Strategy evolution on temporal hypergraphs

TL;DR

It is found that temporal hypergraphs can promote cooperation compared with static networks, and the latter may even underestimate the cooperation-boosting effects of constrained, local interactions.

Abstract

Individuals interact and cooperate in structured systems. Many studies represent this structure using static networks, where each link represents a permanent connection between two nodes. However, real interactions are generally not time-invariant and are often not pairwise. Recently, progress has been made in modeling higher-order interactions using hypergraphs, where a link may connect more than two individuals. Here, we study cooperation on temporal hypergraphs, capturing the time-varying, higher-order interactions seen in empirical systems. We find that temporal hypergraphs can promote cooperation compared with static networks, and the latter may even underestimate the cooperation-boosting effects of constrained, local interactions. We further show that cooperation can be facilitated by temporal hypergraphs with sparse components and higher-order interactions. Importantly, when the size of group interactions (hyperedges) is comparable to the population size, relatively small hyperedge sizes best facilitate cooperation. Synthetic and empirical hypergraphs alike affirm our findings, illuminating how temporal, higher-order interactions profoundly shape the evolution of cooperation.

Figures (6)

• Figure 1: Construction of temporal hypergraphs.a, On classical networks, a link represents pairwise interaction and can connect only two individuals. For higher-order interactions on pairwise networks, each focal individual (black circled nodes) organizes a common pool in which everyone interacts once. b, On hypergraphs, a hyperedge can connect more than two individuals. Individuals within a hyperedge interact together, and the weight is used to indicate the number of interactions. c, Social interactions among $4$ individuals are indicated by nodes with different colours. Each individual is depicted by a line, over which the corresponding nodes will be connected at time $t$, provided two players interact with each other during each time interval. d, The snapshots are generated from all interactions during the time window, $\Delta t$. Individuals are in a given hyperedge if they interact with each other during the window $\left(t-\Delta t,t\right]$. e, The snapshots within each time interval, $\tau$, are then aggregated into subhypergraphs. For example, the green and yellow nodes interact on the first and second snapshots, thus there is a hyperedge between them with weight $2$ on the first subhypergraph, which means they play the game twice. f, All snapshots are aggregated into a static hypergraph.
• Figure 2: Higher-order interactions and sparse components together promote cooperation. We show the fraction of cooperators, $f_C$, as a function of the synergy factor, $r$, on empirical (a, b) and synthetic (f,i) temporal hypergraphs. For synthetic hypergraphs, we set the probability for links to be activated, $p$, to be $0.05$ (f) and $0.6$ (i). Cooperation emerges at a smaller value of $r_C$ when $\Delta t$ is large and $\tau$ is moderate. c illustrates the definition of connected density. For these two networks, with $4$ nodes and $2$ links, the traditional density is $1/3$. However, connected density focuses on the connected components. In the top network, there are two components with densities equal to $1$. The connected density of the whole network is also $1$. The other network (bottom), with one component and one isolated node, has a connected density of $2/3$. We present the average connected density, $\left\langle \rho \right\rangle$, as a function of time interval, $\tau$, on empirical (d) and synthetic (g,j) hypergraphs. The unit of $\tau$ is $1$min in panel d. For the empirical graphs and the synthetic hypergraphs with $p=0.05$, the connected density drops sharply and then increases slowly as $\tau$ increases. However, it increases monotonically on the synthetic hypergraphs when $p=0.6$. e, We plot the average hyperedge size, $\left\langle n \right\rangle$, as a function of the time window, $\Delta t$, for the empirical hypergraphs. Notably, this size increases as the time window gets longer. We also display distributions of the hyperedge size for synthetic hypergraphs (h, k). The synthetic hypergraphs with $p=0.6$ have larger hyperedge sizes compared to those with $p=0.05$. l, We provide intuition for the change of connected density as a function of $\tau$. The temporal network starts with isolated nodes and only a few links. As $\tau$ increases, a giant component emerges, and then the network becomes connected. Beyond this point, increasing $\tau$ can only increase the density. The last subhypergraph is ignored if $\tau$ does not divide $G\Delta t$. The simulations are performed $10^3$ times, and within each simulation, the evolutionary process runs for $10^6$ rounds. All simulations start with an equal number of cooperators and defectors, with the round of interactions $g=100$. The population size $N=327$ for the empirical hypergraphs and $N=100$ for the synthetic hypergraphs. We set $\tau=5$h for panel a and $\Delta t=0.1$h for panel b. The selection intensity is $s=2$ across all simulations.
• Figure 3: Temporal hypergraphs may promote cooperation compared to their static counterparts. We plot the fraction of cooperation, $f_C$, as a function of the synergy factor, $r$, on empirical (a) and synthetic (c, d) hypergraphs. For synthetic hypergraphs, we set the probability $p$ for links to be activated to be $0.6$ (connected, c) and $0.05$ (disconnected, d). The fraction of cooperators is promoted as the round of interactions $g$ increases and eventually exceeds that of the static counterpart. b, We present the critical value of the synergy factor as a function of $g$. The magnitude of the promotion of cooperation (i.e., the reduction in $r_C$) diminishes as $g$ increases. For panel a, $\Delta t=0.1$h, $\tau=1$h. In panels c and d, $\tau=1$ and $\tau=10$, respectively. Other parameters are consistent with those of Fig. \ref{['fig:tem_den']}.
• Figure 4: Advantages of higher-order interactions in promoting cooperation. We show the fraction of cooperation, $f_C$, as a function of the synergy factor, $r$, on empirical hypergraphs and on corresponding traditional networks with pairwise interactions, for $g=100$ (a) and $g=5000$ (b). The value of $r_C$ is always smaller for temporal and static hypergraphs than that for traditional networks. Other parameters are consistent with those of Fig. \ref{['fig:tem_den']}.
• Figure 5: Moderate hyperedge sizes promote cooperation.a, We plot the fraction of cooperation, $f_C$, as a function of the synergy factor, $r$, for synthetic temporal hypergraphs with different hyperedge sizes, $n$. As $n$ increases, cooperation is promoted at first and then impeded as it requires a larger value of $r$ to reach full cooperation. We collect the values of $r_C$ (b) and $r_{99}$ (d) from simulations, calculate the fitted slopes when $0.2\le r\le0.8$ (c), and then plot their relationship with $n$. As $n$ increases, $r_C$ decreases gradually (b), while $r_{99}$ exhibits a slight initial decrease followed by a significant increase (d). This corresponds to a drop in the fitted slopes with increasing $n$ (c). For synthetic temporal hypergraphs, we use a population size of $N=100$ and the number of hyperedges on each subhypergraph to be $m=20$. The number of interactions is $g=100$. Other parameters are consistent with those used in Fig. \ref{['fig:tem_den']}.