Reconstructing Networks from Profit Sequences in Evolutionary Games via a Multiobjective Optimization Approach with Lasso Initialization

Evolutionary games (EG) model a common type of interactions in various complex, networked, natural and social systems. Given such a system with only profit sequences being available, reconstructing the interacting structure of EG networks is fundamental to understand and control its collective dynamics. Existing approaches used to handle this problem, such as the lasso, a convex optimization method, need a user-defined constant to control the tradeoff between the natural sparsity of networks and measurement error (the difference between observed data and simulated data). However, a shortcoming of these approaches is that it is not easy to determine these key parameters which can maximize the performance. In contrast to these approaches, we first model the EG network reconstruction problem as a multiobjective optimization problem (MOP), and then develop a framework which involves multiobjective evolutionary algorithm (MOEA), followed by solution selection based on knee regions, termed as MOEANet, to solve this MOP. We also design an effective initialization operator based on the lasso for MOEA. We apply the proposed method to reconstruct various types of synthetic and real-world networks, and the results show that our approach is effective to avoid the above parameter selecting problem and can reconstruct EG networks with high accuracy.


Supplementary Note 1: Performance Measures
To quantify the performance of our reconstruction method, two standard measurement indices are introduced, namely, the area under the receiver operating characteristic curve (AUROC) and the area under the precision-recall curve (AUPR) [S1].
True positive rate (TPR), false positive rate (FPR), Precision and Recall used to calculate AUROC and AUPR are defined as follows: where k is the cutoff in the edge list, TP(k) is the number of true positives in the top k predictions in the edge list, and G is the number of positives in the gold standard.
where FP(k) is the number of false positive in the top k predictions in the edge list, and Q is the number of negatives in the gold standard.
Reconstruction error (RE) is used to compare the weight matrix of reconstructed network and target network directly,  (S4) where xij is the edge weight between nodes i and j in the target network, xij * is the edge weight between nodes i and j in reconstructed networks, and N is network size.

Supplementary Note 2: Numerical Simulation of EG
Numerical simulation of EG is described as follows. Initially, a fraction of agents is set to choose the strategy of cooperation and the remaining agents are set to choose the strategy of defection. Nodal states are updated in parallel. For agent i of degree k, at round t, the payoff of this agent is calculated. To maximize the payoff of agent i, its strategy is updated. A Monte Carlo round t is referred to the situation where all the states at t+1 have been updated according to their states at t. Random link weights are chosen uniformly from the interval [1.0, Figure S1. The relationships between measurement error, RE, and the sparsity of the solutions on the PF and the position of knee point for three different k. Figure S1 plots the relationship between measurement error and ||Xi||1 when the average degree of ER networks changes. The simulations are conducted on ER networks with 100 nodes.

Supplementary Figures
Here, the degree of networks k is increased from 6 to 18 in steps of 6, NM=1.4, and =0.05. In each case, the left-hand graph is a 2-D plot, graphing the relationship between the measurement error and ||Xi||1. The right-hand graph shows one 2-D views of the data; variation of RE with change in sparsity ||Xi||1. Each graph of Supplementary Fig. S1 shows results for one example trial. We have gathered data for three different cases where  varies from 0.1 to 0.3 in increments of size 0.1. Figure S3. RE as a function of the relative data length NM of time series for (a) BA networks, (b) NW networks, and (c) WS networks. We simulate evolutionary games on different model-based networks, including weighted Barabá si-Albert scale-free networks (BA) [S2], weighted Newman-Watts small-world networks (NW) [S3], and weighted Watts-Strogatz small-world networks (WS) [S4]. The simulations are conducted on network size N=100, k=6 and 12, and =0, 0.05, and 0.3. Here, NM is increased from 0.1 to 1.6 in steps of 0.1. Rewriting probability of small-world networks is 0.3. Each data point is obtained by averaging over 30 independent realizations. each solution of sub-problem is selected from the PF based on knee regions. Figure S4. AUPR as a function of the relative data length NM of time series for (a) BA networks, (b) NW networks, and (c) WS networks. We simulate evolutionary games on different model-based networks, including weighted BA networks, weighted NW small-world networks, and weighted WS small-world networks. The simulations are conducted on network size N=100, k=6 and 12, and =0, 0.05, and 0.3. Here, NM is increased from 0.1 to 1.6 in steps of 0.1. Rewriting probability of small-world networks is 0.3. Each data point is obtained by averaging over 30 independent realizations. each solution of sub-problem is selected from the PF based on knee regions. Figure S5. AUROC as a function of the relative data length NM of time series for (a) BA networks, (b) NW networks, and (c) WS networks. We simulate evolutionary games on different model-based networks, including weighted BA networks, weighted NW small-world networks, and weighted WS small-world networks. The simulations are conducted on network size N=100, k=6 and 12, and =0, 0.05, and 0.3. Here, NM is increased from 0.1 to 1.6 in steps of 0.1. Rewriting probability of small-world networks is 0.3. Each data point is obtained by averaging over 30 independent realizations. each solution of sub-problem is selected from the PF based on knee regions. networks with k=6, (b) BA networks with k=12, (c) NW networks with k=6, (d) NW networks with k=12, (e) WS networks with k=6, and (f) WS networks with k=12. Rewriting probability of small-world networks is 0.3. Here, N=100, =0.05. NM is increased from 0.1 to 0.8 in steps of 0.1. For MOEANet+RE, each solution of sub-problem selected from the PF has the best generalization ability, namely, the smallest value of RE. For MOEANet+KR, each solution of sub-problem is selected from the PF based on knee regions. For the lasso , we set =0.001 for all reconstructions. Each data point is obtained by averaging over 30 independent realizations. networks with k=6, (b) BA networks with k=12, (c) NW networks with k=6, (d) NW networks with k=12, (e) WS networks with k=6, and (f) WS networks with k=12. Rewriting probability of small-world networks is 0.3. Here, N=100, =0.05. NM is increased from 0.1 to 0.8 in steps of 0.1. For MOEANet+RE, each solution of sub-problem selected from the PF has the best generalization ability, namely, the smallest value of RE. For MOEANet+KR, each solution of sub-problem is selected from the PF based on knee regions. For the lasso , we set =0.001 for all reconstructions. Each data point is obtained by averaging over 30 independent realizations. networks with k=6, (b) BA networks with k=12, (c) NW networks with k=6, (d) NW networks with k=12, (e) WS networks with k=6, and (f) WS networks with k=12. Rewriting probability of small-world networks is 0.3. Here, N=100, =0.05. NM is increased from 0.1 to 0.8 in the step of 0.1. For MOEANet+RE, each solution of sub-problem selected from the PF has the best generalization ability, namely, the smallest value of RE. For MOEANet+KR, each solution of sub-problem is selected from the PF based on knee regions. For the lasso , we set =0.001 for all reconstructions. Each data point is obtained by averaging over 30 independent realizations.

Supplementary Table
The details of the real social networks studied in this paper are presented in Supplementary Table S1, which includes the number N of nodes, the number L of links, and description of the networks. Supplementary Table S1. Description of the real social networks analyzed in the paper.

Name
N L Description football [S5] 115 613 The network of American football games, Fall 2000. netscience [S6] 1589 2742 A coauthorship network of scientists working on networks. polbooks [S7] 105 441 A network of books about US politics. dolphin [S8] 62 159 Social network of dolphins. ZK [S9] 34 78 Social network of friendships of a karate club. lesmis [S10] 77 254 Coappearance network of characters in the novel Les Miserables. adjnoun [S6] 112 425 Network of common adjectives and nouns in the novel David Copperfield neuralnet [S4] 297 2359 Represent the neural network of C. Elegans. Data.
The parameters of MOEANet are showed in Supplementary Table S2.