Comprehensive routing strategy on multilayer networks

TL;DR

This paper proposes a comprehensive routing strategy for multilayer networks that considers local node structures and layer transmission speeds, significantly improving traffic capacity and congestion alleviation.

Contribution

It introduces a novel routing strategy that integrates micro- and macro-level controls, optimizing traffic load distribution across multilayer networks.

Findings

Routing strategy effectively redistributes traffic to high-speed layers.

Optimal control parameters maximize network traffic capacity.

Increasing high-speed layer size enhances overall network capacity.

Abstract

Designing an efficient routing strategy is of great importance to alleviate traffic congestion in multilayer networks. In this work, we design an effective routing strategy for multilayer networks by comprehensively considering the roles of nodes' local structures in micro-level, as well as the macro-level differences in transmission speeds between different layers. Both numerical and analytical results indicate that our proposed routing strategy can reasonably redistribute the traffic load of the low speed layer to the high speed layer, and thus the traffic capacity of multilayer networks are significantly enhanced compared with the monolayer low speed networks. There is an optimal combination of macro- and micro-level control parameters at which can remarkably alleviate the congestion and thus maximize the traffic capacity for a given multilayer network. Moreover, we find that increasing the size and the average degree of the high speed layer can enhance the traffic capacity of multilayer networks more effectively. We finally verify that real-world network topology does not invalidate the results. The theoretical predictions agree well with the numerical simulations.

Tables

Table 1. Table 1: Structural properties of Work network and Facebook network, including number of nodes N 𝑁 N , number of edges E 𝐸 E , mean degree ⟨ k ⟩ delimited-⟨⟩ 𝑘 \langle k\rangle , maximum degree k max subscript 𝑘 max k_{\mathrm{max}} , degree heterogeneity H k = ⟨ k 2 ⟩ / ⟨ k ⟩ 2 subscript 𝐻 𝑘 delimited-⟨⟩ superscript 𝑘 2 superscript delimited-⟨⟩ 𝑘 2 H_{k}=\langle k^{2}\rangle/\langle k\rangle^{2} , diameter D 𝐷 D , average shortest distance L 𝐿 L , correlation coefficient r 𝑟 r , clustering coefficient c 𝑐 c , and modularity Q 𝑄 Q .

Network	$N$	$E$	$⟨k⟩$	$k_{\max }$	$H_k$	$D$	$L$	$r$	$c$	$Q$
Work	$60$	$194$	$6.5$	$27$	$1.7$	$4$	$2.4$	$-0.218$	$0.64$	$0.46$
Facebook	$32$	$124$	$7.8$	$15$	$2.3$	$4$	$2.0$	$0.003$	$0.54$	$0.34$

Equations

$$p(s\to t):=s\equiv v_F^0,v_F^1,\cdot \cdot \cdot ,v_F^l,\cdot \cdot \cdot ,v_F^{d-1},v_F^d\equiv t,$$

$$L(p(s\to t),{\alpha }_F,{\beta }_F)=\sum _{l=0}^{n-1}{\alpha }_F[k(v_F^l){]}^{{\beta }_F}.$$

$$H(R)=\underset{t\to \infty }{\lim }\frac{C}{R}\frac{⟨\Delta W⟩}{\Delta t},$$

$$g({\alpha }_F,{\beta }_F,v)=\sum _{s\ne t}\frac{{\sigma }^{st}({\alpha }_F,{\beta }_F,v)}{{\sigma }^{st}({\alpha }_F,{\beta }_F)},$$

$$⟨{\Theta }_F⟩=R\frac{g({\alpha }_F,{\beta }_F,v)}{N_A(N_A-1)}.$$

$$R_c=\frac{N_A(N_A-1)}{g_{\max }({\alpha }_F,{\beta }_F)},$$

$$\lambda =\frac{\sum _{s\ne t}{\sigma }_B^{st}({\alpha }_F,{\beta }_F)}{\sum _{s\ne t}{\sigma }^{st}({\alpha }_F,{\beta }_F)},$$

$$\delta =\frac{\sum _{s\ne t}{e}_B^{st}({\alpha }_F,{\beta }_F)}{\sum _{s\ne t}{e}_A^{st}({\alpha }_F,{\beta }_F)},$$

$$⟨d⟩=\frac{1}{N_A(N_A-1)}\sum _{s\ne t}{d}^{st}({\alpha }_F,{\beta }_F),$$

$$⟨T⟩=\underset{t\to \infty }{\lim }\frac{1}{n}\sum _{p=1}^n{T}_p,$$

Full text

Alleviating the congestion in transportation and communication systems is vital to modern society. For the purpose of redistributing the traffic load in a low speed transportation system such as bus net, we can establish a high speed system (e.g., subway network) in the busy regions or between the stations with high traffic flow, and the two systems make up a new multilayer system (i.e., multilayer network). Recent years, some investigations about traffic congestion on multilayer networks were performed, which mainly focused on the different roles of layers in a macroscopic level (e.g., different transmission speeds), or the local structures of nodes within the same layer in a microscopic level, without taking them into consideration comprehensively. To this end, we propose a comprehensive routing strategy on multilayer networks composed of a low and a high speed network. We introduce a macro- and a micro-level parameter to adjust the roles of network structures played in the routing strategy. Our strategy redistributes the traffic load in low speed layer to the high speed layer reasonably, and the traffic capacity of multilayer networks are thus remarkably enhanced compared with the monolayer low speed networks. For a given multilayer network, an optimal combination of macro- and micro-level parameters is found. Under these parameters, the traffic capacity of the system reaches its maximum value. Moreover, increasing the networks size and the average degree of the high speed layer can enhance the traffic capacity of multilayer networks more effectively. To quantificationally understand the proposed routing strategy, we developed a theoretical approach and a remarkable agreement with numerics is observed in both artificial and real-world networks. Our research may stimulate future studies on designing realistic transportation and communication multilayer networks.

I Introduction

Many systems in modern society can be described by complex networks, such as power grids, transportation networks and social networks Strogatz (2001); Albert and Barabási (2002); Boccaletti et al. (2006); Newman (2010); Barabási (2016). Routing on such networked systems to enhance traffic capacity is a significant issue, and has been widely studied from the perspective of complex network framework over the past decades Arenas et al. (2001); Zhao et al. (2005); Wu et al. (2006); De Martino et al. (2009); Barthélemy (2011); Wu et al. (2013). Most studies about routing are focused on monolayer networks. The studies have revealed that traffic congestion is highly related to the structures of networks Boccaletti et al. (2006); Guimera et al. (2002); Guimerà et al. (2002). Generally, there are three widely used techniques to enhance the throughput of the whole network: (1) modification of network structures Danila et al. (2006, 2007); Liu et al. (2007); Zhang et al. (2007), (2) optimization of traffic resources allocations Xia and Hill (2010); Xiang et al. (2013); Zhang et al. (2011), and (3) designing better routing strategies Yan et al. (2006); De Martino et al. (2009); Tang et al. (2009); Tadić and Mitrović (2009); Ling et al. (2010); Tang and Zhou (2011); Rachadi et al. (2013); Gan et al. (2013). Compare with the first two methods, proposing effective routing strategies seems to be more practical and thus has attracted much interest. Among numerous different kinds of proposed routing strategies, an efficient routing strategy proposed by Yan and his colleagues is widely acknowledged for its simplicity and efficiency Yan et al. (2006). The strategy redistributes traffic load in central nodes to other noncentral nodes and improves the network throughput significantly. Echenique et al. proposed a novel traffic awareness protocol (TAP) by considering the waiting time of packets, in which a node forwards a packet to its neighboring node according to the shortest effective distance Echenique et al. (2004, 2005). Some scholars also proposed strategies for systems with limited band width Wu et al. (2008); Wang et al. (2009).

With the availability of big data, scholars found that modern infrastructures are actually significantly interact with and/or depend on each other, which can be described as multilayer (multiplex) networks Gao et al. (2012); Kivelä et al. (2014); Lee et al. (2015); Boccaletti et al. (2014). For example, to redistribute the traffic load in a low speed transportation network, we can build a new high speed network in the busy regions or between the high flow stations, and the two monolayer networks constitute a multilayer network. Researchers have demonstrated that the dynamics of Boccaletti et al. (2014) and on Buldyrev et al. (2010); Wang et al. (2014); Gómez-Gardenes et al. (2012); Aguirre et al. (2014) multilayer networks are markedly different from monolayer networks. Until very recently, some researchers studied the traffic dynamics on multilayer complex networks, i.e., how to alleviate the traffic congestion in order to enhance the multilayer network capacity Morris and Barthelemy (2012); Zhou et al. (2013); Yagan et al. (2013); Tan et al. (2014); Solé-Ribalta et al. (2016); Li et al. (2016). Interestingly, Solé-Ribalta et al. Solé-Ribalta et al. (2016) developed a standardized model of transportation in multilayer networks, and showed that the structure of multiplex networks can induce congestion on account of the unbalance of shortest paths between layers. Morris and Barthélemy Morris and Barthelemy (2012) analyzed a multilayer network consists of two layers, and showed that it is possible to obtain an optimal communication multiplex by balancing the effects between decreasing the average distance and congestion on a very small subset of edges.

The structures of multilayer networks bring new challenges when we propose effective routing strategies, and the task is markedly different from monolayer networks. On one hand, multilayer networks can relieve the traffic congestion of the low speed layer by using the high speed layer, however congestion may be induced in the high speed layer Brummitt et al. (2012). Although establishing high speed transportation networks can improve the traffic capacity of low speed network, how to reasonably the redistribute traffic loads is an essential issue. On the other hand, when designing effective routing strategies we should (1) take the intra-layer structures into considerations from microscopic perspective, and (2) consider the efficiencies of different layers from a macroscopic view. Previous investigations about traffic congestion on multilayer networks mainly focused on the macroscopic differences between layers Morris and Barthelemy (2012), or the local structure of nodes within the same layer in a microscopic level Tan et al. (2014), without taking both of them into consideration comprehensively. In this work, we propose an comprehensive routing strategy on multilayer networks by incorporating the macroscopic difference of speed between layers and microscopic distinctions among different nodes in the same layer. We find that our routing strategy can redistribute the traffic load in low speed layer to high speed layer reasonably, and the traffic capacity of multilayer networks are remarkably enhanced compared with the monolayer low speed networks. For a given multilayer network, there is an optimal combination of macro-level parameter and micro-level parameter that maximize the traffic capacity. Increasing the size and average degree of the high speed layer can enhance the traffic capacity of multilayer networks more effectively. Numerical results on artificial multilayer networks as well as the real Work-Facebook multilayer network agree well with our analysis.

The outline of the paper is as follows. In Sec. II, we give a detailed description of our routing strategy on multilayer networks. In Sec. III, we suggest theoretical analysis. In Sec. IV, we present our simulation results. Section V summarizes our results and conclusions.

II Model

II.1 Network model

The multilayer network considered is composed by two layers with $N_{A}$ and $N_{B}$ nodes respectively. Layer $A$ represent the low speed network, and layer $B$ is the high speed network. In general condition, the expense of building a high speed network is far more than that of low speed network, thus the size of high speed network is smaller. For example, in the railway-airline multilayer network, where the speed (cost) of the airline network is faster (more) than that of the railway network. Thus, the size of the airline network is smaller than the railway network, and all airline stations are located at points which can be considered as nodes in the railway network, but no vice versa Gu et al. (2011). For simplicity, we assume that the nodes in the high speed layer $B$ are a random subset of the low speed layer $A$ Morris and Barthelemy (2012). We use the uncorrelated configuration model (UCM) Catanzaro et al. (2005) to generate the low speed layer $A$, and use the Erdö-Rényi (ER) networks Erd6s and Rényi (1960) to represent the high speed layer $B$. The multilayer network is generated as follows: (1) Build layer $A$ using the UCM method with power-law degree distributions $P(k)\sim k^{-\gamma}$, where $\gamma$ is the degree exponent. We set the size of layer $A$ as $N_{A}$, the minimum degree is $k_{\rm min}=2$, and the maximum degree is $k_{\rm max}\sim\sqrt{N_{A}}$. (2) Randomly select $N_{B}$ ($N_{B}\leqslant N_{A}$) nodes in layer $A$, and match these nodes one-to-one. This means that each pair of the two matched nodes $v_{A}^{o}$ and $v_{B}^{o}$ are actually the same node but in two different transport manners. Both of them can be denoted as a coupled node $v^{o}$. Or say, a coupled node $v^{o}$ has two replica nodes in layers $A$ and $B$, which are denoted by $v_{A}^{o}$ and $v_{B}^{o}$ respectively. (3) Construct a ER network as the second layer $B$ by using the selected $N_{B}$ nodes in step (2), i.e., each pair of these randomly selected nodes are connected with a probability $p$. According to the above three steps, a multilayer network can be built. Note that every node in layer $B$ has its counterpart node in layer $A$, but the inverse is not true. We denote the degree distribution of the multilayer network as $P(\overrightarrow{K})=P(k_{A},k_{B})$, where $k_{A}$ and $k_{B}$ denote the degrees in layer $A$ and $B$ respectively. For a node $v_{A}$ in layer $A$ without counterpart in layer $B$, we have $k_{B}=0$. An illustration of the multilayer network is shown in Fig. 1(a).

II.2 Routing model

In our model, we assume all nodes in layer $A$ are treated as both hosts and routers for generating and delivering packets, while the nodes in layer $B$ can only deliver packets. We assume that a coupled node $v^{o}$ can deliver packets between layers with infinity bandwidth and no time consumption through two replica nodes $v_{A}^{o}$ and $v_{B}^{o}$. For simplicity, each node in the layer $A$ ($B$) has the same maximum packet delivery ability $C_{A}$ ($C_{B}$). That’s to say, at each time step each node $v_{A}$ can delivery $C_{A}$ packets to its neighbors in layer $A$ if it has no counterpart in layer $B$. Otherwise, its counterpart node $v_{B}$ can also transmit $C_{B}$ packets in layer $B$. We set $C_{A}=C_{B}=C=1$ in this paper for simplicity.

Due to the finite delivery ability of nodes, a queue of buffers is needed for each node to accommodate packets waiting for being delivered. The transport processes is as follows:

Packet generation. At each time step, $R$ number of packets are generated with randomly chosen origins and destinations in layer $A$. For each packet, a path from the source to the destination is chosen according to the comprehensive multilayer routing strategy (to be introduced in the next subsection). If there are several paths between these two nodes, we choose one randomly. Each newly created packet is placed at the end of the queue of its source node $v_{A}$ if the next stop is in layer $A$, or queued at the counterpart node $v_{B}^{o}$ if the next stop is in layer $B$. 2. 2.

Packet processing. The first-in-first-out (FIFO) rule is adopted to hand each queue. At each time step, node $v_{A}$ ($v_{B}$) can process $C_{A}=1$ ($C_{B}=1$) packet from the head of it’s queue and deliver the packet to the next stop in layer $A$ ($B$). So a coupled node $v^{o}$ at most process 2 packets per time step through two replica nodes $v_{A}^{o}$ and $v_{B}^{o}$ in layers $A$ and $B$. For a non-coupled node $v_{A}$ (i.e., node $v_{A}$ has no counterpart in layer $B$), it can process only $C_{A}=1$ packet. When a packet arrives at its destination, it is removed from the system; otherwise it is queued.

II.3 Comprehensive multilayer routing strategy

By integrating different roles of nodes in micro-level, as well as different transmission speed of layers in macro-level, we propose a comprehensive multilayer routing (CMR) strategy, which can remarkably enhance the traffic capacity of multilayer networks. We denote that a path between nodes $s$ and $t$ as

[TABLE]

where $v_{F}^{l}$ is the $l$-st stop and node $v_{F}^{l}$ belongs to layer $F\in\{{A,B}\}$, and $d$ is the number of stops in this path. Similar to Ref. Yan et al. (2006), we denote an ‘efficient path’ for any path between nodes $s$ and $t$ as

[TABLE]

where $k(v_{F}^{l})$ is the degree of node $v_{F}^{l}$ in layer $F$. The efficient path between $s$ and $t$ is corresponding to the route that makes the sum $L(p(s\rightarrow t),\alpha_{F},\beta_{F})$ minimum. If there are several efficient paths between two nodes, we choose one randomly. The efficient path is related to the macro-parameter $\alpha_{F}$ and micro-parameter $\beta_{F}$. The macro-parameter $\alpha_{F}\geq 0$ controls packet transmission speed in layer $F$, and reflects the macro-level transmission speed difference between layers. The smaller value of $\alpha_{F}$, the faster transmission in layer $F$. The parameter $\alpha_{A}$ ($\alpha_{B}$) corresponds to the slower (faster) network and the ratio $\alpha_{B}/\alpha_{A}$ controls the relative time spends each jump in layer $B$ compared with layer $A$. The micro-parameter $\beta_{F}$ determines the tendency of packets’ favour to small-degree or large-degree nodes in layer $F$, and reflects the micro-level difference between nodes in the same layer. Large degree (small degree) nodes in layer $F$ are preferentially to be the next stop when $\beta_{F}<0$ ($\beta_{F}>0$). When $\beta_{F}=0$, nodes with different degrees have the same probability to be the next stop. Fig. 1 illustrates the routing on multilayer networks.

III theoretical analysis

From the perspective of statistical physics, we use the order parameter $H(R)$ to characterize the congestion on multilayer networks Yan et al. (2006),

[TABLE]

where $\Delta W=W(t+\Delta t)-W(t)$, $\langle\cdots\rangle$ is average value over $\Delta t$, and $W(t)$ is the total number of accumulated packets in the system at time $t$. From the varying of $H$ with $R$, we will know the critical point $R_{c}$ (to be computed later) above which the congestion occurs. For a small value of $R$, the number of generated and delivered packets are balanced, i.e., every packet can be transported to their destinations, thus $H(R)=0$. For a large value of $R$ (i.e., $R>R_{c}$), the congestion occurs and the number of accumulated packets increases with time, so $H(R)>0$. The critical traffic capacity $R_{c}$ is the most significant parameter of a transportation network, which can be used to evaluate the performance of a routing strategy, i.e., the larger, the better.

To compute the value of $R_{c}$, we first define the efficient betweenness centralities (EBC) of nodes in multilayer networks as

[TABLE]

where $\sigma^{st}(\alpha_{F},\beta_{F})$ is the number of efficient paths between nodes $s$ and $t$ for given values of $\alpha_{F}$ and $\beta_{F}$, and $\sigma^{st}(\alpha_{F},\beta_{F},v)$ is the number of efficient paths that pass node $v$. The larger value of $g(\alpha_{F},\beta_{F},v)$, the more efficient paths that pass node $v$. As a result, node $v$ needs to process more packets and has a larger probability to be congested. We denote nodes with high values of EBC as high-load (HL) nodes, and similarly denote nodes with low values of EBC as low-load (LL) nodes. A coupled node $v^{o}$ can deliver the packets to its neighbors in both layers $A$ and $B$, and it has two values of EBCs $g(\alpha_{F},\beta_{F},v_{A}^{o})$ and $g(\alpha_{F},\beta_{F},v_{B}^{o})$, where $v_{A}^{o}$ ($v_{B}^{o}$) represents node $v^{o}$ in layer $A$ ($B$). If node $v_{A}^{o}$ or $v_{B}^{o}$ overload, traffic congestion will occur at node $v^{o}$. Thus, node $v^{o}$’s EBC in the system is the maximum value of $g(\alpha_{F},\beta_{F},v_{A}^{o})$ and $g(\alpha_{F},\beta_{F},v_{B}^{o})$, i.e., $g(\alpha_{F},\beta_{F},v^{o})=\max\{g(\alpha_{F},\beta_{F},v_{A}^{o}),g(\alpha_{F},\beta_{F},v_{B}^{o})\}$. For a non-coupled node $v_{A}$ (i.e., node $v_{A}$ has no counterpart in layer $B$), it can only deliver the packets to neighbors in layer $A$, and its EBC can be expressed as $g(\alpha_{F},\beta_{F},v_{A})$.

At every time step, the system will generate $R$ packets in layer $A$. We can get the average number of packets that a node $v$ needs to process as

[TABLE]

When $R\leq R_{c}$, there is no accumulated packets at any node in the system, i.e., $\langle\Theta_{F}\rangle\leq C$. When $R>R_{c}$, traffic congestion will occur at some HL nodes, i.e., $\langle\Theta_{F}\rangle>C$. Since the node with the largest EBC value has the largest probability being congested, and combining the condition $C=1$, the critical packet generating number $R_{c}$ should fulfill

[TABLE]

where $g_{\rm max}(\alpha_{F},\beta_{F})$ is the largest value of EBC in the system for the given $\alpha_{F}$ and $\beta_{F}$.

IV results

We introduce four parameters to investigate the effectiveness of CMR strategy. First, we introduce a generalized parameter coupling based on Ref. Morris and Barthelemy (2012), which is used to describe how well two layers are used to transmit the packets. Here the coupling is defined as

[TABLE]

where $\sigma^{st}(\alpha_{F},\beta_{F})$ is the number of efficient paths between nodes $s$ and $t$ for given values of $\alpha_{F}$ and $\beta_{F}$, and $\sigma_{B}^{st}(\alpha_{F},\beta_{F})$ is the number of efficient paths that contains at least one edge in layer $B$. Specifically, we have $\sigma_{B}^{st}(\alpha_{F},\beta_{F})=0$ when every efficient path between nodes $s$ and $t$ only uses the edges in layer $A$. For the case of $\lambda\approx 0$, most packets are transported only by layer $A$, without using the edges in layer $B$. With the increase of $\lambda$, more packets are transported by using the edges in $B$.

Secondly, we define $\delta$ as

[TABLE]

where $e_{A}^{st}(\alpha_{F},\beta_{F})$ [$e_{B}^{st}(\alpha_{F},\beta_{F})$] is the number of edges belonging to layer $A$ ($B$) in the efficient paths between nodes $s$ and $t$ for given values of $\alpha_{F}$ and $\beta_{F}$. When $\delta\approx 0$, most edges that are used to deliver packets belong to layer $A$. The more edges in layer $B$ are used to transport packets, the larger value of $\delta$. The definitions of $\lambda$ and $\delta$ look similar, the difference is that coupling $\lambda$ represent the proportion of all the efficient paths in system that contain edges in layer $B$, while $\delta$ is that, in all efficient paths, the ratio of edges in $B$ and $A$.

Thirdly, we define the average length of efficient paths to capture the effectiveness of the CMR strategy as

[TABLE]

where $d^{st}(\alpha_{F},\beta_{F})$ is the length or jumps of efficient paths between node $s$ and $t$ for given values of $\alpha_{F}$ and $\beta_{F}$. For example, the length of selected path between nodes $s$ and $t$ in Fig. 1 is 3. The smaller of $\langle d\rangle$, the less average jumps of the packets arrive the destination.

To improve network traffic capacity, the average packet delivery time $\langle T\rangle$ must be minimized. The definition of $\langle T\rangle$ is

[TABLE]

where $n$ is the number of arrived packets at a given time and $T_{p}$ is the packet delivery time of packet $p$. The delivery time of each packet consists of the travelling time from the origin to the destination and the waiting time in the queue of the congested nodes. When $R$ is less than $R_{c}$, $\langle T\rangle$ only depends on the travelling time which is relatively small, while when $R>R_{c}$, $\langle T\rangle$ increases with $R$ rapidly.

IV.1 Artificial multilayer networks

In this subsection, we perform extensive numerical simulations on artificial multilayer networks. We set the size of layer $A$ as $N_{A}=1000$, degree exponents $\gamma=3.0$, the minimum degree $k_{\rm min}=2$, and the maximum degree $k_{\rm max}\sim\sqrt{N_{A}}$. The size of layer $B$ is $N_{B}=500$ and average degree $\langle k_{B}\rangle=6$. All the results are obtained by averaging over 20 different network realizations, with 100 independent runs on each realization.

We first focus on the effects of micro-parameter $\beta_{B}$ on the effectiveness of CMR strategy in Fig. 2. Since $\beta_{A}=1$ is optimal value without layer $B$ for the case of $\alpha_{A}=1$ [see the inset of Fig.2(c)], we set $\alpha_{A}=1$ and $\beta_{A}=1$. Through extensive numerical simulations, we find that other values of $\alpha_{A}$ and $\beta_{A}$ do not qualitatively affect the effectiveness of the proposed CMR strategy. We set $\alpha_{B}/\alpha_{A}<1$ (i.e., $\alpha_{B}\leq 1$) here, which indicates that a journey on the high speed layer $B$ is favored for a journey in layer $A$. From Figs. 2(a) and (b), we find that for different values of micro-parameter $\beta_{B}$, both the order parameter $H$ and average packet delivery time $\langle T\rangle$ monotonically increases with $R$. Above the threshold $R_{c}$, $H$ and $\langle T\rangle$ are finite, and increase with $R$. Importantly, we find that $R_{c}$ exhibits a non-monotonously varying with $\beta_{B}$ as shown in Fig. 2(c), and the system exists an optimal value $\beta_{B}^{o}(\alpha_{B})=0.7$ at which the traffic capacity $R_{c}$ reaches the maximum value $R_{c}^{o}(\alpha_{B})=79$ when $\alpha_{B}=0.5$. The average length of efficient paths $\langle d\rangle$ reaches the minimum value at the same parameters [see Fig. 3(c)]. Specifically, $R_{c}$ first increases with $\beta_{B}$, and peaks at $\beta_{B}^{o}(\alpha_{B})=0.7$, and then decreases. The theoretical predictions agree well with the numerical values of $R_{c}$. To understand the non-monotonous phenomenon, we need to check what happens when varying $\beta_{B}$. When $\beta_{B}$ is small (large), the values of $\lambda$ and $\delta$ are large (small) as shown in Figs. 3(a) and (b). This indicates that packets are more likely to be transmitted in layer $B$ ($A$). For a small value of $\beta_{B}$, many coupled nodes are used to transmit packets. Similar to the effective strategy on monolayer networks, preferentially transmitting the packets through small degree nodes in layer $B$ could improve the traffic capacity of the system Yan et al. (2006), and $R_{c}$ thus first increases with $\beta_{B}$. For a large value of $\beta_{B}$, most packets are transmitted on layer $A$, which decreases the usage of coupled nodes in transmitting the packets, and $R_{c}$ thus decreases. In Fig. 2(d), we further verify in which layer the congestion occurs for different values of $\beta_{B}$. To this end, we set the delivery ability of nodes in layer $A$ ($B$) is infinite when we check the upper limit capacity of layer $B$ ($A$), i.e., $C_{A}\rightarrow\infty$ and $g(\alpha_{F},\beta_{F},v^{o})=g(\alpha_{F},\beta_{F},v_{B}^{o})$ [$C_{B}\rightarrow\infty$ and $g(\alpha_{F},\beta_{F},v^{o})=g(\alpha_{F},\beta_{F},v_{A}^{o})$]. We find that congestion occurs in layer $B$ ($A$) for small (large) values of $\beta_{B}$, since layer $B$ ($A$) has a smaller critical network throughput. From what we discussed above, we can see that compared to the isolated low speed network $A$, the capacity $R_{c}$ of multilayer network is remarkably improved at some parameters, since the traffic load of the low speed layer $A$ is redistributed to the high speed layer $B$ reasonably. The system capacity is affected by both layers $A$ and $B$, and depends non-monotonically on micro-parameter $\beta_{B}$. The theoretical predictions agree well with the numerical simulations in both Figs. 2(c) and (d).

We further study the effects of network size $N_{B}$ and average degree $\langle k_{B}\rangle$ of layer $B$ (i.e., increasing the number of coupled nodes and edges in the high speed network) on system capacity in Fig. 4. As shown in Figs. 4(a) and (c), we find that the maximum traffic capacity $R_{c}^{o}(\alpha_{B})$ when $\alpha_{B}=0.5$ increases with $\langle k_{B}\rangle$ and $N_{B}$, since the number of efficient paths (coupled nodes) increase. That’s to say, increasing the size and average degree of the high speed layer enhance the traffic capacity of multilayer networks effectively. We note that the optimal micro-parameter $\beta_{B}^{o}(\alpha_{B})$ decreases with $\langle k_{B}\rangle$ [see Fig. 4(b)], but does not change with $N_{B}$ [see Fig. 4(d)]. Again, our theoretical predictions agree well with the numerical simulations.

All results above are obtained when macro-parameter $\alpha_{B}=0.5$, we next study the effects of macro-parameter $\alpha_{B}$ in Fig. 5. We find that $R_{c}^{o}(\alpha_{B})$ depends non-monotonically on $\alpha_{B}$ [$R_{c}^{o}(\alpha_{B})$ first increase with $\alpha_{B}$ and then decrease], and the corresponding optimal micro-parameter $\beta_{B}^{o}(\alpha_{B})$ monotonically decreases with $\alpha_{B}$ and reaches a suitable packets’ preference to layer $B$. For small (large) values of $\alpha_{B}$, $\lambda$ and $\delta$ are large (small) as shown in the insets of Figs. 5(a) and (b) respectively. Importantly, we find that the system reaches a maximum traffic capacity $R_{c}^{\star}$ at the optimal macro- and micro-level parameters combination $(\alpha_{B}^{\star},\beta_{B}^{\star})$, and the number of delivered packets by each layer reach to a balance. From Figs. 5(a) and (b), we obtain $R_{c}^{\star}=79$ at $(\alpha_{B}^{\star},\beta_{B}^{\star})=(0.5,0.7)$. Without the high speed network $B$, the maximum traffic capacity of low speed network is 33 [see the inset of Fig. 2(b)]. The maximum traffic capacity of system is improved about 2.5 times once the network $B$ is induced. Although establishing high speed transportation can improve the traffic capacity of low speed network, our results indicate that a reasonable redistribution of traffic load is an essential issue. The theoretical predictions agree well with the numerical simulations.

IV.2 Real-world networks

A wide range of systems in the real world have multiple subsystems and layers of connectivity, which can be described as multilayer networks Gao et al. (2012); Kivelä et al. (2014); Lee et al. (2015); Boccaletti et al. (2014). We verify the effectiveness of our proposed CMR strategy on a real-world multilayer network, which is a social network of Employees of Computer Science Department (ECSD) at Aarhus University Magnani et al. (2013). The multilayer social network consists of five kinds of online and offline relationships (Facebook, Leisure, Work, Co-authorship, Lunch), and we choose the Work and Facebook relationships as layer $A$ and layer $B$, respectively. We denote this real-world network as Work-Facebook multilayer network. Layer $A$ composes of $60$ nodes and $194$ edges, and layer $B$ has $32$ nodes and $124$ edges. Some structural properties of the two networks are presented in Table 1.

We study the effectiveness of the CMR strategy on the Work-Facebook multilayer network in Fig. 6. Since the extremely complicated structures of networks, there has two peaks of $R_{c}$ versus $\beta_{B}$, at which the traffic capacity is very large, and the $R_{c}^{o}(\alpha_{B})$ corresponds to the second peak. Compared to the isolated Work network, the capacity $R_{c}$ of Work-Facebook multilayer network are improved when Facebook network joins in the system [see the inset of Fig. 6(a)], and the system capacity is affected by both Work and Facebook networks [see Fig. 6(b)]. In Fig. 7, we find that $R_{c}^{o}(\alpha_{B})$ versus $\alpha_{B}$ exhibits a nonmonotonic pattern [see Fig. 7(a)], and the corresponding optimal micro-parameter $\beta_{B}^{o}(\alpha_{B})$ monotonically decreases with $\alpha_{B}$ [see Fig. 7(b)]. Similar to the artificial networks, we find that the system reaches a maximum traffic capacity $R_{c}^{\star}=15$ at the optimal micro- and macro-level parameters $(\alpha_{B}^{\star},\beta_{B}^{\star})=(2.1,0.7)$. The fluctuation of curves in Figs. 6 and 7 is caused by the extremely complicated structures of both Work and Facebook networks. We should note that the theoretical predictions markedly well agree with the numerical simulations.

V DISCUSSIONS

For the purpose of alleviating the congestion of a low speed transportation network, an intuitive way is to built a new high speed network in busy regions or among the high flow nodes. The low and high speed networks constitute a multilayer network. How to reasonable redistribution of traffic load to maximize the multilayer network is an essential issue and full of challenges. In this work, we first proposed a comprehensive multilayer network routing (CMR) strategy by considering different transmission speeds of layers from the macroscopic view (by adjusting a macro-parameter $\alpha_{F}$), and different roles of nodes from the perspective of microscopic structure (controlled by a adjustable micro-parameter $\beta_{F}$). We then performed extensive numerical simulations on both artificial and real-world networks. We found that our routing strategy can redistribute the traffic load in low speed layer to high speed layer reasonably, and the traffic capacity of multilayer network are remarkably enhanced compared with the monolayer low speed network. In addition, the system capacity is affected by both layers $A$ and $B$, and depends non-monotonically on micro-parameter $\beta_{B}$ and macro-parameter $\alpha_{B}$. For a given multilayer network, the system reaches a maximum traffic capacity $R_{c}^{\star}$ at the optimal micro- and macro-level parameters $(\alpha_{B}^{\star},\beta_{B}^{\star})$. Moreover, we found that increasing the size and the average degree of the high speed layer $B$ enhances the transport capacity of multilayer networks more effectively. The theoretical predictions agree well with the numerical simulations in both artificial and real-world networks.

A wise way to alleviate traffic congestion for multilayer networks is designing effective multilayer network routing strategy. Our results exhibit a way to reasonable redistribute the traffic load. In this work, we proposed an effective strategy which considers the local structures of different nodes, as well as the transmission speeds of different layers. We study our proposed strategy on multilayer networks including two layers, and it can remarkably improve the systems’ traffic capacity. Our research may stimulate future studies on designing realistic transportation and communication multilayer networks, such as, considering different delivery abilities of nodes, limited traffic resources, transmission cost of layers, and multilayer networks with more than two layers.

Acknowledgements.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11575041 and 61673086), and the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2015J153).