Effective traffic-flow assignment strategy on multilayer networks

An efficient flow assignment strategy is of great importance to alleviate traffic congestion on multilayer networks. In this work, by considering the roles of nodes' local structures on the microlevel, and the different transporting speeds of layers in the macrolevel, an effective traffic-flow assignment strategy on multilayer networks is proposed. Both numerical and semianalytical results indicate that our proposed flow assignment strategy can reasonably redistribute the traffic flow of the low-speed layer to the high-speed layer. In particular, preferentially transporting the packets through small-degree nodes on the high-speed layer can enhance the traffic capacity of multilayer networks. We also find that the traffic capacity of multilayer networks can be improved by increasing the network size and the average degree of the high-speed layer. For a given multilayer network, there is a combination of optimal macrolevel parameter and optimal microlevel parameter with which the traffic capacity can be maximized. It is verified that real-world network topology does not invalidate the results. The semianalytical predictions agree with the numerical simulations.

Figure 1

An illustration of a multilayer network where the nodes of layer $B$ form a subset of the nodes of network $A$, and nodes in common to both layers are considered to be coupled nodes (indicated as dashed lines). Highlighted in red, a path between the “origin” $s$ and the “destination” $t$ is represented, and the arrows show a packet-transporting process on the path without congestion. A pair of coupled nodes can transport packets between layers without time consumption. (a) At $T=0$, a packet is generated with the randomly chosen “origin” $s$ and “destination” $t$ in layer $A$, and a path (highlighted in green) from node $s$ to $t$ is chosen according to the pregiven flow assignment strategy. (b) At $T=1$, “origin” $s$ transports the packet to the next stop $i$ through the edge ($s,i$) in layer $A$. (c) At $T=2$, node $i$ transports the packet to the next stop $j$ through the edge ($i,j$) in layer $B$. (d) At $T=3$, node $j$ transports the packet to its “destination” $t$ through the edge ($j,t$) in layer $B$, and the packet is removed from the system.

Figure 2

The performance of TFA strategy on artificial multilayer networks with ${\alpha }_B=0.5$. (a) The order parameter $H$ and (b) the average delivery time $\langle T\rangle$ of packets versus $R$. In panel (a) the dashed lines show the positions of $R_c$ for different ${\beta }_B$, where the order parameter satisfies $H>2/N_A$ at $R_c$. (c) The traffic capacity $R_c$ of multilayer networks and (d) the upper limit capacity of layer $B\left(A\right)$ as a function of the ${\beta }_B$. The inset of panel (c) is the traffic capacity $R_c$ of monolayer networks (layer $A$) as a function of ${\beta }_A$ when ${\alpha }_A=1$. The semianalytical results are obtained from Eq. (6). We set the other parameters as $N_A=1000$, $\gamma =3.0$, $k_{\min }=2$, $k_{\max }=\sqrt{N_A}$, $N_B=500$, $\langle k_B\rangle =6$, ${\alpha }_A=1$, and ${\beta }_A=1$.

Figure 3

The parameters (a) the coupling coefficient $\lambda$, (b) the effective edge ratio $\delta$, and (c) average length of efficient paths $\langle d\rangle$ vs ${\beta }_B$ on artificial multilayer networks. We set the other parameters as $N_A=1000$, $\gamma =3.0$, $k_{\min }=2$, $k_{\max }=\sqrt{N_A}$, $N_B=500$, $\langle k_B\rangle =6$, ${\alpha }_A=1$, ${\beta }_A=1$, and ${\alpha }_B=0.5$.

Figure 4

The performance of the TFA strategy on artificial multilayer networks with different $\langle k_B\rangle$ and $N_B$. The maximum capacity $R_c^o\left({\alpha }_B\right)$ (top) and the corresponding optimal microlevel parameter ${\beta }_B^o\left({\alpha }_B\right)$ (bottom) as a function of $\langle k_B\rangle$ [(a) and (b)] and $N_B$ [(c) and (d)], respectively. We set the other parameters as $N_A=1000$, $\gamma =3.0$, $k_{\min }=2$, $k_{\max }=\sqrt{N_A}$, ${\alpha }_A=1$, ${\beta }_A=1$, ${\alpha }_B=0.5$, $N_B=500$ (a, b), $\langle k_B\rangle =6$ (c, d).

Figure 5

The performance of the TFA strategy on artificial multilayer networks with different ${\alpha }_B$. (a) $R_c^o\left({\alpha }_B\right)$ and (b) ${\beta }_B^o\left({\alpha }_B\right)$ as a function of ${\alpha }_B$. The insets of panels (a) and (b), respectively, exhibit $\lambda$ and $\delta$ versus ${\alpha }_B$ with ${\beta }_B={\beta }_B^o\left({\alpha }_B\right)$. We set the other parameters as $N_A=1000$, $\gamma =3.0$, $k_{\min }=2$, $k_{\max }=\sqrt{N_A}$, $N_B=500$, $\langle k_B\rangle =6$, ${\alpha }_A=1$, ${\beta }_A=1$.

Figure 6

The performance of the TFA strategy on real-world multilayer networks with ${\alpha }_B=0.5$. (a) The traffic capacity $R_c$, and (b) the upper limit capacity of layer $A$ and layer $B$ as a function of the ${\beta }_B$ on the Railway-Airline coupled networks. (c) and (d) The situations on the Work-Facebook coupled networks. We set ${\alpha }_A=1$, ${\beta }_A=1$, and ${\alpha }_B=0.5$.

Figure 7

The performance of the TFA strategy on real-world multilayer networks with different ${\alpha }_B$. (a) The maximum capacity $R_c^o\left({\alpha }_B\right)$ for a given ${\alpha }_B$, and (b) the corresponding optimal microlevel parameter ${\beta }_B^o\left({\alpha }_B\right)$ as a function of macrolevel parameter ${\alpha }_B$ on the Railway-Airline coupled networks. (c, d) The situations on the Work-Facebook coupled networks. We set ${\alpha }_A=1$, and ${\beta }_A=1$.