Higher-order contagion processes in 3.99 dimensions

Higher-order interactions have recently emerged as a promising framework for describing new dynamical phenomena in heterogeneous contagion processes. However, a fundamental open question is how to understand their contribution from the perspective of the physics of critical phenomena. Based on a mesoscopic field-theoretic Langevin description, we show that: (i) pairwise mechanisms such as facilitation or thresholding are formally equivalent to higher-order ones, (ii) pairwise interactions at coarse-grained scales govern the higher-order contact process and, (iii) the interplay between noise and topology is determined by the network spectral dimension. In short, we demonstrate that classical field theories, rooted on model symmetries and/or network dimensionality, still capture the nature of the phase transition, also predicting finite-size effects in real and synthetic networks.

💡 Research Summary

**
The paper tackles the emerging problem of higher‑order interactions in contagion dynamics by embedding them into a mesoscopic field‑theoretic framework. Starting from the classic SIS/SIR compartmental models, the authors derive a stochastic Langevin equation for the infection density ρ_i that includes linear decay, quadratic and cubic nonlinearities, a diffusive Laplacian term, and multiplicative Gaussian noise. By explicitly expanding the contributions of 1‑simplices (edges) and 2‑simplices (triangles) they obtain a mean‑field equation in which the coefficient
b = βΔ⟨κΔ⟩ – β⟨κ⟩
controls the sign of the ρ² term. When b > 0 the system exhibits a first‑order (discontinuous) transition, while b < 0 yields a continuous (second‑order) transition. Crucially, the authors prove that familiar pairwise mechanisms such as facilitation or thresholding are mathematically equivalent to the effect of 2‑simplex activation; thus higher‑order contagion can be fully captured by an effective pairwise field theory.

Higher‑order simplices with ω > 2 generate terms of order ρ^ω, but a renormalization‑group analysis shows these are irrelevant above the upper critical dimension. Consequently, only edges and triangles matter for universal critical behavior. The diffusive Laplacian couples nodes across the network and introduces the concept of spectral dimension d_S, which plays the role of Euclidean dimension for networks. For d_S ∈ (1, 4) noise and heterogeneity interact nontrivially, while for d_S > 4 the mean‑field description becomes exact.

The theoretical predictions are validated on a suite of synthetic networks (Barabási‑Albert, Kim‑Holme, Erdős‑Rényi, and rewired 2‑D small‑world lattices) and on empirical contact datasets (a rural Malawian village and a French workplace). By varying system size, clustering, or a weight‑threshold h that prunes weak edges, the authors demonstrate that the sign of b changes precisely when the ratio of average triangles ⟨κΔ⟩ to average degree ⟨κ⟩ crosses unity, leading to a switch between continuous and discontinuous epidemic thresholds. Finite‑size effects are shown to be especially pronounced in scale‑free BA networks where clustering vanishes with size, whereas scale‑invariant KH networks keep b constant.

Finally, the paper explores the impact of quenched Gaussian disorder in the infection rates (β, βΔ). Even modest noise amplitudes (δ ≈ 0.05) suppress the discontinuous jump, smoothing the transition and preventing explosive outbreaks. This highlights the pivotal role of stochastic fluctuations in determining whether higher‑order mechanisms manifest as abrupt phase changes.

In summary, the work establishes three key insights: (i) higher‑order contagion can be mapped onto an effective pairwise field theory; (ii) the network spectral dimension governs how topology and noise shape critical behavior; and (iii) only 2‑simplex interactions are relevant for universal properties, with higher‑order terms being renormalization‑group irrelevant. These results bridge the gap between modern simplicial contagion models and classical critical phenomena theory, providing a robust analytical tool for predicting epidemic thresholds and finite‑size effects in real‑world heterogeneous networks.