Evolution of cooperation driven by collective interdependence on multilayer networks
Introduction
Understanding the emergence of cooperation is a fundamental puzzle in evolutionary biology [1], [2]. Cooperation dilemmas occur when incentives for individuals run counter to group interests. For example, the public goods game constitutes a prominent metaphor to elaborate such conflict of interests. Human cooperation in public goods problem such as protecting the global climate is often threatened by the short-term advantages of free riders, which may ultimately lead to collapse of public resources and the tragedy of the commons [3]. As a basic framework to analyse cooperation dilemmas, evolutionary game theory predicts that the free-riders conventionally overcome cooperation by withholding their initial contribution to the public good in well-mixed populations [4], [5]. Therefore, cooperation requires additional mechanisms for natural selection to favor it [1], [2]. The studies of evolutionary game theory in structured populations recognize the fact that cooperative interaction is determined by spatial relationships or social networks rather than randomly interacting [6]. This fact has inspired numerous investigations of suitable extensions that preserve cooperation significantly [6], [7], [8], [9], [10]. In essence, spatial populations only permit individuals to interact with their neighbors, it is possible for cooperators to resist the exploitation of defectors by forming clusters. This process is known as spatial reciprocity or network reciprocity [6].
Population structure for evolutionary games was traditionally modelled as monolayer networks. However, researches on network science recently show that real networks are generally organized as interdependent networks [11], [12]. Interdependent networks are comprised of two or more subnetwork layers that are coupled together. One typical example of a real interdependent network is diverse infrastructures such as communication, transportation and power stations who are coupled together. The investigation of interdependent networks arouses tremendous concerns by numerous scholars. There is a significant progress in understanding the percolation properties and robustness of interdependent networks [12], [13], [14], [15], [16], [17]. Interdependent networks also have already been applied in evaluating seismic risk [18] and gene regulation and metabolism [19], to name a few.
Particularly, another significant development is the study of evolutionary cooperation on interdependent networks [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42]. The studies on interdependent network reciprocity extend the cognition of spatial reciprocity. For the sake of probing how interdependent network reciprocity affects the evolution of cooperation, it is of paramount importance to construct interdependence between subnetwork layers. Mechanisms including coupled evolutionary fitness [21], [22], [23], [24], [25], [26], [27], interconnectedness [28], [29], [30], [31], biased imitation [32], [33], information sharing [34], [35], separation of interaction and learning networks [36], [37], [38], [39], [40], [41], as well as biased resource allocation [42] have shown that the multi-layer network structure facilitates the evolution of cooperation, which enriches the literature on interdependent network reciprocity in the evolutionary game theory.
Group interactions lie as the fundamental among living organisms in describing social dilemmas. However, in the context of the evolutionary game theory, previous works probing pairwise interdependence mainly assume that the functioning of a node in one layer depends on the corresponding node in other layers. The interdependence between groups also exists in interdependent networks. One typical example is the social network of the employees from two companies. The collaboration of product development groups from two companies builds the coordination between different companies. In the context of percolation properties on interdependent networks, Wang et al. [17]. develop a theoretical framework for group percolation. They assume that all nodes are divided into node groups. The functioning of a group in one layer is influenced by the corresponding groups in other layers. The nodes belonging to the same group survive or fail together. Their study concluded that the formation of groups can enhance the resilience of interdependent networks. However, their study assumes that the divided node groups are non-overlapping. Here we would like to extend the scope of evolutionary games on multi-layer networks by introducing collective interdependence. The interdependence induced by collective interaction characterizes the functioning of a group in one layer impacted by the corresponding groups in other layers. We probe how the degree of overlap between the interdependent groups affects the evolution of cooperation. We consider a population organized in two subnetwork layers with equal size, in which a local pubic group consisting of center node and his neighbors on one layer has a probability p to form collective coupling with corresponding local pubic group with a center node on another layer. p quantifies the strength of collective interdependence and decides the degree of overlap between the global groups. Once a collective coupling holds, a global pubic goods game is established between all members of two local groups from different layers, as schematized in Fig. 1. This study aims to probe how collective interdependence between different subnetwork layers affects the evolution of cooperation. Our findings show that the degree of overlap between the global groups impact network reciprocity significantly.
Section snippets
Model
To begin with, interdependent networks are modeled as two square lattices of size 100 × 100 with periodic boundary conditions. Each player on site x in sublayer one and on corresponding site x′ in layer two has equal probability to be a cooperator () or a defector (). The local public goods game is staged on each subnetwork layer, where players are arranged into overlapping groups of size and thus every player is surrounded by its neighbours and belongs to
Evolution in a well-mixed population
Consider the evolutionary dynamics of M interacting infinite well-mixed populations. Every population composes of a fraction xm of cooperators and a fraction () of defectors. Sample group of size N is randomly selected from every population and plays local public goods game. The mean local payoffs of cooperators and defectors in the population m ∈ [1, M] is respectively given by:On the other hand, with a probability p, collective
Discussion
To sum up, by establishing global interactions across interdependent networks, we have studied how collective interdependence between different sublayers affects the evolution of cooperation. By implementing the model on well mixed populations, we show that the effects of collective interdependence in promoting cooperation rely on global synergetic effect. The evolution of cooperation is favored when synergetic effects are larger than a threshold, otherwise cooperation is hindered. However, the
Acknowledgments
This work was supported by the Natural Science Foundation of Anhui Province under Grant No. 1908085QF268. YZ acknowledges the support from National Natural Science Foundation of China under Grant No. 61703323.
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