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Most complex real systems are found to have multiple layers of connectivity and required to be modelled as multiplex networks. One of the extremely critical problems is to reduce the congestion and enhance the transfer capacity, especially in real communication networks with a big data environment. A novel and effective strategy to improve traffic and control congestion is proposed by adding edges according to their weights which are defined by the topology structural properties. Furthermore, which layer is more effective when our strategy is applied is discussed based on its topology structure. Adding edges between nodes whose product of multiplex network betweenness is the highest is confirmed to be more effective, particularly in the layer with stronger community structure. Simulation experiments on both computer-generated and real-world networks demonstrate that our strategy can enhance the transfer capacity of multiplex networks significantly, which is in good agreement with our analysis.

The complex real systems can be depicted as complex networks [_{i} packets at the head of the queue of each node _{c} of the packet-generated rate [_{c} along with the increase of the packet-generated rate. For _{c}, the amount of generated and transferred packets is balanced and the whole network is in a steady free state. For _{c}, it turns into a jammed state because the node cannot transfer the packet beyond its limited transfer capacity. This critical packet-generated rate _{c} can best reflect the maximum packet transfer capacity of a network. Therefore, it is used to estimate which routing strategy is of more validity. It may have a certain reference value for the real-world networks.

Along with the development of studies on the topology of complex networks, a mesoscopic description, community structure, is found in many networks [

Traditional studies of complex networks usually assume that all nodes are linked to each other by a certain type of edge to produce a single-layer network. Recently, it has been recognized that lots of complex real systems are not composed by single network primitively, but by multiplex network [

Adjusting the network topology structure, such as adding or deleting some edges, is effective to improve the network transfer capability of the single-layer network [

The rest of this paper is organized as follows. Section

The node betweenness _{i} is widely used to access the possible traffic through node _{c} is [

When it turns to a multiplex network, there are two different types of shortest paths: intralayer paths and interlayer paths. When we investigate the dynamics of multiplex network, the number of shortest paths must be considered together with the intralayer paths and interlayer paths through a certain node. The critical packet injection rate of the multiplex network is as follows [

where _{i} is the max node processing rate of node _{i} in the previous studies. In general, we suppose that all nodes have the same maximum processing rate _{i} and we set _{i} = 1 for simplicity. _{i} is the multiplex betweenness of node _{c} is used to estimate the maximum transfer capacity of a multiplex network.

In a system divided into _{ij} is the portion of all edges in the system that connect nodes in two different communities, community

Different divisions of the network result in different _{max}. The greater modularity _{max} implies the stronger community structure inside the system. Multiplex networks always contain two or more single-layer networks with different community structures. Our strategies are as follows:

(1) In a certain single-layer network of multiplex network, we calculate the weight of all node pairs, _{ij}, between nodes _{ij} is equal to the product of their multiplex betweenness _{i}∗_{j}_{ij} is equal to the product of their node betweenness _{i}∗_{j} using the shortest-path routing algorithm. In the DDP strategy, _{ij} is equal to the product of their degree _{i}∗_{j}

(2) We sort the weights in decreasing order and add the edge between the node pair whose weight is at the top. If adding an edge will cause multiple edges between the node pair, we will not add the edge, but cope with the node pair rank next.

(3) We renew the weight _{ij} and duplicate step 2 until a certain _{e} of edges are added.

In a simple network with _{e} = 1 means 0.1 ∗ (_{c} of the new multiplex system and record the enhancement as _{c}/_{c0} where _{c0} is the critical packet injection rate of the original multiplex system.

For comparison, we define the RAN strategy as adding edges randomly (also without multiple edges). And, the validity of our strategies on different layers will be discussed by adapting our strategies in different layers.

We generate a multiplex network that only consists of two Erdős–Rényi single-layer networks. In each single-layer network, we utilize a series of pseudorandom networks with _{in}, connected to the nodes in the same community and _{out} connected to the nodes of any other communities, while the average degree <_{in} + _{out} is constant. We can adjust _{in} for different community structures. In all simulations, we generate 100 multiplex networks randomly to calculate the average.

Firstly, we generate a multiplex network with 128 nodes which are separated into 4 equal communities while the average degree <_{in} is fixed to be 8, which means the single-layer network in layer 1 is a totally random network. In layer 2, we change _{in} from 8 to 11 and 14 to get increasingly pronounced community structure. The results are exhibited in Figure

The enhancement of our strategies in multiplex network: (N) = 128, (m) = 4, <(k)> = 16, and (Z)_{in} = 8 in layer 1; in layer 2, (a) (Z)_{in} = 8, (b) (Z)_{in} = 11, and (c) (Z)_{in} = 14.

As shown in Figure _{e} is larger than zero, _{c}/_{c0} is greater than 1. It means that when some edges are added, the critical packet injection rate of the new multiplex network is greater than that of the original multiplex network. Therefore, our strategies can enhance the transfer capacity of the multiplex network. In Figures _{c}/_{c0} of the MBP strategy is the highest. The modularity _{max} of the single network in layer 1 is about 0.2245. In layer 2, when _{in} is 8, the modularity _{max} is close to 0.2245. When _{in} increases to 11, the modularity _{max} is about 0.4404 and 0.6288 for _{in} = 14. By comparing Figures

Then, we apply our strategies in different layers to check the impact of community structure on our strategies. In layer 1, we set _{in} as 14, which means the single-layer network in layer 1 has a strong community structure. In layer 2, we change _{in} from 8 to 14 to obtain the results presented in Figure

The enhancement of our strategies in multiplex network: (N) = 128, (m) = 4, <(k)> = 16, and (Z)_{in} = 14 in layer 1; in layer 2, (a) (Z)_{in} = 8 and (b) (Z)_{in} = 14.

From Figure

Afterwards, we double the average degree <

The enhancement of our strategies in multiplex network: (N) = 128, (m) = 4, <(k)> = 32, and (Z)_{in} = 16 in layer 1; in layer 2, (a) (Z)_{in} = 16 and (b) (Z)_{in} = 28.

Due to the increase of the average degree <

The impact of communities number

The enhancement of our strategies in multiplex network: (N) = 128, (m) = 8, <(k)> = 16, and (Z)_{in} = 8 in layer 1; in layer 2, (a) (Z)_{in} = 8 and (b) (Z)_{in} = 14.

From Figures

Finally, we check the influence of network size. We generate two-layer multiplex network with _{in} are also increased accordingly. Results are shown in Figure

The enhancement of our strategies in multiplex network: (N) = 256, (m) = 8, <(k)> = 32, and (Z)_{in} = 16 in layer 1; in layer 2, (a) (Z)_{in} = 16 and (b) (Z)_{in} = 28.

The enhancement _{c}/_{c0} of the MBP strategy is the highest as usual regardless of the expansion of network scale. The increase of adjusting parameter of community structure _{in} leads to stronger community structure. Figure

The incessant expansion of the network poses a new challenge to our strategies. Since most of the real networks are rather large with massive number of nodes and edges, it is of vital importance to consider the computational cost of the algorithm required to compute the multiplex network betweenness. The computational complexity of our strategies is mainly dominated by the calculation of the multiplex betweenness. The time complexity of computing betweenness centrality of unweighted multiplex networks is

The above simulations are run in computer-generated multiplex networks. Then, we verify our three strategies on real-world systems. We apply our strategies to the multiplex network of relationships among employees of the Computer Science Department of Aarhus University [

The enhancement of our strategies in a real multiplex network. (a) In “Work” network. (b) In “Lunch” network.

As shown in Figure

Adding edges between nodes with the highest product of multiplex betweenness will make the two nodes to connect directly to each other. In general, it will reduce the load of those nodes with high multiplex betweenness. Stronger community structure will result in more nodes with high multiplex betweenness. That is why our strategies are more effective in the network with pronounced community structure. To uncover how our strategies work, we conduct further research on the initial and final multiplex betweenness. We employ the MBP strategy on the computer-generated multiplex network with _{in} = 8 in layer 1 and _{in} = 14 in layer 2 (the same simulation environment as Figure

The multiplex betweenness of each node before and after applying MBP strategy. (a) The computer-generated multiplex network with (N) = 128, (m) = 4, <(k)> = 16, and (Z)_{in} = 8 in layer 1 and (Z)_{in} = 14 in layer 2. (b) The real CS-Aarhus multiplex network.

Initially, in the original multiplex network, the multiplex betweennesses are distributed over a pretty wide range (see the square in Figure

In our strategies, we add edges between nodes which will result in shortcuts between nodes. Therefore, the average path length will be reduced and the small-world phenomenon is still maintained.

In order to enhance the transfer capacity of multiplex systems, we propose some strategies by adding edges according to different weights. By checking the critical packet injection rate of the multiplex network, we discover that our strategies are capable of enhancing the transfer capacity significantly. The MBP which adds edges with the highest product of multiplex betweenness is more effective than the others. The impacts of different topology characteristics, such as the community structure, the average degree, the communities number, and the nodes number, are explored to find that our strategies are more effective in multiplex networks with pronounced community structure. And in a two-layer network with different community structures, applying our strategies in the layer with the stronger community structure can yield better results. Our strategies are proved to be very effective by simulation results of computer-generated networks and real-world systems. The time complexity of our strategies is also acceptable which means our strategies might be helpful in developing more efficient transfer networks and routing strategies.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

F. SHAO thanks Dr. Albert Solé Ribalta of the Complex Systems group at IN3 for sharing the source code of references [