Higher-order contagion processes in 3.99 dimensions

TL;DR

This work builds a bridge between classical phase-transition theory and higher-order contagion by mapping simplicial interactions onto a mesoscopic pairwise Langevin framework. It shows that 2-simplicial (triangle) terms can drive first-order transitions, while higher-order terms with $\omega>2$ are typically irrelevant at coarse scales, and that diffusion plus noise on networks is governed by the spectral (graph) dimension. The authors validate predictions on synthetic and empirical networks, demonstrating finite-size effects and the central role of $d_S$ in constraining fluctuations and heterogeneity. Overall, the study provides a unified, field-theoretic description linking higher-order contagion to established universality classes and highlighting the upper critical dimension $d_c=4$ as a key threshold for when network heterogeneity matters at criticality.

Abstract

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.

Figures (4)

• Figure 1: Spreading dynamics. Different microscopic rules are shown to account for pairwise interactions and higher-order (2-simplex) interactions. Different rates control the spreading process, namely: the activation rate ($\beta$), the inactivation rate ($\mu$), and the 2-simplex activation rate ($\beta_\Delta$).
• Figure 2: Synthetic networks. (a) Averaged value of b versus system size for different networks, BA with $m=5$, and KH with $m=4, ~p=1$, and $m=2,~p=0.5$ (see legend). (b) The averaged value of b versus rewiring probability for a 2D T-SW network (red triangles; $N=2450$) and the giant component of an ER network of different system sizes (see legend). Fraction of infected sites versus rescaled infection probability for (c) a BA network with $m=5$ and different system sizes (see legend) and (d) 2D T-SW networks with $N=2450$ and different rewiring probabilities, $p$ (see legend).
• Figure 3: Contact networks. Fraction of infected sites versus rescaled infection probability for contact networks in (a) a village in rural Malawi for different thresholds (see legend) and (c) in a workplace, both shown for varying threshold values (see legend). The difference between the average number of triangles $\langle \kappa_\Delta \rangle$ and the mean connectivity $\kappa$ as a function of the threshold is shown for (b) Malawi and (d) the workplace. Insets illustrate the resulting network for threshold value h = 10. The black dashed line marks where the giant component contains 50% of the original nodes.
• Figure 4: Dimensionality and discontinuous transitions. (a) Average effective value of $b$ versus $\alpha$ for HMNs with size $N=m_02^s$ for different rates $\beta_\Delta/\beta$ (see legend). Fraction of infected sites versus rescaled infection probability for (b) HMN with $m_0=3,$ and $s=11$ and different values of $\alpha$, for $\beta_\Delta=2\beta$. (c) HMN with $m_0=3,~s=15$ and $\alpha=8$ under the application of a small Gaussian variability $\mathcal{N}(\beta,\delta)$, with $\delta=0.05$. The inset shows the phase transition with $\delta=0$. (d) Temporal evolution of the density of infected nodes at criticality ($\lambda=0.36$) for CLRs obtaining different specified spectral dimensions (see legend). Beyond $d_s=4$, all networks present equivalent critical behavior. Red curves represent an initial density of infected nodes $\rho_0 = 0.5$ while blue ones $\rho_0= 5 \cdot 10^{-3}$. Parameters: $N=10^4$, $\tilde{\mu}\in[\mu-\delta,\mu+\delta]$, with $\delta=0.045$.