Higher-order correlations reveal complex memory in temporal hypergraphs

The study addresses how to analyze and characterize the temporal organization of complex systems exhibiting higher-order (group) interactions that vary over time. Traditional temporal network models track dyadic (pairwise) links and have illuminated effects on dynamics such as spreading, diffusion, and synchronization, but they cannot capture many-body interactions that occur in groups of three or more units. Higher-order structures like hypergraphs and simplicial complexes are known to shape collective phenomena, yet their temporal dimension remains inadequately understood. The research question is to develop a framework that measures and interprets temporal correlations within and across interaction orders (group sizes), revealing memory effects and the dynamics linking mesoscopic structures. The purpose is to bridge the gap by proposing measures to quantify intra-order and cross-order dependencies in time-varying hypergraphs and to model mechanisms, particularly memory, that generate the observed patterns in empirical social interaction data.

Prior work on temporal networks has shown that time-varying interactions impact spreading, diffusion, and synchronization, and that memory in temporal networks is multidimensional with specific microscopic shapes. Extensions include treating network evolution as trajectories in graph space, using correlations and dynamical stability. However, graph-based approaches focus on dyads and neglect higher-order interactions which are prevalent in social, biological, neural, and ecological systems. Higher-order architectures produce novel phenomena in diffusion, synchronization, contagion, and evolutionary dynamics. Early explorations of temporal higher-order interactions found persistent bursts, spatial-temporal correlations, impacts on epidemic thresholds, and consensus convergence times. Modeling frameworks like simplicial activity-driven models and topological data analysis for multivariate time series have emerged. Despite these advances, systematic characterization of temporal organization with higher-order interactions remained open, motivating the present framework.

Systems with higher-order interactions are represented as temporal hypergraphs (V, {H(t)}{t=1}^T), with V the set of N nodes and each H(t) a set of hyperedges present at time t. Hyperedges of order d represent group interactions among d nodes. To analyze temporal organization, the authors construct, for each order d in {2, ..., D}, a sequence of adjacency matrices A^{(d)}(t) where off-diagonal entries a{ij}^{(d)}(t) count the number of d-hyperedges that nodes i and j jointly belong to at time t (diagonal zero). From these, they define the annealed adjacency matrix μ^{(d)} = (1/T) ∑{t=1}^T A^{(d)}(t). Intra-order temporal correlations are quantified via the correlation matrix C^{(d)}(τ) = (1/(T−τ)) ∑{t=1}^{T−τ} (1/(d−1)!^2) [A^{(d)}(t) − μ^{(d)}]^⊤ · [A^{(d)}(t+τ) − μ^{(d)}], and its scalar trace c^{(d)}(τ) = tr(C^{(d)}(τ)). Cross-order dependencies are measured by C^{(d1,d2)}(τ) analogously, yielding c^{(d1,d2)}(τ) = tr(C^{(d1,d2)}(τ)). A normalized interaction matrix K_{d1,d2}(τ) = c^{(d1,d2)}(τ)/√(σ^{(d1)} σ^{(d2)}), where σ^{(d)} = c^{(d)}(0), encodes how interactions of order d1 at time t relate to those of order d2 at t+τ. Asymmetry in temporal dependencies is captured by the cross-order gap δ^{(d1,d2)}(τ) = [c^{(d1,d2)}(τ) − c^{(d2,d1)}(τ)] / [2√(σ^{(d1)} σ^{(d2)})], indicating preferred directions in nucleation or fragmentation. Empirical reconstruction: Social interaction datasets (conference, office, hospital ward, university campus) originally store dyadic contacts with fine temporal resolution. The authors infer group interactions by promoting cliques in the temporal network at each time t to d-hyperedges, assuming that d individuals simultaneously connected pairwise form a group of size d. Modeling: Two synthetic models are introduced. The Discrete Auto Regressive Hypergraph (DARH) model treats each hyperedge’s binary state h ∈ {0,1} as an independent stochastic process that, with probability q^{(d)}, copies uniformly from its last m^{(d)} states (intra-order memory) and, with probability 1−q^{(d)}, draws a random presence/absence via Bernoulli(y_d). DARH generates intra-order correlations but no cross-order correlations. The cross-memory DARH (cDARH) model extends DARH: when copying from memory, a hyperedge of order d may, with probability p^{(d,d')}, copy from the past m^{(d,d')} states of an overlapping hyperedge of order d' (cross-order memory), otherwise from its own past. Overlapping hyperedges are defined by shared nodes (e.g., a {i,j} 2-hyperedge overlapping with {i,j,k} 3-hyperedges). Parameters control intra-order memory lengths m^{(d)}, cross-order memory lengths m^{(d,d')}, and copying probabilities p^{(d,d')}. The models are used to generate synthetic temporal hypergraphs and compute intra- and cross-order correlation profiles for comparison with empirical data.

Empirical analysis of face-to-face interactions at a scientific conference (N=403 participants, 32 hours) shows: 1) Significant long-range intra-order autocorrelations across group sizes d ∈ {2,3,4,5}, with c^{(d)}(τ) decaying slowly over τ (double-log scale), and correlation persistence thresholds decreasing with larger d (larger groups remain autocorrelated for shorter times). A saturation effect with weak periodic peaks appears for d=3 at large timescales. Randomly reshuffled null models remove these correlations. 2) Cross-order correlations are non-trivial and persistent. For example, c^{(4,5)}(τ) and c^{(5,4)}(τ) display structured, significant patterns not reproduced by null models, indicating dynamical interdependence between groups of sizes four and five. 3) The normalized interaction matrix K_{d1,d2}(τ) at τ = 600 s exhibits a banded structure near the diagonal, implying stronger correlations between similar group sizes and suggesting gradual group changes via addition or removal of few members. 4) The cross-order gap δ^{(4,5)}(τ) is positive across most τ, indicating that groups of size four tend to be followed by groups of size five more than the reverse, i.e., a preferred direction toward nucleation rather than fragmentation. The framework suggests that cross-order temporal correlations may underlie overlapping structures in aggregated hypergraphs. Modeling results: The cDARH model (e.g., N=10, D=3, T=3×10^4; with p^{(3)}=0.6 allowing 3-hyperedges to copy from 2-hyperedges, p^{(2)}=0; cross-order memory length m_c^{(2,3)}=60) reproduces intra-order and cross-order correlations, including positive cross-order gaps δ^{(2,3)}(τ). Using heterogeneous intra-order memory lengths sampled from uniform distributions (e.g., m_i^{(2,max)}=40, m_i^{(3,max)}=10) yields slow-decay profiles and loss of correlation consistent with empirical observations and explains the hierarchical correlations across orders. The empirical δ^{(4,5)}(τ) showing two peaks suggests multiple temporal memory scales.

The findings demonstrate that social systems with higher-order interactions possess long-range temporal memory manifested in coherent mesoscopic structures across group sizes, organized hierarchically. Crucially, interactions across different orders are temporally interdependent, revealing cross-order correlations and asymmetries that signify preferred directions of group evolution (e.g., nucleation from size four to five). The cDARH model identifies cross-order memory as a key mechanism behind these phenomena, linking microscopic memory processes to emergent mesoscopic dynamics. In social contexts, these cross-order dependencies connect to schisming dynamics in conversations, where groups fluctuate, nucleate, and split. The framework extends naturally to other complex systems (brain activity, ecosystems) where higher-order interactions are relevant, enabling the study of interactions among emergent structures and multi-scale phenomena.

The study introduces a general framework to quantify intra-order and cross-order temporal correlations in time-varying hypergraphs, revealing complex, multi-scale memory in real social systems and coherent mesoscopic structures. It shows that cross-order memory mechanisms can generate observed empirical patterns, including hierarchical organization and asymmetric cross-order gaps. This advances the analysis of temporal higher-order interactions and opens avenues for investigating coherent structures in diverse domains, such as neuroscience, ecology, and physical systems with multi-fragmentation or vortex interactions. Future work may refine models to incorporate multiple memory scales, improve computational scalability for larger realistic systems, and validate the framework across broader datasets and domains.

Synthetic generation with the cDARH model for realistic system sizes can be computationally costly, so demonstrations used small N (e.g., N=10), which may limit direct scalability. Initial model correlation profiles did not match empirical power-law-like decays until heterogeneous memory lengths were introduced, indicating simplified memory assumptions. The reconstruction of higher-order interactions from dyadic data relies on promoting cliques at each time step, which may introduce biases if true group interactions deviate from clique structures. Cross-order results were illustrated for specific orders and time lags (e.g., τ=600 s), and additional datasets and parameter ranges are discussed in Supplementary Information.