The partial information decomposition framework allows for the characterization of the information-sharing interdependencies between groups of variables [25–27]. Qualitatively, these relations can be of three types: redundant, synergistic, or unique. Consider three variables , , and . Information is redundant if it is replicated over the variables (i.e., recoverable from ), synergistic if it can only be recovered from their joint state ( ), and unique if it can only be recovered from one variable and nowhere else. In this framework, mutual information has been extended to groups of three or more variables by the O-information [18]. To generalize the O-information of multivariate time series from equal-time correlations to time-lagged correlations—similarly to how transfer entropy extends mutual information [28]—Stramaglia et al. proposed dynamical O-information [24]. This quantity is defined by (i) considering variables on which we compute the O-information , (ii) adding a new variable , and (iii) computing the variation of O-information: . To remove shared information due to common history, the dynamical O-information is defined by conditioning on the history of the target variable :

Here, , is the conditional mutual information, the past and present of , and its next instance. The parameter is the temporal horizon of the time series, usually set to a relevant time scale of the process. To quantify the dynamical O-information regardless of source-target assignments, we define the total dynamical O-information as

Total dynamical O-information inherits from O-information the property of being a signed metric: indicates that information sharing among the units of is dominated by redundancy, while indicates that it is dominated by synergy (see Sec. II of Supplemental Material [29] for a more extended presentation of these observables).

We consider two discrete higher-order dynamical models: a simplicial Ising model and the simplicial model of social contagion [23]. Both are defined on simplicial complexes, a class of hypergraphs [15] that encode multinode interactions as simplices and respect downward closure (we refer to SM Sec. III for an extended presentation of the two models).

The first model we consider is a simplicial Ising model. This model is an extension of the Ising model [21,22] with group interactions of different strengths for simplices of different sizes. We consider a simplicial complex with average generalized degrees , where each of the nodes has two possible states: spin-up ( ) or spin-down ( ). The model is defined by the Hamiltonian

where is the maximal order of and

is the Kronecker delta for an arbitrary number of binary arguments. Inserting the Kronecker delta—instead of the product [38–40]—in the coupling terms is necessary to preserve the symmetry under spin flip at all sites of the dyadic model with no magnetic field ( ). We consider the dynamics of this system to be the sequence of Monte Carlo moves performed with the Metropolis-Hastings acceptance-rejection rule [41] at temperature .

The second model we consider is the simplicial model of social contagion [23]. Following the Susceptible-Infected-Susceptible (SIS) framework [42], we associate to each of the nodes of a simplicial complex a binary variable , corresponding to the susceptible or infected state of agent at time . At the initial time step , a fraction of infected agents is placed in the population. At each time step, susceptible agents [ ] become infected with a probability if they belong to a -simplex where all other nodes are infected. Infected agents [ ] recover independently with probability . We introduce the usual rescaled infectivity parameter of order : .

For computational feasibility, we limit ourselves to group mechanisms and interdependencies up to three nodes (i.e., ). The results shown are obtained in random simplicial complexes with nodes and average degrees , —results are qualitatively consistent across a large range of such parameters. The Ising model was simulated for time steps, and the contagion model for time steps. Other parameters were set to (Ising) and and (contagion).

We simulate the two systems for different values of the control parameters and compute the total dynamical O-information on the resulting time series for different types of node triplets: 2-simplices, 3-cliques, and uniformly randomly chosen triplets of nodes (that are not connected in a 2-simplex or a 3-clique). The 2-simplices are the true higher-order interactions as the nodes belonging to them will interact through the three-body ferromagnetic coupling in the Ising model and the group infection rate in the contagion model. The 3-cliques can be thought of as “spurious” higher-order interactions as the nodes belonging to them are all interconnected but through pairwise couplings only. In both cases, we set the delay of the dynamical O-information to as both systems are Markovian. Increasing the strength of group interactions ( , the three-body coupling in the Ising model, and , the group rescaled infection rate in the contagion model), we observe an increasing cooccurrence of higher-order mechanisms and synergistic higher-order behaviors (Fig. 2). In both systems, all types of triplet groups display synergistic higher-order behaviors ( ); however, and crucially, as we increase the relative strength of the higher-order mechanisms, we see that 2-simplices, i.e., the genuine higher-order interactions, display significantly stronger synergistic behaviors than the other groups.

FIG. 2.

To go further, we now show (Fig. 3) how the total dynamical O-info changes as the strength of group mechanisms is increased ( and , respectively), relatively to its value without group mechanisms ( and , respectively). We see that, as we increase this group coupling strength, the measured on 2-simplices (solid lines) shows a relative increase with respect to the case without group mechanisms, regardless of the (color-coded) pairwise coupling strength ( and , respectively). Moreover, total dynamical O-info measured on 3-cliques (dashed lines) stays roughly constant as and increase. These two facts confirm the results from Fig. 2, showing that group mechanisms promote higher-order synergistic behavior. More importantly, we see that the relative response of 2-simplices is qualitatively different in the two systems. First, the response appears to be roughly linear (until the transition to the ferromagnetic phase occurs, see SM Sec. IV where we show the numerical values of the total dynamical information in the two systems) in the simplicial Ising model, but nonlinear in the simplicial contagion model. Second, although the relative depends on the pairwise coupling strength ( and , respectively) in both systems, it does so in opposite ways: an increase in the pairwise coupling yields a larger relative in the simplicial Ising model, but yields a lower relative in the simplicial contagion model. These results indicate a complex and system-dependent interplay between low- and higher-order mechanisms and behaviors, which requires further investigation.

Despite the strong synergistic behaviors displayed by genuine higher-order interactions, we still do not know the extent of this correspondence, nor whether low-order observables could already detect—and to what degree—the presence of higher-order interactions. Moreover, we need to determine whether group behaviors are truly higher-order, or the byproduct of low-order interdependencies. To answer these questions, we compare our higher-order metric with a lower-order metric over the parameter space of both systems. For the latter, for each triplet, we compute the sum of the transfer entropies between the time series of the three possible node pairs. For both metrics, we quantify the difference in behavior between 2-simplices and 3-cliques via the statistical distance [43,44]. For two distributions—here, for 2-simplices and for 3-cliques—over a common alphabet , is defined as

which we denote for short. The distance quantifies the overlap of the two distributions. By definition, it takes values in [0, 1]: if the two distributions are identical, and if the two distributions take nonzero values on nonoverlapping subsets of [45].

In both systems, we find two main results. First, the low-order behavioral metric does not see differences between the lower- (3-cliques) and higher-order mechanisms (2-simplices), whereas the higher-order metric does (the statistical difference between 3-cliques and random triplets is shown in SM Sec. V). Indeed, this is indicated by the uniformly low values of with low-order metric [Figs. 4(a) and 4(c)] with respect to the large values exhibited by with the higher-order metric, the total dynamical O-information [Figs. 4(b) and 4(d)]. The latter is consistent with the synergetic signature results shown in Fig. 2. So, the higher-order mechanisms can be identified—and distinguished from low-order mechanisms—by the higher-order behavioral metric and not by the low-order one.

FIG. 3.

FIG. 4.

Second, focusing on , we see that the difference between the 2-simplices and 3-cliques is large (large , dark blue) over a finite region of parameter space [Figs. 4(b) and 4(d)]. This region corresponds to the cooccurrence of higher-order mechanisms and synergistic higher-order behavior (Fig. 2). This occurs for sufficiently large values of the strength of the higher-order mechanisms.

These findings apply to both models, but each model has its specificities. While explaining the full shape of the dark blue region is a hard task, we can explain some of its features. In the Ising model [Fig. 4(b)], the large region (dark blue) does not extend above (dashed line), which is the magnetization threshold of the pairwise model with no magnetic field [46]. This is because, above that value, the system magnetizes, and no information can be recovered. Below ( ), the region appears above a certain strength of the three-body coupling , and that value seems to decrease as increases.

In the contagion model [Fig. 4(d)] the large region displays larger values ( ). It does not appear to be bounded from above. However, it is bounded from below. First, for (left of the vertical dashed line), the region appears only above (dashed-dotted line). These values are known: is the epidemic threshold of the pairwise version of the model (SIS on an Erdős-Rényi graph), and is the value where the discontinuous transition occurs [23]. Below that value, only the epidemic-free state exists, for which no higher-order behaviors are expected. Second, the region does not appear to extend below (horizontal dashed line), below which no discontinuous transition can exist, and we thus expect the system to behave more like its low-order variant. Finally, for larger , the region starts above values of that are larger as increases, suggesting that their ratio plays a role.

These synergistic signatures of group interactions can be useful for many downstream tasks. For example, we present in SM (Sec. VI) a simple method leveraging these synergistic signatures for detecting higher-order interactions, given the temporal sequence of the states of nodes.

In conclusion, by exploring the relation between mechanisms and behaviors in two systems with higher-order interactions, we uncovered emergent synergistic signatures characterizing group mechanisms. Quantifying higher-order behaviors using , we showed that in both models, an increase in the strength of the parameter controlling the group mechanisms in 2-simplices led (nonlinearly) to significantly larger synergistic values of . We have also shown that the synergistic behavioral response of the groups of nodes to the variation of the driving higher-order mechanisms is nonlinear and shows system-dependent characteristics. Crucially, low-order observables did not capture the behavioral signatures of groups, supporting the importance of higher-order observables to study group interdependencies. By exploring the control parameter spaces of the two systems, we showed that synergistic signatures are not ubiquitous and can be overshadowed by other emergent phenomena (e.g., the magnetization transition in the Ising model). A full characterization of the relation between these group signatures and other emergent phenomena in the systems, as well as the effect of variations in the structure of the system under study, is an interesting venue for future research. We expect our results to be relevant for any attempts at reconstructing [16,40,47,48] and predicting [7,49] complex interacting systems from signals, and for the ongoing discussion about the nature and importance of higher-order systems [50,51].