A complex network is usually represented by a graph. A graph consists of nodes and edges. In the literature, a node also can be called a vertex while an edge can be called a link. Mathematically, a graph is generally denoted by G = {V, E}. In this kind of notation, the symbol V represents the node set and the symbol E represents the edge set. The nodes of a graph represent the entities of a network that the graph models, while the edges represent the relationships between the entities.

The relationships between the nodes of a graph can be depicted using its adjacency matrix which is usually denoted by A. Figure 1 shows an example of a simple network together with its corresponding adjacency matrix.

For the graph G shown in Figure 1 with n = 9 nodes, its adjacency matrix is normally with size n × n. The element of A is usually denoted by a _ij , representing the relationship between nodes i and j. Note that the value of a _ij is problem-specific. For a binary network with a _ij ∈ {1, 0}, the element a _ij only represents whether there is connection between nodes i and j.

A network rewire mechanism is usually used in the network science domain. For a given network G, a network rewire mechanism aims to generate a new network G′. Figure 2 shows an example of a simple network rewire mechanism.

In Figure 2, the original network has 10 nodes and 11 edges. After adopting a simple rewire mechanism, network G has been changed into network G′. It can be seen from the figure that the network rewire mechanism just changes the edge connections between the nodes and does not change the total number of edges of the original network.

Many real-world problems can be reformulated as optimization problems. Because real-world problems could be very difficult, therefore when turning a real-world problem into an optimization problem, the optimization formula may not have mathematical properties like gradient. In this case, mathematical methods cannot solve such kind of optimization problems.

In order to solve optimization problems that cannot be solving using traditional mathematical methods, scientists have developed the so called bio-inspired algorithms [25,26] and amongst which is the particle swarm optimization algorithm (PSO) [27–29].

Suppose that one aims to maximize a function f(x) with x being the argument by using PSO. Then a PSO algorithm works with a swarm or a population of individuals. Each individual is a feasible solution x to the optimized problem f(x). Let us use p _i to denote the ith population and x _j ∈p _i to denote the jth individual in p _i . Then a PSO algorithm iteratively evolves p _i to approximate the optimal solutions to f(x). During each iteration, the algorithm updates individual x _j in the following way: x j = x j + v j (1) where v _j is the velocity of individual x _j . The velocity v _j is calculated as follows v j = w v j + c 1 r 1 L j − x j + c 2 r 2 G − x j (2) where w, c ₁ and c ₂ are three constants. r ₁ and r ₂ are two random number with r ₁, r ₂ ∈ [0, 1]. The symbol L _j is the personally best individual of x _j , i.e., the best solution that individual x _j has so far found. The symbol G is the globally best individual found by all the individuals.

Note that an individual updates its velocity v _j by referring to its historical information and the information from the whole population. The new velocity v _j then could help individual x _j to explore to promising area. During each iteration, the algorithm updates L _j and G . By iteratively evolving p _i for a prescribed number of iteration, the algorithm stops and G therefore is regarded as the optimal solution(s) to the original optimization problem.

The research aims to enhance the robustness of a given network by proposing a novel network rewire mechanism. To do so, first we need to know how to quantify the robustness of a given network. In the literature, there are many studies on network robustness [16,30]. As a consequence, many methods have been developed by scientists to quantify the robustness of complex networks [13,31] and amongst which is the method based on spectral analysis [31].

Given a network G with its adjacency matrix being denoted by A. Let us use d _i to represent the degree of node i, i.e., the number of edges attached to i. Then one can easily obtain a diagonal matrix D in which its ith element is d _i . In the literature, the Laplacian matrix L of network G is defined as L = D − A (3) For the matrix L we can work out all its eigenvalues λ . An eigenvalue λ satisfies the following relation L x = λ x (4) in which x is the corresponding eigenvector. Let λ ₂ ∈ λ be the second smallest non-negative eigenvalue of matrix L. It has been found that λ ₂ has a positive relationship with the network’s connectivity [31]. As a consequence, in the literature scientists use it as a metric to quantify the network’s robustness. Due to its computationally friendly feature, in this study we use λ ₂ as the network robustness quantification metric. Normally, the larger the value of λ ₂, the more robust the given network.

In this work, we propose a new network rewire mechanism to enhance the robustness of a given complex network. Figure 3 takes a simple network as an example to show the basic idea of the proposed network rewire mechanism.

On the left hand side of Figure 3 is a simple network G ₀ with 10 nodes. The network G ₀ originally carries with 11 edges. In order to enhance its robustness, existing rewire mechanisms change the edge connections. Specifically, they reconnect the nodes and aim to find out a new network G ₁ with exactly the same number of edges and higher network robustness.

In this work we propose a novel rewire mechanism as shown on the right hand side of Figure 3. The proposed mechanism does not require that G ₁ has exactly the same number of G. As can be seen from the right hand side of Figure 3, we delete some edges from G. We also add new edges between the nodes. As long as G ₁ has higher network robustness than G, then we can both delete edges from and add new ones to G. As compared to existing rewire mechanisms, the newly proposed mechanism is more flexible and could be more of practical usage.

Note that for a given network there are many choices for edge deletion and addition. In order to determine which edges can be deleted and added, we propose a multiobjective optimization model which will be described in detail in the subsequent subsection.

For a given network G we adopt the λ ₂ metric to quantify its robustness. The proposed network rewire mechanism aims to enhance G’s robustness by removing and adding edges. Specifically, we propose the following multiobjective optimization model. f 1 = min λ 2 0 − λ 2 1 f 2 = min | E − | f 3 = min | E + | (5) in which λ 2 0 is the robustness of G, E ⁻ is the set of edges to be deleted from G and E ⁺ is set of edges to be added to G. For network G the proposed rewire mechanism removes edges in the set E ⁻ and adds edges in the set E ⁺. Then G becomes G ₁ with respect to E ⁻ and E ⁺ and λ 2 1 is its corresponding robustness.

In the above model, objective f ₁ aims to maximize the robustness of G ₁. Both objectives f ₂ and f ₃ aim to minimize the edge operations. For a network in reality, it is not good to remove a large amount of edges from the network as it will change the network’s structure a lot. Objective f ₂ then constraints this operation. Similarly, it is not practical to add a large amount of edges to the network as adding new edges needs extra workload. Objective f ₃ is such a constraint to avoid this behaviour.

Note that existing rewire mechanisms also do edge addition and removal, i.e., they consider both objectives f ₂ and f ₃. However, they require to remain the total number of edges in the network, i.e., they require that f ₂ = f ₃. In the newly proposed rewire mechanism, we do not pose such constraints. As a consequence, the proposed mechanism is of more practical use. For networks in which adding edges is more convenient than that of deleting edges, then we can minimize f ₂ but relax f ₃. Similarly, if deleting edges is much easier than adding new ones, then we can pay more attention to f ₃.

One may clearly see from Eq. 5 that the proposed optimization model does not have a direct relationship between the decision variable (E ⁺ and E ⁻) and the objective function values. A concrete solution of E ⁺ and E ⁻ cannot directly be substituted into the calculation of f ₁. Meanwhile, E ⁺ and E ⁻ are edge sets.

The proposed model contains three objectives. In order to solve it, traditional mathematical methods will apply weighted sum method to do it. However, since the edge sets are composed of links, therefore mathematical methods cannot solve Eq. 5 as the decision variables are discrete and the objectives do not have any gradient information. Because the proposed optimization model given in Eq. 5 involves three discrete objectives, in the literature, scientists have developed nature-inspired algorithms such as genetic algorithms (GAs) [32], particle swarm optimization algorithms (PSOs) [33], etc., to solve such kind of problems. Both GAs and PSOs are designed for single objective optimization. In order to solve multiobjective optimization, representative algorithms like the non-dominated sorting genetic algorithm-version II (NSGA-II) [34], multiobjective particle swarm optimization (MOPSO) [35], multiobjective evolutionary algorithm based on decomposition (MOEA/D) [36], etc., have been developed by scientists and been applied to solve many engineering problems.

In order to solve the model shown in Eq. 5, we in this work adopt the MOPSO algorithm proposed in [27]. The MOPSO algorithm is chosen because of its simplicity in terms of algorithm understanding and implementation. We redesign some operators of the MOPSO algorithm to make it fit for the model in Eq. 5. The whole algorithm framework for solve Eq. 5 is given in Algorithm 1.

Step 4 of Algorithm 1 involves the individual representation method which will be described in the following subsection. Step 7a of Algorithm 1 evaluates a particle based on the proposed multiobjective optimization model. Step 10 of Algorithm 1 is the main for loop which mainly contains the particles’ position and velocity information update and this will be described in the following subsection. Note that the details for generating a testing network will be provided in the experiment section and all the parameter settings will also be given in the experiment section.

The algorithm proposed in [27] is for network clustering. The authors therefore developed an integer based particle position representation method. Since in this work we aim to do network rewire to enhance network robustness, we thus need to reconsider the way to represent a particle position.

Keep in mind that the purpose of the work is to find out the edge sets E ⁺ and E ⁻ so as to improve the robustness of a given network. For a given network G, we know its adjacency matrix A and its edge set E. We then can remove edges from E and E ⁻ can be determined. Based on matrix A, we also know the possible new edges between the nodes and therefore can determine E ⁺. Considering this, we thus propose the following way to represent the position of the ith particle of a PSO algorithm. x i = ( x i 1 , x i 2 , … , x i m ) (6) In Eq. 6, x _ij ∈ x _i is a binary variable, i.e., x _ij ∈ {0, 1}. The length of x _i is m and m = n × (n − 1)/2. Figure 4 takes a simple network as an example to show how Eq. 6 works.

In Figure 4, the simple network G has five nodes and seven edges. Its adjacency matrix is given as A. In the figure there is a binary sequence x which represents a particle’s position. This binary sequence represents a possible edge rewire solution. We first set up an empty matrix A′. Then we turn the binary sequence x into the elements of the upper triangle matrix of matrix A′. The upper triangle matrix then represents the new edge connection relationship and corresponds to the new network G′.

As mentioned earlier, the purpose of this work is to find E ⁺ and E ⁻. So the most straightforward way is to represent x _i as an adjacency matrix A′. Based on the difference between A and A′ we can easily work out E ⁺ and E ⁻ and therefore can calculate the values of the objective functions. However, this kind of representation requires too much computer memories when n is very big. As can be seen from Figure 4, the proposed representation mechanism can reduce the memory size by more than half.

How to represent the velocity of a particle is related to the way how the position of the particle is represented, since the velocity vector help a partite to update its position information. As compared to the matrix representation method, the proposed particle position representation method is more convenient for the velocity representation.

Since in this work we propose to encode the edges in the upper triangle of the adjacency matrix of a given network using binary coding schema, therefore we decide to use binary representation of the velocity vector of a given particle. Specifically, for a given velocity vector v _i = (v _i1, v _i2, … , v _im ) corresponding to particle x _i , its element v _ij ∈ {0, 1}. The element v _ij represents whether the jth element in the vector x _i will be changed or not.

For the ith particle with x _i being its position vector, we update its position information with the following equation x i ′ = x i ⊕ v i ′ (7) in which the operator ⊕ defines the operation for x _i ’ as follows x i j ′ = x i j i f v i j ′ = 0 (8) x i j ′ = ∼ x i j i f v i j ′ = 1 (9) The velocity vector v _i ′ of the ith particle is updated as follows v i ′ = S v (10) in which the speed vector v is calculated as v = w v i + c 1 r 1 L i − x i + c 2 r 2 G r − x i (11) where G _r ∈ G is a solution randomly chosen from the global best solution set G and the symbol S () is a confining function which maps vector v into v _i ′. Specifically, S () works as follows v i j ′ = 0 i f 1 1 + e − v j ≤ 0.5 , v j ∈ v (12) v i j ′ = 1 i f 1 1 + e − v j > 0.5 , v j ∈ v (13) It can be seen from the above equations that v is a real valued vector. By implementing the S () operation we turn v into a binary vector based on which we thus can update the particle’s position information. Note that other functions also can be adopted to replace the S () function. However, there is no guarantee that other functions would perform better than the S () function does. The S () function is adopted because it is smooth in the objective space.

Note that PSO is a population based algorithm. During the iterations we need to update the information of the local best individual L _i for all i ∈ [1, psize] and the global best individual G since they help the entire swarm to explore for better solutions. Algorithm 2 presents the pseudocode for updating the local and global best particles in the adopted MOPSO algorithm.

As can be seen from Algorithm 2, during each iteration, when a new individual x _i is generated, we then evaluate its fitness. If x _i dominates its historical best L _i (step 1c of Algorithm 2), then we replace L _i with x _i , otherwise, we keep L _i as it is. When a new population of individuals are generated, we then update G . Specifically, we use the Pareto solutions obtained from the current population to update those in the history. When any of the new Pareto solutions dominate any historical Pareto solutions, we then replace the old solutions with the new one (step 3 of Algorithm 2).

As mentioned above, when generating the networks we need to determine some parameters first. Meanwhile, for the introduced algorithm it also carries with several parameters. In the experiments, we set all the needed parameters as what are shown in Table 1.

Note that the main idea of this work is to validate if the proposed optimization based edge rewire mechanism is effective or not. Therefore, we do not set the value of n large enough. For a network with n nodes, there are maximum n × (n − 1)/2 edges. If n is too large, then the optimization process will take a long time to finish. In the experiments we set it to be small so as to quickly check if the proposed idea is feasible or not. This is the main reason why the tested real-world networks are small in size.

In the experiments, we generate two types of artificial networks. For each type of networks, we generate 12 networks with the configurations of the number of nodes and the degree control parameter, i.e., d for the Erdős–Rényi network and λ for the scale-free network. Figure 8 and Figure 9 respectively visualize the Pareto fronts obtained by the introduced multiobjective particle swarm optimization algorithm when tested on the Erdős–Rényi and scale-free networks. Note that for each tested network we run the introduced multiobjective particle swarm optimization algorithm for 20 times. After each run we save the Pareto solutions. We eventually merge all the Pareto solutions over the 20 runs and filter out the final Pareto solutions based on Pareto dominance mechanism.

Both in Figure 8 and Figure 9 each row represents the results for networks with different sizes. In Figure 8, each column corresponds to different d, while each column in Figure 9 corresponds to different λ. We can clearly see from those two figures that the values of f ₁ for all the solutions are negative. This means that the robustness of the original networks have being improved based on the corresponding edge rewire mechanism. One may further see from the figures that the proposed multiobjective optimization based edge rewire mechanism can yield many different solutions to improve the robustness of a given network. Actually, this is one of the biggest advantages of the proposed edge rewire mechanism as compared to existing ones which can only provide single solutions for the decision makers. As can be seen from the two figures, each solution corresponds to edge deletions and additions but the number of added edges does not need to be equal to that of the deleted edges, which makes the proposed mechanism more flexible for real use.

The above experiments are carried out on Erdős–Rényi networks and scale-free networks which are two important types of networks. In the next, we show the results on the four real-world networks. Figure 10 visualizes of the Pareto fronts obtained by the introduced algorithm when tested on the four networks.

It also can be seen from Figure 10 similar phenomenon occurs to the tested real-world networks as compared to that of the computer-generated networks. More specifically, there are multiple Pareto solutions for improving the robustness of each of the tested real-world networks. The above experiments prove that the proposed edge rewire mechanism can reconnect those tested networks to improve their network robustness. In what follows, we analyze the robustness of the enhanced networks with respect to the obtained Pareto solutions.

The above experiments indicate that the proposed multiobjective optimization based edge rewire mechanism can help to improve the robustness of a given network. Note that by optimizing the proposed mechanism, a set of rewire solutions can be obtained. In this section we analyze the robustness improvement of two extreme solutions for each tested network.

As can be seen from Figures 8–10, the Pareto front for each network contains a set of solutions. Here, we choose two extreme solutions from each Pareto front. For all the solutions, we sum up f ₂ and f ₃. Then we choose the two extreme solutions as the two solutions that require the minimum number of edge operations including addition and deletion. We then calculate the ratio of robustness improvement. The results for all the tested artificial networks are shown in Table 2.

In Table 2, the robustness improvement is calculated as ( λ 2 1 − λ 2 0 ) / λ 2 0 . In the table, we can see that for all the tested artificial networks, the robustness of the original networks have been greatly improved. We also observe from the table that when the size of a network is increased, then the robustness improvement will also be increased. This is because that a network with n nodes has n × (n − 1)/2 maximum number of edges. Thus, a larger network has larger maximum number of edges and there is larger probability to improve its robustness given the fact that the most robust network is a full connected network.

From Table 2 we also notice that the robustness improvement for scale-free networks are more significant than that of Erdős–Rényi networks. This is because that a scale-free network has relatively less edges than an Erdős–Rényi network with the same number of nodes. Consequently, the proposed edge rewire mechanism can work better for improving the robustness of scale-free networks.

Table 3 records the robustness of the tested four real-world networks (recorded by λ 2 0 ) as well as the robustness of the corresponding optimized networks (recorded by λ ₂′ and λ 2 ′ ′ ) with respect to the selected extreme solutions. For the Karate network, its original robust is 0.4685 as can be seen from Table 3. By rewiring its edges with respect to the two selected solutions, the robustness of the two optimized networks are respectively 8.9665 and 9.0996, which are about 20 times of its original robustness. For the remaining three networks, their robustness also have been significantly improved based on the proposed edge rewire mechanism. The experiments on real-world networks also validate the effectiveness of the proposed rewire mechanism.

Tables 2, 3 show that the robustness of the tested networks have been greatly improved. In this section we graphically visualize the optimized structures of the networks against the original ones. For simplicity, here we only visualize the structures of Erdős–Rényi networks for d = 4 and scale-free networks for λ = 2.

Figures 11, 12 respectively visualize the original network structures (first column) and the optimized networks structures (second and third columns) for Erdős–Rényi networks for d = 4 and scale-free networks for λ = 2. We can see from Figures 11, 12 that the proposed edge rewire mechanism makes a given network have more edges to improve its robustness. As compared to Erdős–Rényi networks, the optimized scale-free networks seem to be much denser than their original networks. As explained earlier, this is because that a scale-free network has relatively less edges than a same sized Erdős–Rényi network.

Figure 13 visualizes the original network structures (first row) and the optimized networks structures (second and third rows) for the four tested real-world networks. It can be clearly seen from Figures 11–13 that the selected two solutions suggest to add more links to each of the tested network to improve its network robustness.

One may notice from Figures 11–13 that the proposed edge rewire mechanism tends to make a given network be denser in order to improve its robustness. This is not necessarily the feature of the proposed model. Actually, the proposed multiobjective optimization based rewire mechanism is very flexible. Note that in the proposed model we do not add in any constraints. We do not limit the number of deleted and added edges. As long as the edge operations can improve the robustness of a given network, then the multiobjective optimization algorithm will determine the goodness of the edge operations based on Pareto dominance mechanism. If the operations are non-dominated with each other, then those operations will be saved as the possible choices.

Note that in the experiments for a tested network, its number of edges is far less than that of the maximum number of edges it can has. This is especially true for the tested scale-free networks. A scale-free network is normally sparse in its connections. Therefore there could be many choices to improve its robustness. In order to avoid making a network be denser, we provide below some possible solutions.

1) One may consider to increase the population size of the particle swarm optimization algorithm and run it for more than 20 times to possibly obtain more Pareto solutions. Then one may expect to select Pareto solutions that do not make many edge reconnections. It is also suggested that one can try to redesign the algorithm operators especially the status update principles for the particle swarm optimization algorithm.