Structure-aware imitation dynamics on higher-order networks

TL;DR

The paper addresses how imitation-based updating interacts with group-structured higher-order networks to shape cooperation. It introduces structure-aware update rules on hypergraphs, parameterized by $(s,q)$, and derives a weak-selection condition $\sum_{j=0}^{m-1}[\eta_{(s,q)}F_j(p)+G_j(p)](a_j-b_{m-1-j})>0$ with $\eta_{(s,q)}$ depending on network structure and $(s,q)$. A key outcome is the information diversity metric $\mathcal{D}=(sq-q)/(sq-1)$, which the authors prove governs the effectiveness of update rules across three canonical dilemmas (LPGG, MSG, TPGG) and extends to heterogeneous hypergraphs and general dilemmas via a containment framework. The results show that higher information diversity lowers the critical payoff thresholds $r^*$ for cooperation and that the simple rule $(s,q)=(2,1)$ often maximizes this benefit by maximizing diversity. This work provides a unified design principle for promoting cooperation on higher-order networks and offers practical guidance for designing efficient, group-aware update rules in social and technological systems.

Abstract

Imitation is a basic updating mechanism for strategy evolution in structured populations, determining how individuals sample social information and translate it into behavioral changes. Higher-order networks, such as hypergraphs, generalize pairwise links to hyperedges and provide a natural representation of group interactions. Yet existing studies on higher-order networks largely emphasize structural effects, while the impact of imitation-based update rules and how they interact with group structures remains poorly understood. Here, we introduce a class of structure-aware imitation rules on hypergraphs that explicitly parameterize how many groups are sampled and how many peers are consulted within each sampled group. Under weak selection, we derive an analytical condition for the success of cooperation for any multiplayer social dilemmas on homogeneous hypergraphs. This analysis yields an interpretable metric, information diversity, which quantifies how an update rule diversifies the sources of social information across groups. Analytical predictions and numerical simulations show that cooperation is more effectively promoted by update rules that induce higher information diversity for three representative dilemmas. Further simulations demonstrate that this principle extends to non-homogeneous hypergraphs and a broad class of multiplayer social dilemmas. Our work thus provides a unifying metric that links microscopic updating to evolutionary outcomes in higher-order networked systems and establishes a general design principle for promoting cooperation beyond pairwise interactions.

Figures (5)

• Figure 1: Schematics of structure-aware imitation dynamics on higher-order networks.a, We plot a schematic illustration of a higher-order network (hypergraph), which is composed of nodes (marked by solid circles) and hyperedges (marked by light-yellow shaded areas). Here, nodes represent individuals and hyperedges the range of higher-order interactions (i.e., the set of individuals that engage in a higher-order interaction). In addition, each individual can choose either to cooperate or to defect. Correspondingly, they are called cooperators (marked by blue circles) or defectors (marked by red circles). b, We present a general payoff matrix of an $m$-player game that occurs on hyperedges of size $m$. In such a game, when a cooperator (defector) faces $j$ cooperators among the remaining $m-1$ players, its payoff is denoted as $a_j$ ($b_j$). On a hypergraph, every individual engages in multiplayer games across all hyperedges it belongs to. The (overall) payoff of an individual is calculated as the average payoff over all the games it participates in, denoted by $\pi$. As illustrated, individual $j$ chooses to cooperate in the two games organized in the hyperedges it belongs to. In these two games, individual $j$ faces $0$ and $2$ other cooperators among its neighbors, respectively. Thus, the payoff of $j$ is $\pi_j=(a_0+a_2)/2$. c, After game interactions, a random individual $l$ is chosen as the focal individual (marked by a dashed circle) to update its strategy. It forgoes its own strategy, selects several role models from its neighbors, and tries to imitate one of them. For the role-model selection process, we explicitly take the higher-order (i.e., group) structure of hypergraphs into account, and it has two steps: in the first step, the focal individual randomly chooses $s$ hyperedges (marked by light-blue shaded regions) from its $k_l$ hyperedges ($1\le s \le k_l$); in the second step, it selects $q$ role models (marked by blue outlined circles) randomly from each chosen hyperedge ($1 \le q \le n-1$), where $n$ denotes the smallest group size among the $k_l$ hyperedges. After all $sq$ role models are selected, the focal individual then adopts the strategy of one of them with a probability proportional to their fitness. Here, $s$ and $q$ parameterize a structure-aware imitation process, and each pair $(s,q)$ represents a specific update rule.
• Figure 2: Update rules with high information diversity promote the evolution of cooperation.a, Information diversity $\mathcal{D}$ is defined as the probability that any two randomly selected role models among the $sq$ ones come from different hyperedges. b-d, Depending on the values of $s$ and $q$, four distinct cases can be identified: (1) $s=1$, $q>1$; (2) $s>1$, $q>1$; (3) $s>1$, $q=1$; and (4) $s=1$, $q=1$. The first three cases correspond to panels b-d, which respectively present the values of information diversity $\mathcal{D}$ under each setting. The fourth case is excluded in our study since it represents neutral drift. e-j, We plot the difference between the fixation probability of cooperators $\phi_C$ and that of defectors $\phi_D$, i.e., $\phi_C-\phi_D$, as a function of the key game parameter in three distinct social dilemmas, which are linear public good games (LPGG, e,h), multiplayer snowdrift games (MSG, f,i), and threshold public goods games (TPGG, g,j). Under these games, we examine update rules with seven $(s,q)$ combinations: $(6,1),~(2,1),~(3,2),~(2,2),~(2,3),~(1,2)$, and $(1,3)$. In detail, panels e, f, and g show the effect of the number of role models from each hyperedge, $q$ on the evolutionary outcomes. And panel h, i, and j present the effect of the number of selected hyperedges, $s$, on the evolution of cooperation. In these panels, each data point is the fixation probability difference obtained from $10^7$ independent simulations; each solid line is the result obtained by the linear fit to the corresponding data and is used to guide the eye. The vertical dashed lines indicate the critical values predicted by our theoretical results. k, We plot the values of information diversity $\mathcal{D}$ for different $(s, q)$ pairs. l, Theoretical analysis reveals that, across all three games, the critical value decreases with increasing information diversity. Parameter values: $N=500$, $k=6$, $m=4$, $d=2$, $p=1/N$, $w=0.01$.
• Figure 3: Effect of hyperdegree and order (i.e., hyperedge size) on evolutionary outcomes. The top and bottom panels show the effects of the node hyperdegree $k$ and the hyperedge order $m$, respectively, on the theoretically predicted critical values. Results show that update rules with higher information diversity $\mathcal{D}$ consistently promote cooperation across hypergraphs with varying hyperdegree $k$ and order $m$. Here, the black horizontal dashed lines in a-c indicate the theoretical critical values for the $4$th-order complete hypergraph lin2025evolutionary in three social dilemmas. Our results show that as the hyperdegree $k$ increases, the hypergraph $\mathcal{H}$ approaches a complete hypergraph, and the differences in critical values under different update rules become smaller. Parameter values: $m=4$ (a,b,c), $k=6$ (d,e,f), $d=2$ (c,f).
• Figure 4: Critical values for the evolution of cooperation on heterogeneous higher-order networks. We run a series of simulations and calculate the critical values under various update rules associated with different information diversity $\mathcal{D}$ on two types of heterogeneous hypergraphs. The top panels (a-c) show results for order-heterogeneous hypergraphs, where each node belongs to $k = 6$ hyperedges and the order of hyperedges follows a power-law distribution with a mean of six. The bottom panels (d-f) show results for hyperdegree-heterogeneous hypergraphs, where all the orders of hyperedges are set to be six while the hyperdegrees follow a power-law distribution with a mean of six. Here, to consider update rules with different values of information diversity, we use seven $(s,q)$ combinations: $(1,2),~(1,3),~(2,3),~(2,2),~(3,2),~(2,1)$, and $(3,1)$. Our results indicate that enhancing information diversity during the strategy updating process promotes cooperation on heterogeneous higher-order networks. This means that our findings on homogeneous higher-order networks also apply to non-homogeneous ones. Parameter values: $N=500$, $p=1/500$, $w$ = 0.01, $d$=2.
• Figure 5: Positive impacts of update rules with high information diversity on cooperation in a broad class of multiplayer social dilemmas. We define $\mathcal{S}_{\mathcal{D}}$ as the set of games (or game space) where cooperation prevails over defection under an update rule with information diversity $\mathcal{D}$. To determine whether the update rule with $\mathcal{D}_1$ is more effective in promoting cooperation than that with $\mathcal{D}_2$ in general social dilemmas, we compare $\mathcal{S}_{\mathcal{D}_1}$ with $\mathcal{S}_{\mathcal{D}_2}$: if $\mathcal{S}_{\mathcal{D}_1} \supset \mathcal{S}_{\mathcal{D}_2}$, the former one is more effective; otherwise, the latter one is more effective. a, To provide an illustration of $\mathcal{S}_{\mathcal{D}}$, we take three-player games as an example (i.e., $m=3$) and plot the game space $\mathcal{S}_{\mathcal{D}}$ that supports cooperation with the information diversity $\mathcal{D}=0,~2/3,~4/5,$ and $1$. Here, $\mathcal{S}_{\mathcal{D}}$ is obtained via linear programming by jointly considering the constraints of the social dilemmas and the condition for the success of cooperation shown in Eq. (\ref{['phiC-phiD>0']}). Results show that $\mathcal{S}_1\supset \mathcal{S}_{4/5}\supset \mathcal{S}_{2/3}\supset \mathcal{S}_0$. This aligns with our findings that the greater the information diversity, the easier it is for the evolution of cooperation. b-d, The three panels show cross-sections of panel a at $\Delta_2=1,0.6,0.3$, respectively. The grey areas represent games that are not social dilemmas. e-n, We calculate $H_{\text{max}}(k,m,p)$ (see Eq. (\ref{['Hmax']})) for general social dilemmas on higher-order networks with different hyperdegree $k$ and order $m$ under various initial fractions of mutants $p\in \{0.01,0.02,0.03,0.04,0.05,0.1,0.2,0.3,0.4,0.5\}$. Note that if $H_{\text{max}}(k,m,p)\le 0$, we have $\mathcal{S}_{\mathcal{D}_1}\supset \mathcal{S}_{\mathcal{D}_2}$ whenever $\mathcal{D}_1>\mathcal{D}_2$. Our numerical results confirm that $H_{\text{max}}(k,m,p)\le 0$ holds for all the sets of parameters we consider here. This means that beyond canonical social dilemmas such as LPGG, MSG, and TPGG, our finding that enhancing information diversity promotes cooperation also holds for a larger class of other social dilemmas, which demonstrates the generality of our results. Parameter values: $k=3,m=3,p=0.1$ (a-d).