Evolutionary game dynamics for higher-order interactions

TL;DR

The paper addresses how cooperation evolves under higher-order interactions that occur in hypernetworks, moving beyond traditional pairwise-network models. It introduces a theoretical framework for evolutionary game dynamics on hypernetworks, deriving a simple condition for cooperation via a closed-form threshold $(b/c)^*$ and showing that higher-order interactions can lower this threshold, especially in large populations. A unifying perspective connects higher-order dynamics to a replacement-network description, enabling cross-paradigm comparisons with pairwise and group interactions, and a simple rule $b/c > dg/(d+1)$ is established for uniform-uniform hypernetworks. The findings provide principled tools to analyze and promote cooperation in systems with higher-order interactions, with implications for social, biological, and engineered networks and potential extensions to temporal hypernetworks and multi-strategy scenarios.

Abstract

Cooperative behaviors are deeply embedded in structured biological and social systems. Networks are often employed to portray pairwise interactions among individuals, where network nodes represent individuals and links indicate who interacts with whom. However, it is increasingly recognized that many empirical interactions often involve triple or more individuals instead of the massively oversimplified lower-order pairwise interactions, highlighting the fundamental gap in understanding the evolution of collective cooperation for higher-order interactions with diverse scales of the number of individuals. Here, we develop a theoretical framework of evolutionary game dynamics for systematically analyzing how cooperation evolves and fixates under higher-order interactions. Specifically, we offer a simple condition under which cooperation is favored under arbitrary combinations of different orders of interactions. Compared to pairwise interactions, our findings suggest that higher-order interactions enable lower thresholds for the emergence of cooperation. Surprisingly, we show that higher-order interactions favor the evolution of cooperation in large-scale systems, which is the opposite for lower-order scenarios. Our results offer a new avenue for understanding the evolution of collective cooperation in empirical systems with higher-order interactions.

Figures (5)

• Figure 1: Illustration of the evolutionary process. (a) In the pairwise network, an edge can connect only two individuals. (b) Pairwise interactions are based on such edges. Each individual plays the donation game with their neighbors and accumulates benefits from all games. (c) Group interaction is also based on pairwise networks, where individuals form a group with their neighbors to play a public goods game. (d) The strategy updates for these two interactions are based on pairwise networks where individuals learn the strategy of a particular neighbor. (e) Higher-order interactions involve multiple individuals interacting simultaneously, and hypernetworks can effectively characterize such interactions by allowing edges, known as hyperedges, to connect multiple individuals. In the hypernetwork, each hyperedge corresponds to a public goods game. When performing strategy updates for higher-order interactions, the focal individual learns the strategies of individuals who have shared hyperedges with the focal individual. Indeed, if an individual shares more hyperedges with a particular neighbor, there is a higher probability of learning that neighbor’s strategy.
• Figure 2: Verification of theoretical predictions. (a) For the verification of the theoretical predictions, we take these four typical hypernetworks as examples. (b) We present the product of the fixation probability of cooperation and the number of individuals ($N\rho_C$) as a function of the benefit-to-cost ratio ($b/c$) under death-birth update across various hypernetworks depicted in (a). The simulation conditions are set as follows: $N=48$, $\delta=0.01$, $c=1$, with the number of numerical simulation iterations set to $5\times 10^6$. Inclined scatter points represent the results of the numerical simulation. Vertical dashed lines indicate theoretical predictions. (c) We calculate the critical thresholds of these four hypernetworks for the three interactions described in Fig. \ref{['fig:figure1']}.
• Figure 3: Some intuitive explanations of games on hypernetworks. (a) The pair-approximation is based on this special uniform-uniform hypernetwork that satisfies that all node hyperdegrees are equal and all hyperedge orders are equal. We visualize the hypernetwork and its corresponding replacement network (orange pairwise edges). (b) We consider a focal individual (grey node) selected to update its strategy, it will learn the strategy of one of its neighbors. Pair-approximation calculation shows that for weak selection the cooperator has one more cooperator among its $d(g-1)-1$ other neighbors than the defector. Hence, the focal individual has a higher probability of learning the strategy of a cooperator neighbor, if $b/c>dg/(d+1)$. (c) We present the evolutionary dynamics of an infinite population. Parameters are $\delta=0.01$, $c=1$, $d=3$, $g=4$, $b=3.3$ for the top panel and $b=2.7$ for the bottom panel. The direction of selection dynamics is indicated by the arrow, where the small solid circle represents a stable equilibrium and an empty circle represents an unstable equilibrium.
• Figure 4: Effect of the Simpson degree on the critical threshold. We number 100 ring-arranged individuals in (a). Based on sequential selection approach, the first hyperedge of model ‘[1 2 3 4]’ consists of four individuals numbered 1, 2, 3, 4, the second hyperedge consists of four individuals numbered 2, 3, 4, 5, and continues in this manner. The last hyperedge consists of four individuals numbered 100, 1, 2, and 3. The other three baseline models are similar. Meanwhile, we demonstrate the equivalent swapping approach. The hypernetwork is first represented as a bipartite network. Two links are randomly selected, disconnected, and then exchanged. The proportion of swapped links to total links is defined as the swapped rate, which ranges $5\%, 10\%,\cdots, 50\%$. In (b)-(e), we show the distributions of the critical values for the average Simpson degree. The four different colors indicate the four baseline models and their corresponding equivalent swapping hypernetworks. The small network in each panel indicates the local replacement network of the baseline model, where the colored nodes indicate the individual we are focusing on. The colored solid lines indicate the edges between the focused node and its neighbors, and the grey solid lines indicate the edges between the neighbors of the focused individual. (f) We present the change in the average topological degree with the equivalent swapping ratio for these four models. The simulation conditions are: $N=100, d=4,g=4$.
• Figure 5: The effect of linking group methods on the evolution of cooperation. (a) We denote three segregated groups of nine individuals represented by the grey nodes. In (b) and (c), we show different ways of linking groups and the corresponding critical benefit-to-cost $(b/c)^*$ that favors cooperation, respectively. In (b), the three groups are linked by a single individual, denoted by a red node. And in (c), the three groups are linked by an additional hyperedge, indicated by a dark gold solid triangle, consisting of individuals from each group. The panels (d)-(f) indicate the trend of the threshold for cases shown in (b) and (c) over different numbers of groups and group sizes, and the vertical dashed lines in (e) and (f) indicate when $(b/c)^*_{\mathrm I}=(b/c)^*_{\mathrm H}$. (g) We plot the magnitude of the threshold, and the blue line indicates $(b/c)^*_{\mathrm I}=(b/c)^*_{\mathrm H}$. The grey solid line indicates that the group size is equal to the number of groups.