An evolutionary game approach for determination of the structural conflicts in signed networks

Social or biochemical networks can often divide into two opposite alliances in response to structural conflicts between positive (friendly, activating) and negative (hostile, inhibiting) interactions. Yet, the underlying dynamics on how the opposite alliances are spontaneously formed to minimize the structural conflicts is still unclear. Here, we demonstrate that evolutionary game dynamics provides a felicitous possible tool to characterize the evolution and formation of alliances in signed networks. Indeed, an evolutionary game dynamics on signed networks is proposed such that each node can adaptively adjust its choice of alliances to maximize its own fitness, which yet leads to a minimization of the structural conflicts in the entire network. Numerical experiments show that the evolutionary game approach is universally efficient in quality and speed to find optimal solutions for all undirected or directed, unweighted or weighted signed networks. Moreover, the evolutionary game approach is inherently distributed. These characteristics thus suggest the evolutionary game dynamic approach as a feasible and effective tool for determining the structural conflicts in large-scale on-line signed networks.

w ij can either be positive or negative.
The game captures the interplay between components. Each node possesses a strategy and play games with all its neighbors. A payoff is obtained after the game interactions. Considering that the relationships of node pairs are homogeneous in most cases, it is usually assumed that all pairs of nodes play a unified game in previous works [3], [4]. Yet, in this work, we consider two different games in the networks.
The nodes choose a strategy from {+1, −1}. If the relationship between two nodes is positive, then the following game is played: And if the relationship between two nodes is negative, then an opposite game is played: 2 Thus, according to the above game interactions, the payoff of each node is That is, the node payoff depends not only on the strategies of its neighbors but also on the relationships with its neighbors.
The strategy updating rule describes the microscopic evolving process of the system. Each agent updates its strategy based on the information of the payoff and strategies of its neighbors and itself. Different kinds of updating rules have been introduced. In this work, we consider a selection-mutation rule: if the fitness of a node is less than a threshold, then selection happens; otherwise, if the fitness of a node is larger or equal than the threshold, then mutation happens with some rate. Let v i (t), F i (t), and R i (t) denote the strategy, payoff, and mutation rate of node v i at time t. Let Θ = 0 denote the threshold. Then, the selection-mutation rule can be written as follows: Mutation rate. The value of mutation rate can greatly influence the pace and trajectory of the evolution process. Appropriate mutation rate can lead to a better performance. In this work, the mutation rate is set to be where the noise parameter T = α k T 0 . Here, T 0 > 0 is the initial noise parameter and 0 < α < 1 is the decaying exponential. Moreover, k = ⌊t/K⌋, where t is the iteration steps, K is the damping period, and the notion ⌊t/K⌋ denotes the maximal integer less than t/K.
The above setting of mutation rate has two novel features. First, the mutation rate depends on the node fitness. Nodes with larger fitness possess a smaller mutation rate, while those with smaller fitness possess a larger mutation rate. Such a setting is intuitively reasonable. Second, the mutation rate is eventually decreasing with time. The noise parameter is large initially and decreases to zero eventually, which is very similar to simulated annealing algorithms [5]. As a result, Eq. 5 is referred to heterogeneous decaying mutation rate in the following.
For comparisons, consider two other mutation rates: (i) Constant mutation rate R i = R. The mutation rate keeps unchanged during the evolutionary process.
Moreover, it is the same for all nodes.
(ii) Heterogeneous mutation rate R i = 0.5e −F i /T 0 . The mutation rate only depends on the node fitness.
In the following, we explore the effect of the above settings of mutation rate on the evolutionary process in the yeast (gene regulatory) network [6]. Fig. S1 shows the evolutionary trajectories of the network fitness with different constant mutation rates.
Since the mutation rate is a small constant, the network fitness can rapidly increase to a stable value.
Yet, after that, the network fitness fluctuates around the stable value due to the non-zero mutation rate.
Moreover, compared with that in the heterogeneous decaying mutation, the network fitness in constant mutation is much lower.   S2 shows the evolutionary trajectories of the number of structural conflicts with different heterogeneous mutation rates. In the heterogeneous mutation rate, the noise parameter T is fixed. If the noise is large, then the evolutionary trajectory will fluctuate. However, if the noise is too small, then the evolutionary trajectory will end with a local optimal stable states with more structural conflicts. Thus, compared with the heterogeneous decaying mutation rate, the problem of the heterogeneous mutation rate lies in the difficulties in determination of the noise parameter.
Assignments of the initial noise parameter, decaying exponential, and damping period can also greatly affect the performance of the evolutionary game dynamic approach. Figure S3 shows the evolutionary trajectories of the network fitness with different parameter assignments. It can be found that the parameters can affect the convergence pace and state of the evolutionary process. Luckily, mutation happens on each node and thus the assignments of the mutation parameters are irrespective with the network size. That is, proper assignments of mutation parameters can be applied to various kinds of networks. The network fitness T 0 =100, =0.9,K =600 T 0 =100, =0.9,K =1000 T 0 =100, =0.9,K =100 T 0 =100, =0.6,K =600 T 0 =10, =0.9,K =600