Unveiling the impact of cross-order hyperdegree correlations in contagion processes on hypergraphs

TL;DR

This work addresses contagion dynamics on hypergraphs with both pairwise and higher-order interactions by explicitly modeling cross-order hyperdegree correlations. It introduces a configurational hyperdegree model and two linked abstractions—the effective hyperdegree model (EHDM) and its compact form—to capture how node participation across orders shapes epidemic thresholds, bistability, and spreading pathways. The framework reveals that positive cross-order correlations lower the epidemic threshold and synchronize transmission, while anti-correlations desynchronize and shift hub roles, with clear implications for designing targeted interventions. The results provide a scalable, physics-informed route to analyze and control complex contagions in systems where group interactions are essential, with potential extensions to temporal networks and empirical inference of cross-order correlations.

Abstract

Contagion processes in social systems often involve interactions that go beyond pairwise contacts. Higher-order networks, represented as hypergraphs, have been widely used to model multi-body interactions, and their presence can drastically alter contagion dynamics compared to traditional network models. However, existing analytical approaches typically assume independence between pairwise and higher-order degrees, and thus study their roles in isolation. In this paper, we develop an effective hyperdegree model (EHDM) to describe Susceptible-Infected-Susceptible (SIS) dynamics on hypergraphs that explicitly captures correlations between the distribution of groups with different sizes. Our effective hyperdegree model shows excellent agreement with stochastic simulations across different types of higher-order networks, including those with heterogeneous degree distributions. We explore the critical role of cross-order degree correlations, specifically, whether nodes that are hubs in pairwise interactions also serve as hubs in higher-order interactions. We show that positive correlation decreases the epidemic threshold and anti-correlation temporally desynchronizes infection pathways (pairwise and group interactions). Finally, we demonstrate that, depending on the level of correlation, the optimal control strategy shifts -- from one that is purely pairwise- or higher-order-focused to one in which a mixed strategy becomes optimal.

Figures (8)

• Figure 1: Representation of hub distribution for correlated and anti-correlated systems: This diagram depicts an illustrative representation of how cross-order hyperdegree correlations influence the distribution of highly connected nodes in a heterogeneous hypergraph. In (a) we show the positively correlated case, where the hub (red node) is highly connected in both group interactions and pairwise links. In (b) we show the negatively correlated case, where there are two distinct hubs (red nodes): one is highly connected in group interactions with very few pairwise links while the other is highly connected in pairwise links with few group interactions.
• Figure 2: EHDM vs simulation comparison: We compare the model against stochastic simulations for three different classes of hypergraphs: a regular one where all nodes have $k_1=5$ and $k_2=3$; a hypergraph with Poisson hyperdegree distribution with average $\langle k_1 \rangle = 5$ and $\langle k_2 \rangle = 3$; lastly, a hypergraph exhibiting a truncated power-law hyperdegree distribution with characteristic exponent $\nu_1=2.5$ for pairwise interactions and $\nu_2=2.25$ for three-body interactions. We consider $N=1000$, and a recovery rate $\gamma = 1$ for all cases. Panels (a--c) compare the temporal evolution of the proportion of infected individuals between the compact effective hyperdegree model (red line) and Gillespie simulations (orange lines). For these cases $\lambda_1=1.5$ and $\lambda_2=3$. Panels (d--f) show the phase diagram of the final proportion of infected individuals $\rho$ as a function of $\lambda_1$, showing the bistable regime. In these cases $\lambda_2=2.5$.
• Figure 3: Role of hyperdegree heterogeneity: (a) shows the final epidemic size from both stochastic simulations (diamonds) and the solution of the compact effective hyperdegree model (continuous line) as a function of the effective infection rate $\lambda_1$ for five different hypernetworks, all exhibiting negative binomial hyperdegree distributions for both pairs and triples with $N=1000$, $\langle k_1\rangle \approx 5$, $\langle k_1\rangle \approx 3$. The structures differ in the variance of their hyperdegree distributions, taking values $10$, $30$, $60$, $100$, and $300$. In panels (b--e), we display heat-maps of the difference in prevalence $\Delta\rho$ between two epidemic processes on hypernetworks with the same properties and the same infection and recovery rates, but different initial numbers of infected individuals. Each panel corresponds to a higher-order network with increasing variance $10$, $60$ and $80$.
• Figure 4: Effect of cross-order hyperdegree correlation: (a) displays the final epidemic size from both simulations (diamonds) and the solution of the compact effective hyperdegree model (solid lines) as a function of the infection rate $\lambda_1$ for heterogeneous hypernetworks with varying cross-order hyperdegree correlation, from $\sigma = -1$ to $\sigma = 1$ in different colors, for all curves $\lambda_2=3$. In panel (b), we show the value of the critical infection rate $\lambda_1^*$ found through the EDHO model as a function of cross-order hyperdegree correlation for higher-order networks with different levels of heterogeneity. Panel (c) showcases the difference between temporal centroids of pairwise infections $I_{PW}$ and group infections $I_{HO}$ as a function of cross-order hyperdegree correlations for higher-order networks with different levels of heterogeneity. Panels (d--h) display the hyperdegree distributions of hypernetworks with a different cross-order hyperdegree correlation, specifically $\sigma = -1, -0.5, 0, 0.5, 1$. Panels (i--m) show the temporal evolution of the number of infected individuals $I$ and the pairwise and group contributions $I_{\rm PW}$ and $I_{\rm HO}$ for simulations on hypernetworks with hyperdegree distributions shown in panels (d--h). All hypergraphs have size $N=1000$ and exhibit negative binomial hyperdegree distributions for both pairwise and triple interactions, with average hyperdegrees $\langle k_1\rangle \approx 6$ and $\langle k_2\rangle \approx 6$, and a hyperdegree variance of approximately 30 for both interaction orders (except in panel (c), where the variance differs for each curve as indicated in the legend).
• Figure 5: Hierarchical Spread in Higher-Order Networks: In panel (a), we show the temporal evolution of the average 1-hyperedge degree of newly infected nodes, and in panel (b) the corresponding average 2-hyperedge degree. Panels (c) and (d) display the time evolution of the participation ratio for pairwise and group interactions, respectively, as defined in \ref{['eq_inv_part']}. All hypergraphs have size $N=1000$ and exhibit negative binomial hyperdegree distributions for both pairwise and three-body interactions. In all four panels, each curve corresponds to a distinct value of the cross-order hyperdegree correlation, $\sigma \in {-1, -0.5, 0, 0.5, 1}$. The hyperdegree distributions used in the simulations have mean values $\langle k_1\rangle \approx 6$ and $\langle k_2\rangle \approx 6$, with a variance of 30 for both interaction orders. Disease parameters are fixed at $\lambda_1 = 0.9$ and $\lambda_2 = 3$.