Constructing Robust Cooperative Networks using a Multi-Objective Evolutionary Algorithm

The design and construction of network structures oriented towards different applications has attracted much attention recently. The existing studies indicated that structural heterogeneity plays different roles in promoting cooperation and robustness. Compared with rewiring a predefined network, it is more flexible and practical to construct new networks that satisfy the desired properties. Therefore, in this paper, we study a method for constructing robust cooperative networks where the only constraint is that the number of nodes and links is predefined. We model this network construction problem as a multi-objective optimization problem and propose a multi-objective evolutionary algorithm, named MOEA-Netrc, to generate the desired networks from arbitrary initializations. The performance of MOEA-Netrc is validated on several synthetic and real-world networks. The results show that MOEA-Netrc can construct balanced candidates and is insensitive to the initializations. MOEA-Netrc can find the Pareto fronts for networks with different levels of cooperation and robustness. In addition, further investigation of the robustness of the constructed networks revealed the impact on other aspects of robustness during the construction process.


Supplementary Note 1: Other network robustness measures
Some other robustness measures are introduced in this section, and these measures are employed to evaluate the robustness of networks on the Pareto fronts for further depict the performance of the networks on Pareto fronts. Zeng et al. extended R to the case of link-attacks in [S1] as follows, where M is the number of links, s(P) is the fraction of largest connected component after removing P links. 1/M is the normalization factor.
Besides, Wu et al. [S2] proposed the robustness measure based on the average eigenvalue of networks, termed as natural connectivity (  ), where  i is the i-th eigenvalue of adjacency matrix.
Moreover, the communication efficiency between nodes in networks may also be influenced by the fluctuation of network structure, which is important in real-world communication and transportation networks. As Latora et al. proposed in [S3], the communication efficiency C(G) considers the reciprocal of shortest path between nodes i and j in networks, where c ij = 1/d ij and d ij is the length of the shortest path between nodes i and j. When the two nodes belong to different components, c ij = 0. Being similar with R, we take the sum of C(G) during the process of attack as the measure: where C(G q ) is the communication efficiency of networks after removing q nodes, and 1/N is the normalization factor.  Figure S8. Topologies of extracted networks from the Pareto fronts in Fig. 2. Images in the first row ((a), (b), and (c)) are extracted from Pareto front initialized by ER networks, in the second row ((d), (e), and (f)) are initialized by SF networks, and in the last row ((g), (h), and (i)) are initialized by SW networks. Images in the first column represent G l , in the second column represent G m , and in the last column represent G r . In the figures, the size of nodes is proportional to the degree of this node. Links between nodes with same degree are highlighted.   Figure S10. Original structures of the three real world networks and topologies of extracted networks from the Pareto fronts in Fig. 5. Images in the first column represent the original structures of real world networks: (a) is WU Power grid network, (e) is Dolphin social network, and (i) is Scotland corporate interlock network. Other images in the first row ((b), (c), and (d)) are extracted from Pareto front initialized by network in (a), in the second row ((f), (g), and (h)) are initialized by network in (d), and in the last row ((j), (k), and (l)) are initialized by network in (i), and they represent for G l , G m , and G r separately. Also, in the figures, the size of nodes is proportional to the degree of this node, and links between nodes with same degree are highlighted.

Supplementary Table
The detail of the real world networks studied in this paper are presented in Supplementary Table S1, which includes the number N of nodes, the number M of links, and description of the networks.
Supplementary Table S1. Detail information of the real world networks analyzed in the paper.  [1904][1905] The parameters used in MOEA-Net rc are described in Supplementary Table S2.
Supplementary Table S2. Parameter settings of MOEA-Net rc .
("-" means determined by the algorithm in the running process)