Evolution of cooperation driven by collective interdependence on multilayer networks

doi:10.1016/j.amc.2020.125532

Applied Mathematics and Computation

Volume 388, 1 January 2021, 125532

https://doi.org/10.1016/j.amc.2020.125532 Get rights and content

Highlights

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The effects of collective interdependence on the evolution of cooperation are studied.
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Global and local synergetic effects affect the evolution of cooperation significantly.
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A moderate collective interdependence facilitates the propagation of cooperation best.
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Collective interdependence is a double-edged sword in promoting cooperation.
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The effects of collective interdependence impact network reciprocity significantly.

Abstract

Group interactions, formulated in terms of a public goods game, can not be deduced by the corresponding sum of pairwise interactions. This study proposes a collective interdependence characterizing the functioning of interdependent groups between subnetworks. Via the establishment of global group interactions across subnetwork layers, we show a double-edged sword of collective interdependence in promoting cooperation. Enhancement of collective interdependence hinders the evolution of cooperation whenever global synergy factor is small, while an optimal collective interdependence emerges most favoring the evolution of cooperation for high global synergy factors. However, for such high global synergy factors a low level of interdependence puts cooperators to a most disadvantaged place. A combination of low levels of collective interdependence and high local synergy factors shows that a moderate global synergetic effect most favors the evolution of cooperation. Our work reveals that collective interdependence impacts interdependent network reciprocity significantly and highlights the importance of network reciprocity in enhancing the evolution of cooperation.

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 ( $C, s_{x} = 1$ ) or a defector ( $D, s_{x} = 0$ ). The local public goods game is staged on each subnetwork layer, where players are arranged into overlapping groups of size $N_{L} = 5,$ and thus every player is surrounded by its $k = N_{L} - 1$ neighbours and belongs to $n_{L} = N_{L}$

Evolution in a well-mixed population

Consider the evolutionary dynamics of M interacting infinite well-mixed populations. Every population composes of a fraction x_m of cooperators and a fraction ( $1 - x_{m}$ ) 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: $f_{C_{m}}^{l} (x_{m}) = \frac{r_{L} c}{N} ((N - 1) x_{m} + 1) - c$ $f_{D_{m}}^{l} (x_{m}) = \frac{r_{L} c}{N} ((N - 1) x_{m})$ 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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It is an arduous task to explore the internal mechanism of the widespread existence of cooperative behaviors in diverse biological systems. The influence of multigames and interdependent networks on the evolution of cooperation is multifaceted. In this work, we attempt to perform multigames on interdependent networks with different topologies and understand how cooperation evolves. In the model, the modulation of the sucker’s payoff makes each individual have an equal probability to engage in the prisoner’s dilemma and the snowdrift game. The basic composition of interdependent networks includes the Barabási–Albert scale-free network, the Erdös–Rényi random graph and the classical square lattice. Numerous simulations demonstrate that the evolution of cooperation is closely related to the topological structure of the sub-networks. For instance, the interdependent networks constructed by two square lattices favor the diffusion of cooperation, while the coupling between two Barabási–Albert scale-free networks inhibits cooperation. The internal evolutionary mechanism of cooperation is analyzed by conducting the multigames on the interdependent networks coupled by two square lattices. Furthermore, the simulations on the interdependent networks with heterogeneous sub-network topologies are also performed. Our research may be beneficial to stimulate further explorations of the spread of cooperation on multi-layer complex networks.
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Motivated by the realistic situation that local and global information feedbacks maybe influence on the cooperative behavior, a new model of spatial prisoner’s dilemma game is proposed to consider the local popularity and global popularity based on the number of players who hold the same strategy with the central or target agent in the local and global environments. The weight of local popularity and the scaling index of the popularity are introduced to the fitness calculation. Numerical experiments show that the information of local popularity and global popularity have different effects on cooperative behavior when the scaling index of the popularity changes from positive to negative. And when the scaling index of the popularity is negative, the greater the absolute value of the scaling index is, the more conducive it is to the cooperation; but there exists the optimal value of the scaling index for the cooperative behavior when the scaling index is positive. These results maybe offer sights to a comprehensive understanding of the influence of the popularity on promoting cooperation.
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Finally, we conclude and discuss potential directions for future work. In this section we introduce our model of networked multiplayer public goods game that generalizes several previous models [38,39]. In the one-shot PGG, the rational strategy for an individual is to avoid any contribution and to free-ride on the contributions of others.
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Evolution of cooperation driven by collective interdependence on multilayer networks

Highlights

Abstract

Introduction

Section snippets

Model

Evolution in a well-mixed population

Discussion

Acknowledgments

J. Theor. Biol.

Trends Ecol. Evol.

J. Theor. Biol.

Philos. T. R. Soc. B

Nature

Sci. Rep.

New J. Phys.

PLoS ONE

Phys. Rev. E

Roy. Soc. Open Sci.

Five rules for the evolution of cooperation

Science

Evolutionary dynamics of group interactions on structured populations: a review

J. R. Soc. Interface

A simple rule for the evolution of cooperation on graphs and social networks

Nature

Evolutionary dynamics of social dilemmas in structured heterogeneous populations

P. Natl. Acad. Sci.

Evolutionary games on graphs

Phys. Rep.

Cooperation with both synergistic and local interactions can be worse than each alone

Sci. Rep.

The structure and dynamics of multilayer networks

Phys. Rep.

Interdependent networks: reducing the coupling strength leads to a change from a first to second order percolation transition

Phys. Rev. Lett.

Avalanche collapse of interdependent networks

Phys. Rev. Lett.

Robustness of a network of networks

Phys. Rev. Lett.

Percolation in real interdependent networks

Nat. Phys.

Group percolation in interdependent networks

Phys. Rev. E

Seismic risk assessment of interdependent critical infrastructure systems: the case of european gas and electricity networks

Earthq. Eng. Struct. D.

The interdependent network of gene regulation and metabolism is robust where it needs to be

Nat. Commun.