Spreading processes over a given topology are ubiquitous across many social and physical systems. From epidemics to information diffusion, many studies aim to understand the governing rules of spreading processes in social contexts [1–4]. Nevertheless, the influence of the underlying topology on which spreading occurs can sometimes be as important as the mechanistic rules governing the spreading process [5,6]. This consideration is crucial in the study of contagion and epidemics on social networks, where local properties like degree and clustering can mediate nontrivial dynamics such as super-spreader events and endemic states [6,7].

Recent approaches propose the inclusion of interactions along higher-order topologies such as simplices (e.g., triangles and tetrahedrons) to account for social interactions that include more individuals than pairwise interactions [5,8,9]. The inclusion of higher-order interactions has been shown to promote critical transitions and bistability in epidemics [3,10–12]. However, the importance that they play in altering the overall epidemic trajectory remains an active area of research. Current work has also explored the effect of simplicial contagion processes on infection paths on hypergraphs [13]. Several analytical methods have also been developed to better understand the governing factors of unique simplicial contagion behaviours such as hysteresis, bistability, and reduced epidemic thresholds [14,15].

In this work, we present a discrete numerical model that generalizes the classical SIS/SIR compartmental model with homogeneous mixing to account for hyperedge interactions of arbitrary order. Comparing dynamics with and without higher-order interactions, we find that topology primarily affects epidemic transients. If the network structure is known, one can calculate normalization values related to the potential network activity. These values may be used to define disease parameters for first-order pairwise interactions that replicate the steady state dynamics of those that include higher-order interactions, but are insufficient for reproducing epidemic transients. However, we find a duality where the effects of network topology can be overcome by allowing for temporally varying disease parameters, allowing first-order dynamics to closely approximate both transient and steady-state trajectories of higher-order systems.

In addition to requiring knowledge of the order scaling of interaction dynamics, the normalization approach makes two assumptions regarding the hypergraph topology. Namely, that (1) all hypergraphs are defined as clique complexes, and (2) macroscale topological features are uniformly distributed across the network. To investigate the robustness of the results, tests are conducted on cases where the scaling function is incorrect. We show robustness to model misspecification provided the size scales of the maximal infection amplification between the true and misspecified scaling functions are of similar order. These results are broadly aligned with the recent analytical results by [16] where mean-field approximations of pairwise interactions were able to capture features of simplicial contagion. We also extend analyses to artificial and empirical networks that do not adhere to Assumptions (1) and (2) and show robustness of results when disease parameters are allowed to temporally vary. Further inspection of the network structure and component hyperedge activities suggests that nontrivial macroscale heterogeneities in hypergraph topologies limit the approximation of higher-order dynamics by pairwise interactions.

We begin by presenting a variation of the classical compartmental ODE model [17], focusing on the three-compartment case with susceptible-infected-removed (SIR) states,

The simple SIR model assumes that populations homogeneously mix. However, this is not true for human populations where disease spread occurs on a social contact network. Here, individuals—represented as nodes—interact within a small neighborhood with occasional branching, clustering, or long-range connections [5,18,19]. One can simulate more realistic spread dynamics by employing an agent-based approach where randomly selected nodes interact with all or a collection of their neighbors every time step [20]. Whilst this node-based approach is more realistic for modeling the diffusion of abstract quantities such as information, it is not necessarily adequate for disease contagion as it treats individuals rather than interactions as facilitators of infection. It is unlikely for an individual to interact with all of their neighbors in a given time step. Instead, we propose simulating diseases on networks using an edge-based approach.

Let be a hypergraph consisting of a set of hyperedges and nodes . A -th order hyperedge is a connection between different nodes. Each simulation time step consists of iterations. A random hyperedge is selected in each iteration. Each susceptible node in has a probability of getting infected and transitioning to state ,

where and are the number of nodes in that are infected and susceptible respectively, is a base infection probability for a pairwise interaction, and is a scaling function that controls the extent to which multiple infected nodes in a given hyperedge amplify the probability of infection. In simplicial social contagion models (SCM) where interactions occur up to order , susceptible nodes whose neighbors are all infected experience an amplified probability of infection [5]. Whilst SCM has only been applied to social dynamics (e.g., knowledge spreading), we argue that a similar principle is also relevant when studying diseases where the infection probability may be mediated by the concentration of disease particles within an individual's immediate vicinity during an interaction. The allowance for variable infectivity has been explored previously by Fu et al. [21]. Inspired by this, we use the following scaling function to describe the disease spread dynamics on hyperedges:

where is the maximum amount that a hyperedge interaction can amplify the infection probability. After interaction, all infected nodes recover or die with probabilities and .

The proposed numerical model achieves two goals. First, it unifies the classical homogeneous ODE disease model with the existing SCM up to order . For and a fully connected graph , this model is a numerical approximation for the SIR/SIS model using the following disease parameters:

where is the simplicial infection rate. Second, this generalizes simplicial contagion to account for simplices of arbitrary order and varying proportions of infected individuals within each hyperedge.

FIG. 1.

Identifying dynamical differences between networks with and without interactions on higher-order topologies requires an appropriate normalization that allows for comparison. Thus, we consider two candidate quantities related to the rate of new infection .

The first quantity, called the combinatorial network activity calculated up to order for a hypergraph with respect to a baseline infection rate , is defined as

where

and approximates under the assumption that individuals are equally likely to be infected or susceptible. Therefore, depends only on the network topology and the scaling function .

Similarly, an exact network activity up to order can be calculated as

where is the set of all hyperedges of order up to , and are the nodes' states. Unlike , does not assume a distribution of node states and is instead a function of the full network state and approximates the instantaneous value of .

In this work, we quantify the effect of higher-order topologies on the spreading dynamics on networks. Focusing on the single disease case, we have presented an agent-based model that unifies the classical compartmental SIR model of homogeneous mixing with pairwise simplicial contagion models on networks, and extends it to account for interactions on hyperedges of arbitrary order. We use this model in conjunction with a normalization method based on the notion of network activity to compare spread dynamics across hypergraphs of different maximal orders.

We first consider the restricted case of clique complexes with uniformly distributed macroscale topological features for both SIS and SIR dynamics. The inclusion of higher-order topology is found to primarily affect the transient dynamics in epidemic trajectories. Using carefully chosen disease parameters such that initial combinatorial network activities are approximately equal, simulations restricted to pairwise (order 1) interactions are able to replicate the steady-state behaviours resulting from higher-order interactions. Furthermore, a sufficiently accurate normalization can be calculated purely based on the network topology, network state, and the scaling function describing the higher-order interaction. We find that allowing disease parameters to dynamically vary over time is sufficient for pairwise simulations to reproduce full trajectories of higher-order interactions. Thus, we conclude that there is a duality between topological effects and the stationarity of disease parameters.

The proposed normalization method based on network activities makes several assumptions, such as having knowledge of the scaling function, and having a clique complex structure with relatively mild macroscale heterogeneities. While the findings are theoretically interesting, these assumptions impose heavy constraints on applicability as real-world systems rarely fulfill these conditions. To this end, we test the robustness of the network activity approach toward model misspecification and non-clique hypergraphs with nontrivial topology taken from empirical data. Overall, all results are relatively robust to model misspecification due to scale and functional form errors. In contrast, we find that network topology plays a larger role where networks with macroscale heterogeneities—such as social networks or those arising from preferential attachment mechanisms—where higher-order interactions are not as easily or robustly approximated by pairwise interactions. This result is particularly true for cases where disease parameters are kept static.

The ability for pairwise interaction models to reproduce epidemic trajectories from models containing higher-order complexity features raises interesting questions on the need for higher-order features to be included in disease models. Furthermore, it encourages a reconsideration of whether dynamics, observed or modeled, should be attributed as the result of complex topological structure or temporally varying spreading mechanisms. The above tests focus on epidemic trajectories, which represent summaries of the disease averaged across the entire network. These results invite further questions on the similarity of spreading dynamics on the local scale of a given network and whether the order of infection events between individuals differs between interactions of different orders. One can also consider the effect of incubation and immunity refractory periods, which may impart a form of time delay in the infection process, resulting in more dynamic disease parameters. On application, network activities offer a potential way to construct adaptive rewiring strategies that can be used to maintain a sustainable level of infection.

The data that support the findings of this article are openly available [27].