Value of peripheral nodes in controlling multilayer scale-free networks

We analyze the controllability of a two-layer network, where driver nodes can be chosen randomly only from one layer. Each layer contains a scale-free network with directed links and the node dynamics depends on the incoming links from other nodes. We combine the in-degree and out-degree values to assign an importance value $w$ to each node, and distinguish between peripheral nodes with low $w$ and central nodes with high $w$. Based on numerical simulations, we find that the controllable part of the network is larger when choosing low $w$ nodes to connect the two layers. The control is as efficient when peripheral nodes are driver nodes as it is for the case of more central nodes. However, if we assume a cost to utilize nodes that is proportional to their overall degree, utilizing peripheral nodes to connect the two layers or to act as driver nodes is not only the most cost-efficient solution, it is also the one that performs best in controlling the two-layer network among the different interconnecting strategies we have tested.

Figure 1

Illustration of the coupling strategies for two layer networks. (a) The main scenario where we aim to control the system of networks using only driver nodes from layer 0. (b) The PP interconnecting strategy. (c) The CP interconnecting strategy. (d) The CC interconnecting strategy. Nodes colored in red are coupled by interlayer links denoted by red dashed lines. In the illustration, the two layer networks are interconnected by $L=3$ links.

Figure 2

Plot of $n_b$ as a function of the fraction of interlayer links and the number of driver nodes. The characteristics of $n_b$ as a function of the fraction of interlayer links $q$ are shown for (a) $\alpha =0.5$ and (b) $\alpha =0.2$. The results are produced with 30 driver nodes. Also shown are the characteristics of $n_b$ as a function of the number of driver nodes $N_c$ for $q=0.4$ and (c) $\alpha =0.5$ and (d) $\alpha =0.2$. A change of the number of driver nodes to other values smaller than $N_d$ does not alter our findings. All the results are produced with $N_0=N_1=2000$. The data points are obtained over 100 simulations, with error bars representing standard deviation.

Figure 3

(a) Color map encoding the fraction of controlled network $n_b$ on the $\alpha \text{−}q$ parameter plane for the PP strategy. (b)–(d) Color map encoding the difference between $n_b$ for the PP and $n_b$ for the CP, CC, and PC strategies. A positive value indicates that $n_b$ for the PP strategy is greater than the corresponding strategy. The results are obtained with $N_c=30$ driver nodes.

Figure 4

(a) Plot of $n_b$ as a function of the number of driver nodes $N_c$ for $q=0.4$. Increasing the number of driver nodes $N_c$ to $N_0=2000$ confirms that the PP strategy to link peripheral nodes in both layers allows one to better control the whole network. The data points are obtained over 100 simulations, with error bars representing standard deviation. (b) Ratio of average $n_b$ with driver nodes of low $w$ over the average $n_b$ with driver nodes of high $w$. This panel shows that the difference in $n_b$ is marginal and can be safely neglected. The figure is produced using the two-layer network configuration that is discussed in the text.

Figure 5

Plot of $n_b$ as a function of the fraction of interlayer links. (a) Driver nodes are sampled from nodes of high $w$. (b) Driver nodes are sampled from nodes of low $w$. The results are produced with $N_c=30$ driver nodes.

Figure 6

Plot of $n_b$ as a function of the fraction of interlayer links and the number of driver nodes. The characteristics of $n_b$ are shown as a function of (a) and (b) the fraction of interlayer links $q$ and (c) and (d) the number of driver nodes $N_c$, for $L=800$ interconnecting links, with (a) and (b) $\gamma =2.5$ and the two network layers connected by bidirectional interconnecting links and (c) and (d) $\gamma =2.5$ and the two network layers connected by interconnecting links whose directions were assigned randomly. The results are produced with 30 driver nodes and $\alpha =0.5$ and are consistent with the results shown in Fig. 2.

Figure 7

Plot of $n_b$ as a function of the fraction of interlayer links with (a) and (b) $\gamma =2.5$ and two layers of networks connected by bidirectional interconnecting links and (c) and (d) $\gamma =2.0$ and two layers connected by interconnecting links whose directions were assigned randomly. Driver nodes are sampled from nodes of (a) and (c) high $w$ and (b) and (d) low $w$. All the results are produced with $N_c=30$ driver nodes and are consistent with the results shown in Fig. 4.