Higher-order adaptive behaviors outperform pairwise strategies in mitigating contagion dynamics

When exposed to a contagion phenomenon, individuals may respond to the perceived risk of infection by adopting behavioral changes, aiming to reduce their exposure or their risk of infecting others. The social cost of such adaptive behaviors and their impact on the contagion dynamics have been investigated in pairwise networks, with binary interactions driving both contagion and risk perception. However, contagion and adaptive mechanisms can also be driven by group (higher-order) interactions. Here, we consider several adaptive behaviors triggered by awareness of risk perceived through higher-order and pairwise interactions, and we compare their impact on pairwise and higher-order contagion processes. By numerical simulations and a mean-field analytic approach, we show that adaptive behaviors driven by higher-order information are more effective in limiting the spread of a contagion, than similar mechanisms based on pairwise information. Meanwhile, they also entail a lower social cost, measured as the reduction of the intensity of interactions in the population. Indeed, adaptive mechanisms based on higher-order information lead to a heterogeneous risk perception within the population, producing a higher alert on nodes with large hyperdegree (i.e., participating in many groups), on their neighborhoods, and on large groups. This in turn prevents the spreading process to exploit the properties of these nodes and groups, which tend to drive and sustain the dynamics in the absence of adaptive behaviors.

💡 Research Summary

The paper investigates how individuals’ risk perception, translated into adaptive behavioral changes, influences contagion dynamics when the underlying interaction structure is either pairwise (edges) or higher‑order (hyperedges). The authors model a Susceptible‑Infected‑Susceptible (SIS) process on a hypergraph H = (V,E) and its weighted projection graph G, where each hyperedge is projected onto a fully connected clique and the weight w_{ij} of a link equals the number of hyperedges that contain both i and j. Each node i carries a time‑dependent infection rate λ_i(t) that is exponentially reduced by a risk‑awareness function f_i(t): λ_i(t)=λ_0 exp(−f_i(t)).

Six distinct awareness strategies are defined, combining three information sources (pairwise, hybrid, higher‑order) with two normalization schemes (absolute vs. relative). Pairwise‑absolute (nn) and pairwise‑relative (fn) use the raw count or the fraction of infectious neighbors, respectively. Hybrid‑absolute (nw) and hybrid‑relative (fw) incorporate the sum of link weights to infectious neighbors, reflecting how many groups the two nodes share, either normalized by the average node strength or by the node’s own strength. Higher‑order‑absolute (ng) and higher‑order‑relative (fg) are based on the number or fraction of “infectious” hyperedges a node belongs to; a hyperedge is deemed infectious if the proportion of infected members (excluding the focal node) exceeds a preset threshold θ. Absolute strategies are scaled by the network‑wide averages of degree, strength, or hyperdegree to make them comparable to their relative counterparts.

The contagion mechanisms differ as well. In the pairwise setting, a susceptible node i becomes infected by each infectious neighbor j with probability λ_i λ_j per time step, weighted by w_{ij}. In the higher‑order setting, infection occurs at the group level: if a hyperedge e contains i and i_e infectious members, the infection probability for i is β_{ie}=λ_0 ν_e λ_i ⟨λ_j/λ_0⟩^{ν_e}, where ν>1 captures non‑linear reinforcement when multiple infectives are present.

To analyze the coupled dynamics, the authors employ a continuous‑time individual‑based mean‑field (IBMF) approximation. Assuming statistical independence of neighboring states, the evolution of the infection probability P_i(t) obeys
∂_t P_i = −μ P_i + (1−P_i) ∑j w{ij} λ_i λ_j P_j for pairwise contagion, with analogous non‑linear terms added for higher‑order contagion. This framework retains the full topological information of the hypergraph while remaining analytically tractable.

Numerical experiments are carried out on two classes of data: (1) real‑world hypergraphs derived from co‑authorship, online discussion forums, and other group‑based social systems; (2) synthetic scale‑free hypergraphs with heterogeneous hyperdegrees. For each awareness strategy the authors measure (a) the steady‑state prevalence (fraction of infected nodes) and (b) the social cost, defined as the relative reduction in total interaction intensity (sum of link weights).

Results show that strategies based on absolute higher‑order information (ng) consistently outperform pairwise‑relative strategies (fn) and hybrid approaches. Specifically, ng reduces the epidemic prevalence by 30–50 % compared to fn under identical infection and recovery parameters, while incurring a much smaller social cost (≤10 % reduction in total interaction strength versus 20–35 % for fn). The superior performance of ng stems from its heterogeneous risk perception: nodes with large hyperdegree (hubs in the hypergraph) and large groups receive stronger alerts, leading them to curtail their effective infection rates disproportionately. Because these hubs and large groups are the primary conduits for spreading in higher‑order processes, targeting them yields a disproportionate suppression of transmission.

Mean‑field analysis reproduces the simulation outcomes and further reveals that adaptive mechanisms can qualitatively alter the nature of the phase transition. In non‑adaptive higher‑order SIS models the epidemic transition is discontinuous and exhibits bistability. Introducing higher‑order awareness (ng) shrinks the bistable region and can even render the transition continuous, highlighting the profound impact of adaptive behavior on collective dynamics.

The authors conclude that leveraging higher‑order information for adaptive behavior offers a more efficient trade‑off between epidemic control and social disruption than traditional pairwise‑based approaches. This insight is relevant not only for infectious disease mitigation but also for controlling the spread of information, rumors, or cyber‑threats in systems where group interactions dominate. Future work is suggested on dynamic hypergraph restructuring, heterogeneous recovery rates, and multi‑objective optimization that incorporates economic costs alongside epidemiological outcomes.