Anzhi Sheng

Graphs provide a natural framework to describe processes shaped by pairwise interactions among agents. But many dynamical systems involve interactions within groups of three or more agents. Here, we develop the “ ℓ -hyperedge approximation,” an analytical framework for stochastic processes on regular hypergraphs, in which each individual belongs to k groups of size ℓ . The framework accommodates both higher-order interactions that determine payoffs and higher-order processes for updating states in response to payoffs. For evolutionary games on hypergraphs, our analysis generalizes the classical b / c > k rule for cooperation to the ℓ -player donation game; and it yields critical benefit-to-cost ratios for the nonlinear ℓ -player public goods game, which remains bounded as the degree grows. Applied to neutral complex contagions, where inheritance of states occurs within hyperedges rather than along parent–offspring edges, the framework gives a closed-form fixation probability, showing how a single complexity parameter governs the spread of rare types. Coupling the two processes produces a unified stochastic model of payoff-biased complex contagions in structured populations. Together, these results extend pair approximation from graphs to hypergraphs, accommodating multiway interactions and group-level inheritance with no pairwise analog.

Cooperation is typically seen as the ideal outcome in a social dilemma. Because cooperators are vulnerable to exploitation, much of the literature has focused on mechanisms, such as spatial structure, that support prosocial behaviour and the production of public goods^1-11. Yet the rules for distributing these goods also shape behaviour and long-term prosperity. Here we study policies for allocating public goods, comparing equitable allocation, in which returns are proportional to potential contributions, with uniform allocation, in which all individuals receive equal shares. For most social networks, we find that uniform allocation facilitates the spread of cooperation compared with equitable allocation. But this success comes with a cost. Uniform allocation concentrates resources in a small number of highly connected individuals¹², whereas peripheral individuals receive fewer benefits and may even be worse off than in a non-cooperative society. We develop a theoretical analysis of the tension between cooperation and equality, and we identify this conflict across diverse empirical social networks. Our results show that inequality may be an unavoidable consequence of allocation policies designed to foster cooperation in spatially heterogeneous populations. The question of how to promote cooperation is therefore incomplete: because policies that facilitate cooperation can also generate social stratification, we must weigh the benefits of cooperation against the inequality that accompanies it.

Cooperation is key to prosperity in human societies. Population structure is well understood as a catalyst for cooperation, where research has focused on pairwise interactions. But cooperative behaviors are not simply dyadic, and they often involve coordinated behavior in larger groups. Here we develop a framework to study the evolution of behavioral strategies in higher-order population structures, which include pairwise and multi-way interactions. We provide an analytical treatment of when cooperation will be favored by higher-order interactions, accounting for arbitrary spatial heterogeneity and nonlinear rewards for cooperation in larger groups. Our results indicate that higher-order interactions can act to promote the evolution of cooperation across a broad range of networks, in public goods games. Higher-order interactions consistently provide an advantage for cooperation when interaction hyper-networks feature multiple conjoined communities. Our analysis provides a systematic account of how higher-order interactions modulate the evolution of prosocial traits.

Human social interactions tend to vary in intensity over time, whether they are in person or online. Variable rates of interaction in structured populations can be described by networks with the time-varying activity of links and nodes. One of the key statistics to summarize temporal patterns is the inter-event time, namely the duration between successive pairwise interactions. Empirical studies have found inter-event time distributions that are heavy-tailed, for both physical and digital interactions. But it is difficult to construct theoretical models of time-varying activity on a network that reproduce the burstiness seen in empirical data. Here we develop a spanning-tree method to construct temporal networks and activity patterns with bursty behavior. Our method ensures any desired target inter-event time distributions for individual nodes and links, provided the distributions fulfill a consistency condition, regardless of whether the underlying topology is static or time-varying. We show that this model can reproduce burstiness found in empirical datasets, and so it may serve as a basis for studying dynamic processes in real-world bursty interactions.

Population structure is a well-known catalyst for the evolution of cooperation and has traditionally been considered to be static in the course of evolution. Conversely, real-world populations, such as microbiome communities and online social networks, frequently show a progression from tiny, active groups to huge, stable communities, which is insufficient to be captured by constant structures. Here, we propose sequential temporal networks to characterize growing networked populations, and we extend the theory of evolutionary games to these temporal networks with arbitrary structures and growth rules. We derive analytical rules under which a sequential temporal network has a higher fixation probability for cooperation than its static counterpart. Under neutral drift, the rule is simply a function of the increment of nodes and edges in each time step. But if the selection is weak, the rule is related to coalescence times on networks. In this case, we propose a mean-field approximation to calculate fixation probabilities and critical benefit-to-cost ratios with lower calculation complexity. Numerical simulations in empirical datasets also prove the cooperation-promoting effect of population growth. Our research stresses the significance of population growth in the real world and provides a high-accuracy approximation approach for analyzing the evolution in real-life systems.