Leveraging statistical physics to improve understanding of cooperation in multiplex networks

Understanding how public cooperation emerges and is maintained is a topic of broad interest, with increasing contributions coming from a synergistic combination of evolutionary game theory and statistical physics. The comprehensive study by Battiston et al (2017 New J. Phys. 19 073017) improves our understanding of the role of multiplexity in cooperation, revealing that a significant edge overlap across network layers along with benign conditions for cooperation in at least one of the layers is needed to facilitate the emergence of cooperation in the multiplex.

We live in a remarkably interconnected world, and we weave our colorful lives across different social domains [2]. The prosperity of almost every facet of our group endeavors, ranging from scientific collaboration to online e-commerce, depends crucially on our ability of brewing and fostering cooperation not just between friends and people we know, but also even between strangers we do not know. Despite the ubiquity of cooperation in human societies, specific mechanisms are needed for cooperation to thrive [3].

In a public goods game of G players, each player decides whether to contribute d to a common pool, which is then multiplied by an enhancement factor $r\lt G$, and distributed evenly among all group members regardless of their contributions. Since each contribution by oneself gives $(r/G-1)d\lt 0$ in return, public cooperation becomes a problem: everyone would abstain from contributing and free-ride on the efforts of others. This leads to 'the tragedy of the commons'. Therefore, understanding the puzzle of cooperation is key to solving challenging issues such as climate change and environmental pollution.

Proposed as an effective means to promoting cooperation, network reciprocity—where interactions between individuals are not random but quite exquisitely structured (as described by graphs or networks)—has been extensively studied over the past decade. Until recently, games in multiplex networks that take into account the multitude of individual interactions across different yet linked social contexts have attracted widespread attention [4]. At first glance, such multiplicity of interactions arising from different social domains appears to be a natural extension of games on single-layered networks, and thus might help enhance the resilience of cooperation. However, it remains largely unclear under what conditions cooperation can actually be promoted further by the presence of multiplexity, rather than solely by traditional network reciprocity.

To address this question, the work by Battiston et al quantifies the determinants of public cooperation in multiplex networks using extensive Monte Carlo simulations [1]. The study unveils that enhanced resilience of cooperation owing to the presence of multiplexity requires significant edge overlap ω across network layers, in combination with at least one layer being able to sustain cooperation by means of a sufficiently high enhancement factor r. These results shed light on the complexity of cooperation in the multiplex through the lens of statistical physics. Moreover, their findings can have practical implications for promoting prosocial behavior in multiple, interdependent social domains.

In the present model of multiplex networks, individuals are initially connected by exactly the same regular random graph on a given number of layers M, and by edge rewiring, the study is able to tune the degree of edge overlap of the multiplex from complete overlap ($\omega =1$) to no overlap at all ($\omega =0$). Noteworthy, one advantage of using regular random graphs for each layer is to make it feasible to compare and contrast the impact of multiplexity on cooperation without the confounding impact of network degree heterogeneity, which could otherwise be present.

Individuals play the public goods games with their immediate neighbors on each layer, and can have different strategies across network layers. Payoffs obtained from each social layer are accumulated and determine the overall performance of individuals. The strategy updating occurs in a random sequential order for each layer and for each individual; based on the accumulated payoffs from all layers, individuals can imitate the particular strategy of their neighbors in a given network layer. To quantify the coherence of strategies across layers, the study defines a second order parameter of coherence ξ, in complement to the average cooperation level c on each layer as the first order parameter.

The study shows that the number of layers M has the most favorable impact on cooperation when the multiplex is completely overlapping across the layers ($\omega =1$), while reverting to traditional network reciprocity (namely, no multiplex network reciprocity) if there exists zero network edge overlap across the layers ($\omega =0$). Moreover, by continuously varying the edge overlap ω for two-layered networks and allowing different enhancement factors r of the public goods games across the two layers, Battiston et al demonstrate that the resilience of cooperation requires significant edge overlap ($\omega \gt 0.25$), or benign conditions for cooperation in at least one layer ($r\gt 3.75$). In general, however, if there is substantial edge overlap among large numbers of layers, public cooperation can thrive well below the critical r = 3.75 threshold for single-layer networks. Their study also shows how the spontaneous symmetry breaking of cooperation levels and the coherence of individual strategies across network layers depends on the model parameters ω and r.

One can intuitively understand these results as follows. As a consequence of social imitation, an increase in the edge overlap strengthens the formation of cooperative clusters across different layers, as individuals are then more likely to learn from the same neighbors, even though they are located on different layers. In other words, the presence of edge overlap hinders the spontaneous symmetry breaking of cooperation yet promotes coherence of strategies across social layers.

Most importantly, the present paper inspires future research. In the first place, the mathematical framework of games on multiplex networks allows a straightforward extension that incorporates potential strategy interference across network layers, that is, the so-called spillover effect in social psychology. Second, more generally, different types of games, for example, one-shot games versus repeated games, can occur in different network layers. In doing so, one can explore the evolution of social heuristics across different social domains in the setting of multiplex networks [5]. Last but not least, despite the tremendous difficulty of studying multiplex networks analytically, it is possible to derive theoretical results at least for certain particular classes of simplified multiplex network structures, such as lattice populations [4] and set-structured populations [6].

These extensions of games in multiplex networks, combined with multiplex strategy updating, will enhance our understanding of human cooperation, particularly across different social domains and their potential interference with each other. We are delighted to see such promising developments and advancements along this line in the near future.

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