Abstract
Many realworld complex systems are characterized by interactions in groups that change in time. Current temporal network approaches, however, are unable to describe group dynamics, as they are based on pairwise interactions only. Here, we use timevarying hypergraphs to describe such systems, and we introduce a framework based on higherorder correlations to characterize their temporal organization. The analysis of human interaction data reveals the existence of coherent and interdependent mesoscopic structures, thus capturing aggregation, fragmentation and nucleation processes in social systems. We introduce a model of temporal hypergraphs with nonMarkovian group interactions, which reveals complex memory as a fundamental mechanism underlying the emerging pattern in the data.
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Introduction
Temporal networks, where links connecting pairs of nodes are not continuously active, provide a framework to model how the interactions of a complex system evolve in time^{1,2,3}. They have revealed key in understanding how the timevarying interaction network of realworld social and biological systems affects the properties of dynamical processes, such as epidemic spreading^{4,5,6,7}, diffusion^{8,9,10,11,12}, synchronization^{13,14}, and others^{15,16,17,18}. Recent results have highlighted the complex way in which the activity of each link depends on the activities of all other links, showing that memory^{19,20,21,22} in temporal networks is inherently a multidimensional concept with a well defined microscopic shape^{23}. Different approaches have aimed to describe the time evolution of a network as a trajectory in graph space, by naturally extending to the case of graphs notions such as correlations^{24} or even dynamical stability^{25} traditionally used for scalar or vectorial timeseries.
Temporal network approaches, however, have a strong limitation. They are based on a graph description and, as such, they can only describe how dyadic interactions (i.e., links) vary in time, neglecting manybody interactions. Indeed, many realworld social^{26,27,28,29,30}, biological^{31,32}, neural^{33,34} or ecological^{35,36} systems also exhibit higherorder interactions, i.e., interactions involving groups of three or more units at the same time. Such manybody interactions are better modeled by higherorder networks, such as hypergraphs and simplicial complexes, where hyperedges and simplices encode interactions among an arbitrary number of units^{37,38}. Interestingly, taking into account the higherorder architecture of realworld systems is known to produce novel collective phenomena in a variety of dynamical processes, including diffusion^{39,40}, synchronization^{41,42,43,44,45}, contagion^{46,47} and evolutionary games^{48,49}.
Some early works have already started to explore the temporal dimension of higherorder interactions. For instance, group interactions in realworld social systems have been found to occur in persistent bursts of activity^{28}, with events of different sizes close in time also spatially correlated in the network^{50}. Such persistent temporal higherorder interactions have been shown to anticipate the onset of endemic states in epidemic processes^{51}, and to affect the convergence time of nonlinear consensus dynamics^{52}. Theoretical frameworks for modeling temporal group activation data^{53,54}, and for constructing simplicial complexes based on topological data analysis of multivariate timeseries from brain functional activity, financial markets and disease spreading^{55} have also been recently developed. However, how to analyze and characterize the temporal organization of realworld complex systems with higherorder interactions is, to this day, an open problem.
In this article, we bridge this gap by introducing a general framework to study higherorder temporal dependencies in complex systems. We represent a complex system with interactions in groups whose size and composition can change in time as a temporal hypergraph, i.e., a hypergraph with timevarying hyperedges of different orders. We then define a set of measures to extract higherorder temporal correlations, namely to characterize how the dynamics of hyperedges of different orders are correlated. We test our framework on a variety of empirical social systems, where patterns are amenable to intuitive interpretations and validation. Results show the existence of longrange correlations at different group sizes and their hierarchical organization. Furthermore, we uncover the presence of temporal correlations between groups of different sizes, i.e., between hyperedges of different orders, unveiling the existence of persistent dynamical relationships between coherent mesoscopic structures previously unaccounted for. Finally, to gain intuition about the underlying microscopic mechanisms, we introduce novel theoretical models of temporal hypergraphs with higherorder memory, able to explain the observed empirical patterns. Beyond networked systems, our measures and models open the door to investigate interactions among emergent coherent structures and other multiscale phenomena in complex systems.
Results
Temporal correlations in hypergraphs
To represent the temporal evolution of systems with higherorder interactions we rely on temporal hypergraphs^{28,56}. A temporal hypergraph is a tuple \(({{{{{{{\mathcal{V}}}}}}}},{\{{{{{{{{\mathcal{H}}}}}}}}(t)\}}_{t=1}^{T})\), where \({{{{{{{\mathcal{V}}}}}}}}\) is a set of N nodes, and \({\{{{{{{{{\mathcal{H}}}}}}}}(t)\}}_{t=1}^{T}\) is a sequence of T sets. Each \({{{{{{{\mathcal{H}}}}}}}}(t)\) is a set of M(t) hyperedges, representing the interactions among the system units at time t. Each hyperedge represents an interaction among multiple units. A hyperedge of order 2, or 2hyperedge, is a set of two nodes representing a twobody interaction, a 3hyperedge is a set of three nodes representing a group interaction among three units, and so on, up to order D. To study the temporal organization of systems with higherorder interactions, we represent the temporal hypergraphs as a set of D − 1 sequences \({\{{{{{{{{{\bf{A}}}}}}}}}^{(d)}(t)\}}_{t=1}^{T}=\{{{{{{{{{\bf{A}}}}}}}}}^{(d)}(1) ,{{{{{{{{\bf{A}}}}}}}}}^{(d)}(2),\ldots ,{{{{{{{{\bf{A}}}}}}}}}^{(d)}(T)\}\) where the element \({a}_{ij}^{(d)}(t)\) of matrix A^{(d)}(t) counts the number of dhyperedges nodes i and j belong to at time t, while \({a}_{ii}^{(d)}(t)=0\,\forall i\). See Methods for more details on how to represent temporal hypergraphs.
The presence of higherorder interactions makes the analysis of temporal correlations a multifaceted problem. First, to quantify temporal correlations in interactions of a given order d, we introduce the intraorder correlation matrix
with d ∈ {2, …, D}. Here, τ is the temporal lag, A^{⊤} denotes the transpose of A, and we have defined the annealed adjacency matrix of order d as \({\mu }^{(d)}=\frac{1}{T}{\sum }_{t=1}^{T}{{{{{{{{\bf{A}}}}}}}}}^{(d)}(t)\). Note that, for d = 2, Eq. (1) recovers the correlation matrix for temporal networks^{24}. The diagonal terms of \({{{{{{{{\mathcal{C}}}}}}}}}^{(d)}(\tau )\) capture how hyperedges of order d are temporally autocorrelated, whereas the offdiagonal terms quantify crosscorrelations. When the latter are negligible, one can focus on the diagonal terms and define an intraorder correlation function
that provides a scalar measure of how hyperedges of order d are autocorrelated at lag τ.
Second, we can inquire whether interactions of two different orders d_{1} and d_{2} display temporal interdependence, i.e., whether mesoscopic structures are dynamically interrelated or, conversely, evolve independently. To this aim, we introduce the crossorder correlation matrix
where d_{1}, d_{2} ∈ {2, …, D}. Note that, when d_{1} = d_{2} = d, we recover the intraorder correlation matrix \({{{{{{{{\mathcal{C}}}}}}}}}^{(d)}\). We then define a scalar crossorder correlation function as
All the information about intraorder and crossorder correlations can be encoded in a (D − 1) × (D − 1) normalized interaction matrix \({{{{{{{{\mathcal{K}}}}}}}}}_{{d}_{1}{d}_{2}}(\tau )={c}^{({d}_{1},{d}_{2})}(\tau )/2\sqrt{{\sigma }^{({d}_{1})}{\sigma }^{({d}_{2})}}\), where σ^{(d)} = c^{(d)}(0), whose entry \({{{{{{{{\mathcal{K}}}}}}}}}_{{d}_{1}{d}_{2}}(\tau )\) describes how interactions of order d_{1} at a given time are correlated with those of order d_{2} occurring τ time steps later. Notice that matrix \({{{{{{{{\mathcal{K}}}}}}}}}_{{d}_{1}{d}_{2}}(\tau )\) is not symmetric, as the quantity \({c}^{({d}_{2},{d}_{1})}(\tau )\) measures how order d_{1} is correlated with order d_{2} at τ time steps before, and is in general different from \({c}^{({d}_{1},{d}_{2})}(\tau )\). The presence of a significant discrepancy between these two quantities captures asymmetries in the temporal dependencies between different orders of interaction. We quantify such an asymmetry in terms of a crossorder gap function
A positive value of \({\delta }^{({d}_{1},{d}_{2})}(\tau )\) indicates that the presence of groups of size d_{1} correlates with the presence of groups of size d_{2} after a time lag τ, more than the other way around.
Analysis of human interaction data
To explore intraorder and crossorder correlations in complex systems, we consider different social systems, for which we have highresolution empirical data about their temporal evolution. We first focus on a dataset describing facetoface interactions over a period of 32h among the N = 403 participants of a scientific conference^{57,58} (three further cases, namely the social interactions occurring in an office^{59}, in a hospital ward^{60} and in a university campus^{61} are described in the Supplementary information). We encode the finegrained temporal information of the dataset in a temporal hypergraph \(({{{{{{{\mathcal{V}}}}}}}},{\{{{{{{{{\mathcal{H}}}}}}}}(t)\}}_{t=1}^{T})\), with \( {{{{{{{\mathcal{V}}}}}}}}=403\). Each set \({{{{{{{\mathcal{H}}}}}}}}(t)\) is constructed by assuming that d individuals in contact at a given time t interact together in a group of size d, thus corresponding to a hyperedge of order d at time t. See Methods for details on how to reconstruct higherorder interactions from empirical data.
We begin by studying how and if groups of a given size are temporally correlated, i.e., if mesoscopic persistent structures emerge. Figure 1 reports the intraorder correlation functions c^{(d)}(τ) for orders d ∈ {2, …, 5} (circles). Significant longrange temporal autocorrelations are found for different orders of interaction, as indicated by the slow decays of c^{(d)}(τ) with τ in a double logarithmic scale, up to a threshold, which typically decreases with d. This indicates that groups, i.e., coherent structures, of larger sizes generally remain autocorrelated for shorter times. Interestingly, we also observe a saturation effect for interactions in groups of size three, with a series of peaks revealing a weak periodicity at large timescales (see Supplementary information for further details). Empirical results are also compared with a null model (squares) obtained by reshuffling the sequence defining the temporal hypergraph.
We then investigate whether interactions in groups of a given size d_{1} can also be correlated to interactions in groups of size d_{2} ≠ d_{1}, i.e., whether mesoscopic structures are related to each other. The crossorder correlation functions c^{(4, 5)}(τ) (cyan circles) and c^{(5, 4)}(τ) (olive circles) for groups of sizes four and five, are reported in Fig. 2a (see Supplementary information for an analysis of other group sizes). For clarity of presentation, we display a binned average of the crosscorrelations functions, as well as the corresponding standard deviation. Remarkably, both crossorder correlation functions show precise patterns that can not be reproduced by the corresponding null model (squares), indicating nontrivial relationships between mesoscopic coherent structures. Namely, groups of sizes four and five in this social system show a nontrivial and persistent dependence.
Figure 2b shows the normalized interaction matrix \({{{{{{{\mathcal{K}}}}}}}}(\tau )\) at time lag τ = 600s (see Supplementary information for an analysis of different τ). We observe a banded structure around the main diagonal, meaning that crossorder correlations are higher between groups of similar sizes. This indicates that, in the interactions at a scientific conference analyzed here, groups change gradually, with the loss or the addition of one or few members (see Supplementary information for the interaction matrix of different social systems, including the social contacts in a university campus where large groups reveal a more complex correlation pattern). Finally, Fig. 2c shows the crossorder gap function δ^{(4, 5)}(τ) (purple circles). Positive values of δ^{(4, 5)}(τ) in almost the entire range of the time lag τ considered indicate that groups of size four at a given time are correlated to those of size five occurring τ time steps later, more than the other way around. This result, which again cannot be reproduced by the null model (green squares), suggests that the formation of a group of five individuals from a group of four is more likely than the loss of one member in groups of five individuals, indicating a preferred temporal direction in the dynamics of group nucleation/fragmentation of this social system (see Supplementary information for an analysis of \({\delta }^{({d}_{1},{d}_{2})}(\tau )\) for other group sizes).
The temporal patterns revealed in realworld systems by our framework can be related to other properties of such systems. For instance, the existence of crossorder temporal correlations might explain the presence of overlapping structures, namely the tendency of different hyperedges to share nodes or to be included one within another, observed in temporallyaggregated hypergraphs^{26,29,62,63}.
Models of hypergraphs with higherorder memory
To investigate the mechanisms shaping intraorder and crossorder correlation profiles, we introduce two models to generate synthetic temporal hypergraphs with higherorder memory, inspired by DAR processes^{21,23,64}. The first model, named Discrete Auto Regressive Hypergraph (DARH) model, treats the binary states of each hyperedge h^{α} ∈ {0, 1} (absent/present) as independent stochastic processes. Each hyperedge updates its state either drawing a state from its past, or randomly sampling a new state. With probability q^{(d)}, where d ∈ {2, …, D}, a hyperedge of order d samples its state uniformly at random from its \({m}_{s}^{(d)}\) previous states, while with probability 1 − q^{(d)}, the hyperedge state is drawn randomly following a Bernoulli process with probability y^{(d)}. In this way, the tuning parameter q^{(d)} controls the memory strength^{21} of the hyperedges of order d. See Methods for a detailed description of the DARH model. Our first model displays intraorder correlations (i.e., the existence of mesoscopic persistent structures), but no crossorder correlations (i.e., these coherent structures do not interact). See Supplementary information for a characterization of the DARH model.
We then introduce a second model, the crossmemory DARH (cDARH) model, a variation of the DARH model where a hyperedge of order d can update its state by drawing not only from its \({m}_{s}^{(d)}\) previous states but also from the \({m}_{c}^{({d}^{{\prime} },d)}\) previous states of a hyperedge of a different order \({d}^{{\prime} }\). This updating mechanism, to which we will refer as crossorder memory, is what ultimately allows the model to account for interactions among coherent structures. The parameter \({m}_{s}^{(d)}\) represents the intraorder memory length of the hyperedges of order d, while \({m}_{c}^{({d}^{{\prime} },d)}\) is the crossorder memory length. When copying from memory, each hyperedge draws from the past of other hyperedges with probability p^{(d)}, and from its own past with probability 1 − p^{(d)}. We assume that hyperedges can copy from the memory of overlapping hyperedges only. This choice is motivated by previous studies on higherorder interactions in social networks pointing out a tendency of groups to progressively add or remove members, one step at a time^{28}. For illustration, let us consider the case of groups of size two and three: a 2hyperedge {i, j} selects one of the (N − 2) possible 3hyperedges containing nodes i and j and draws from its previous \({m}_{c}^{(3,2)}\) states. Similarly, a 3hyperedge {i, j, k} selects one of the three 2hyperedges that can be formed from it, i.e., {i, j}, {j, k}, and {i, k}, and copies a state from its previous \({m}_{c}^{(2,3)}\) steps (see Fig. 3a–b for a schematic illustration of the model). Such a mechanism can be straightforwardly extended to hyperedges of other orders. See Methods for more details about the cDARH model.
We generate with the cDARH model temporal hypergraphs with N = 10 nodes, maximum hyperedge order D = 3 and a temporal range of T = 3 ⋅ 10^{4} time steps. Note that realworld systems are usually characterized by a larger number of units. However, generating a temporal hypergraph with a realistic size using the cDARH model can be computationally costly. Yet our model is able to describe the patterns observed in the data even considering a few nodes. See Supplementary information for an analysis of larger hypergraphs. We set p^{(2)} = 0 and p^{(3)} = 0.6, meaning that hyperedges of order three can copy from the past of hyperedges of order two, while hyperedges of order two evolve independently. We also set \({m}_{c}^{(2,3)}=60\), for the crossorder memory length of 3hyperedges. Considering a single couple of intraorder memory lengths \({m}_{s}^{(2)},{m}_{s}^{(3)}\) for all hyperedges of order two and three, respectively, the cDARH model displays both intraorder and crossorder correlations. Yet the profiles of c^{(d)}(τ) do not match exactly those of empirical data. In particular, the functions c^{(2)} and c^{(3)} remain constant for \(\tau \le {m}_{s}^{(d)}\) and decay exponentially after that value, with the same rate of decay (see Supplementary information for a deeper analysis), while in empirical systems the correlation functions follow a powerlaw decay. This is not surprising, as realworld social interactions can be shaped by different scales of memory^{23}. Hence, we sample the intraorder memory length of a hyperedge of order d from a uniform distribution, with maximum values for the interactions of orders two and three set to \({m}_{s,\max }^{(2)}=40\), and \({m}_{s,\max }^{(3)}=10\), respectively. In Fig. 3c we observe that the profile of the intraorder correlation functions c^{(d)} shows a slow decay followed by a loss of correlation, which is in good agreement with what we see in the empirical networks. Such a minimal model reveals that memory can be the driving mechanism for the emergence of intraorder temporal correlations, with different orders possessing different degrees of memory, also explaining the hierarchical structure of correlation observed in the data. Figure 3d shows that δ^{(2, 3)}(τ) > 0 for different values of τ (purple circles), meaning that hyperedges of order two are correlated to hyperedges of order three occurring later in time more than the other way around. We observe a striking similarity between this trend and that observed in Fig. 2c for the empirical data. This result indicates that crossorder memory is a fundamental factor for the emergence of crossorder correlations among different orders of interaction as well as crossorder gaps in realworld social systems. The two peaks for δ^{(4, 5)}(τ) observed in the empirical system suggest again a more complex dependence on memory, possibly due to multiple temporal scales.
Discussion
In this article, we have introduced a framework to characterize different dimensions of memory in networked systems with higherorder interactions. We have shown that realworld social systems display longrange temporal correlations at different group sizes –i.e., that coherent mesoscopic structures emerge–, organized in a hierarchy across multiple scales. Moreover, we have found that group interactions are characterized by nontrivial crossorder correlations, with crossmemory being a fundamental mechanism underlying such a complex behavior. In the context of social systems, such crossorder interactions can be interpreted in terms of the schisming phenomenon^{65,66,67,68}, where groups in human interactions, e.g. conversations, fluctuate, nucleate, and display complex dynamics.
In conclusion, our work sheds light on the multifaceted nature of memory that emerges at different scales in realworld interacting systems. The analyses presented here can be naturally extended to other higherorder complex systems traditionally modeled in terms of networks of interactions, such as the human brain and biological ecosystems. Beyond the scope of network science, we hope that our framework can open new avenues to reveal the higherorder dynamics of coherent structures in a variety of physical systems, from multifragmentation in nuclear physics to vortexvortex interaction in the atmosphere or other fluid dynamical systems.
Methods
Representation of timevarying systems with higherorder interactions
Systems with higherorder interactions can be represented as hypergraphs^{56}. A hypergraph is a tuple \(({{{{{{{\mathcal{V}}}}}}}},{{{{{{{\mathcal{H}}}}}}}})\), where \({{{{{{{\mathcal{V}}}}}}}}\) is a set of N nodes, and \({{{{{{{\mathcal{H}}}}}}}}\) is a set of M hyperedges. Each hyperedge represents an interaction among two or more units. A hyperedge of order 2, or 2hyperedge, is a set of two nodes representing a twobody interaction, a 3hyperedge is a set of three nodes representing an interaction among three units, and so on, up to order D. While hypergraphs are usually represented by adjacency tensors of different ranks, to capture dynamical dependencies within and among orders we will rely instead on a set of adjacency matrices of the same rank. First, we consider a set of incidence matrices [E^{(2)}, E^{(3)}, …, E^{(D)}] where the element \({e}_{i\alpha }^{(d)}\) of matrix E^{(d)} is one if node i belongs to the dhyperedge α, while it is zero otherwise. For each order of interaction d, we can then construct an adjacency matrix A^{(d)} as
The offdiagonal elements \({a}_{ij}^{(d)}={\sum }_{\alpha }{e}_{i\alpha }{e}_{j\alpha }\) represents the number of dhyperedges nodes i and j belong to, while \({a}_{ii}^{(d)}=0\,\forall i\).
To represent a system with higherorder interactions evolving in time we rely on temporal hypergraphs^{28}. A temporal hypergraph is a tuple \(({{{{{{{\mathcal{V}}}}}}}},{\{{{{{{{{\mathcal{H}}}}}}}}(t)\}}_{t=1}^{T})\), where \({{{{{{{\mathcal{V}}}}}}}}\) is again a set of N nodes, and \({\{{{{{{{{\mathcal{H}}}}}}}}(t)\}}_{t=1}^{T}\) is a sequence of T sets. Each \({{{{{{{\mathcal{H}}}}}}}}(t)\) is a set of M(t) hyperedges, representing the interactions occurring at time t. For each order of interaction d, we can define a sequence \({\{{{{{{{{{\bf{A}}}}}}}}}^{(d)}(t)\}}_{t=1}^{T}=\{{{{{{{{{\bf{A}}}}}}}}}^{(d)}(1),{{{{{{{{\bf{A}}}}}}}}}^{(d)}(2),\ldots,{{{{{{{{\bf{A}}}}}}}}}^{(d)}(T)\}\), where A^{(d)}(t) is an adjacency matrix encoding the interactions of order d occurring at time t. Hence, we can fully represent the temporal evolution of the system using a set of D − 1 sequences \([{\{{{{{{{{{\bf{A}}}}}}}}}^{(2)}(t)\}}_{t=1}^{T},{\{{{{{{{{{\bf{A}}}}}}}}}^{(3)}(t)\}}_{t=1}^{T},\ldots,{\{{{{{{{{{\bf{A}}}}}}}}}^{(D)}(t)\}}_{t=1}^{T}]\).
Reconstruction of higherorder social interactions from empirical data
To investigate the temporal organization of social interactions, we rely on four datasets, three coming from the SocioPatterns project^{57,58,59,60} and one from the Copenhagen Network Study^{61}. These datasets store the interactions among the individuals as a temporal network, namely they contain dyadic interactions only. However, as people often engage in groups where more than two individuals interact at the same time, a network description of the system might result in an inadequate representation of the system. Still, the finegrained temporal information of the datasets allows us to extract group interactions from the data. In particular, we assume that d individuals that are in contact through dyadic interactions at a given time t interact together in a group of size d. For instance, if at time t individual i is in contact with individuals j and k, while individual j is also interacting with individual k, we assume individuals i, j and k to be engaged in a group interaction. Mathematically, if at time t a set of d nodes form a clique in the temporal network, we promote the clique to a dhyperedge in the temporal hypergraph.
The DARH and the cDARH models
To investigate the mechanisms shaping the onset of intraorder and crossorder correlations in temporal hypergraphs, we introduce two theoretical models that generate temporal hypergraphs with higherorder memory. The first model, called the Discrete Auto Regressive Hypergraph (DARH) model, treats the binary states of each hyperedge h^{α} ∈ {0, 1} (absent/present) as independent stochastic processes. The state of each hyperedge is updated either randomly or by drawing one of the previous states of the hyperedge. In particular, with a probability 1 − q^{(d)}, the hyperedge state is drawn randomly according to a Bernoulli process, i.e., the hyperedge is present with a probability y^{(d)}, or it is absent with a probability 1 − y^{(d)} . Note that d ∈ {2, …, D} represents the order of the hyperedge, meaning that for each order of interaction separately we can tune the sampling probabilities q^{(d)} and y^{(d)}. With probability q^{(d)}, instead, the next state is sampled uniformly at random from the \({m}_{s}^{(d)}\) previous states of the hyperedge. Formally, the dynamics of the state of a hyperedge α of order d, h^{α}, is given by
where Q_{t} ~ Bernoulli(q^{(d)}) is a random variable selecting how the state of the hyperedge is updated, Y_{t} ~ Bernoulli(y^{(d)}) defines whether the hyperedge is present/absent when its state is selected randomly, while \({\mu }_{t} \sim {{{{{{{\rm{Uniform}}}}}}}}(1,{m}_{s}^{(d)})\) determines which state is drawn when the update is done by sampling from the hyperedge past.
The second model, named the crossmemory DARH (cDARH) model, is a variation of the DARH model where a hyperedge of order d can update its state by drawing not only from its past but also from that of a hyperedge of a different order. Similar to the DARH model, with a probability 1 − q^{(d)}, the hyperedge state is drawn randomly according to a Bernoulli process with probability y^{(d)}, while the state is copied from past states with a probability q^{(d)}. When copying from memory, with probability 1 − p^{(d)} the state of the hyperedge is sampled uniformly at random from its \({m}_{s}^{(d)}\) previous states. With probability p^{(d)}, instead, the state of the hyperedge is drawn from the \({m}_{c}^{({d}^{{\prime} },d)}\) previous states of an overlapping hyperedge of order \({d}^{{\prime} }\) that overlaps. The order \({d}^{{\prime} }\) of the hyperedge can be drawn according to a given probability distribution \({\rho }^{(d)}({d}^{{\prime} })\). Formally, we can write the dynamics of the state of a hyperedge α of order d, h^{α}, as
As for the DARH model, Q_{t} ~ Bernoulli(q^{(d)}) is a random variable selecting how to update the state of the hyperedge, while Y_{t} ~ Bernoulli(y^{(d)}) determines if the hyperedge is present/absent when sampling randomly. ε_{t}(α) is a random variable that defines if the update of the hyperedge is done by sampling from its own past or from that of another hyperedge. Mathematically, it follows the equation
where P_{t} ~ Bernoulli(p^{(d)}) selects whether to copy from the past of hyperedge α or from that of another hyperedge, indexed as β. β is sampled from the set of hyperedges of order \({d}^{{\prime} }\) overlapping with α, with \({d}^{{\prime} }\) drawn from \({\rho }^{(d)}({d}^{{\prime} })\). Finally, the variable μ, determining which state from the past is sampled when copying from memory, is sampled according to the value of ε_{t}(α). In particular, when ε_{t}(α) = α, i.e., for the intraorder memory process, we have \({\mu }_{t} \sim {{{{{{{\rm{Uniform}}}}}}}}(1,{m}_{s}^{(d)})\), while we have \({\mu }_{t} \sim {{{{{{{\rm{Uniform}}}}}}}}(1,{m}_{c}^{({d}^{{\prime} },d)})\) when ε_{t}(α) ≠ α, i.e., for the crossorder memory process.
Reporting summary
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
Data availability
The SocioPatterns data on the contacts in the scientific conference, the office and the hospital ward are available at https://www.sociopatterns.org/datasets. The Copenhagen Network Study data on the contacts in the university campus are available at https://doi.org/10.6084/m9.figshare.11283407.
Code availability
The measures described here are implemented as part of the HGX library^{69} and are available at https://github.com/HGXTeam/hypergraphx.
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Acknowledgements
L.G. acknowledges C. Zappalà for the insightful discussions and comments. L.G. and F.B. acknowledge support from the Air Force Office of Scientific Research under award number FA86552217025. L.L. acknowledges funding from the Spanish Research Agency MCIN/AEI/10.13039/501100011033 via projects DYNDEEP (EUR2021122007), MISLAND (PID2020114324GBC22), and the María de Maeztu project CEX2021001164M. V.L. acknowledges support from the European Union, NextGenerationEU, GRINS project (GRINS PE00000018  CUP E63C22002120006).
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L.G. conceptualized the work, developed the methodology, carried out the analysis, curated the data, the code, and the visualization. L.L. provided methodological insights and carried out the formal analysis. V.L. provided methodological insights. F.B. conceptualized and supervised the work. All authors wrote, reviewed and edited the paper.
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Gallo, L., Lacasa, L., Latora, V. et al. Higherorder correlations reveal complex memory in temporal hypergraphs. Nat Commun 15, 4754 (2024). https://doi.org/10.1038/s41467024485786
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DOI: https://doi.org/10.1038/s41467024485786
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