes is proportional to the fraction of nodes present in both layers. The size of a node is proportional to the number of airports in which the corresponding company operates, while the color (from yellow to red) corresponds to the node strength, which in this case is proportional to the total node overlap with other airlines.

Figure 7

The distribution of the normalized Hamming distance $H_{\alpha ,\beta }$ between all the possible pairs of layers on various multiplex networks. Notice that $P\left(H_{\alpha ,\beta }\right)$ increases exponentially for the continental airlines networks.

Figure 8

Distribution of pairwise multiplexity (a) and Zipf's plot of node activity (b) for the European airlines multiplex network (solid black line) and the corresponding synthetic networks obtained by four different models, namely, HM (red circles), MDM (orange squares), MSM (green diamonds), and LGM (blue triangles). Notice that LGM fits well the distribution of pairwise multiplexity and performs better than HM in reproducing the rank distribution of node activity. The shape of $P\left(B_i\right)$ in MDM and MSM is identical to that of the original multiplex by construction.

Figure 9

The rank distribution of node-activity vectors in APS (a) and IMDb (b), compared with those of synthetic multiplex networks generated using MDM and MSM.

Figure 10

Density plots of overlapping degree, participation coefficient, and node activity for APS (top panels) and IMDb (bottom panels). On average, node activity is positively correlated with both overlapping degree and participation coefficient (the solid line shows the average $\langle B_i\rangle$ computed over all the nodes having a certain value of $o_i$). However, the fluctuations in the values of $B_i$ are quite large in all the cases.

Figure 11

Different degree correlation coefficients, namely (a) Pearson's $r$, (b) Spearman's $\rho$, and (c) Kendall's $\tau$ for different couples of layers, and the corresponding distributions (d) are reported for the APS and show that interlayer correlations in this system tend to be assortative. A similar pattern is observed in IMDb [panels (e)–(h)]. However, some movie genres, like adult and talk show (respectively corresponding to layers number 2 and number 25 in the diagram) have marked negative interlayer correlations with almost all the other layers.

Figure 12

The Zipf's plots of the distribution of multidegree in (a) APS and (b) IMDb have a power-law tail with exponent close to $1.0$. However, the multidegree distribution might be affected by large fluctuations. In fact, in both cases around $90\%$ of the multidegree vectors are present only once, and more than $95\%$ are observed fewer than four times.

Figure 13

The interlayer pairwise degree correlation function $\overline{k^{\left[\beta \right]}}\left(k^{\left[\alpha \right]}\right)$ is shown for (a) C. elegans and BIOGRID and for various couples of layers $\alpha$ and $\beta$, respectively, in (b) APS and (c) IMDb. The lines reported are fit obtained by a power law of the form $\overline{k^{\left[\beta \right]}}\left(k^{\left[\alpha \right]}\right)\sim {\left(k^{\left[\alpha \right]}\right)}^{\mu }$. The plots are vertically displaced to enhance readability.

Figure 14

The interlayer correlation pattern of (a) APS and (b) IMDb is evident by considering a graph whose nodes correspond to layers and the weight of the edges is the value of the interlayer correlation exponent $\mu$. In the figure blue weights correspond to positive correlations, while red weights correspond to negative ones. (c) The distribution of the values of the interlayer correlation exponent $\mu$ in APS (solid black line) and in IMDb (dashed red line). Notice that while interlayer degree correlations are always positive in APS, the layers of IMDb might be either positively or negatively correlated.

Figure 15

The values of the Spearman correlation coefficient in the original multiplex (left panels) and in that obtained through Algorithm 7 (middle panels), respectively, for APS (top) and IMDb (bottom). In the rightmost panel of each row we show the difference between the original distribution of $\rho$ and that obtained in the synthetic network. In both cases, the overall shape of the distribution of interlayer correlations in the synthetic multiplex looks very similar to the original one. However, the differences in the obtained value of $\rho$ might be quite high. This is due to the fact that Algorithm 7 allows to set only $M-1$ pairs or correlations over the total $M(M-1)/2$.

Figure 16

The values of the interlayer degree correlation exponent $\mu$ in the APS multiplex (left) and in a synthetic multiplex network generated through Algorithm 8 (middle). The rightmost panel shows the difference between the exponents observed in the original system and those measured in the synthetic network. Although the left and the middle panels look qualitatively similar, the right panel reveals that the difference in the actual interlayer degree correlation exponent $\mu$ of the synthetic network might be as high as $0.7$.