Emerging Activity Temporal Hypergraph: a model for generating realistic time-varying hypergraphs

TL;DR

The paper tackles the challenge of generating realistic synthetic time-varying hypergraphs to study dynamical processes on higher-order interactions. It introduces the Emerging Activity Temporal Hypergraph (EATH), a two-phase framework where independent node activity dynamics drive system-level group activation through memory-enabled edge formation, enabling surrogate hypergraphs that reproduce empirical temporal and topological properties. By extracting parameters from real face-to-face interaction datasets, EATH can produce surrogate hypergraphs and also hybrid/artificial variants, facilitating higher-order contagion analyses and scenario testing in data-sparse contexts. The work demonstrates accurate replication of node and hyperedge burstiness, weight and modularity distributions, and higher-order SIR dynamics, highlighting memory's crucial role in shaping group interactions and spreading processes.

Abstract

Time-varying group interactions constitute the building blocks of many complex systems. The framework of temporal hypergraphs makes it possible to represent them by taking into account the higher-order and temporal nature of the interactions. However, the corresponding datasets are often incomplete and/or limited in size and duration, and surrogate time-varying hypergraphs able to reproduce their statistical features constitute interesting substitutions, especially to understand how dynamical processes unfold on group interactions. Here, we present a new temporal hypergraph model, the Emerging Activity Temporal Hypergraph (EATH), which can be fed by parameters measured in a dataset and create synthetic datasets with similar properties. In the model, each node has an independent underlying activity dynamic and the overall system activity emerges from the nodes dynamics, with temporal group interactions resulting from both the activity of the nodes and memory mechanisms. We first show that the EATH model can generate surrogate hypergraphs of several empirical datasets of face-to-face interactions, mimicking temporal and topological properties at the node and hyperedge level. We also showcase the possibility to use the resulting synthetic data in simulations of higher-order contagion dynamics, comparing the outcome of such process on original and surrogate datasets. Finally, we illustrate the flexibility of the model, which can generate synthetic hypergraphs with tunable properties: as an example, we generate "hybrid" temporal hypergraphs, which mix properties of different empirical datasets. Our work opens several perspectives, from the generation of synthetic realistic hypergraphs describing contexts where data collection is difficult to a deeper understanding of dynamical processes on temporal hypergraphs.

Figures (10)

• Figure 1: Properties of empirical temporal hypergraphs. In panel a we report the evolution of the number of active hyperedges $E_t$ for the SFHH dataset, while in panel b we show the hyperedge size distribution $\Psi(m)$ for all the datasets. In panels c-h, the first and second rows show respectively the temporal properties of nodes and hyperedges in different datasets. c,f: distribution of event durations $P(T)$; d,g: distribution of inter-event times $P(\tau)$; e,h: distribution of the number of events in a train of events $P(E)$, where a train is defined with $\Delta = 15 \delta t$ for all datasets, except for LH10 and SFHH where we consider $\Delta = 60 \delta t$, with $\delta t$ the hypergraph resolution.
• Figure 2: Nodes and hyperedges temporal heterogeneity. For each node and hyperedge we compute the burstiness values $\Delta B_{\tau}$ and $\Delta B_{T}$ of their event and inter-event duration distributions. In each panel, each point corresponds to a node (a,c,e,g,i,k,m,o) or a hyperedge (b,d,f,h,j,l,n,p) and we show the scatterplot between $\Delta B_{\tau}$ and $\Delta B_{T}$. The gray dashed lines correspond to the reference $\Delta B=0$. Panels q, r show the distributions $P(\Delta B_{\tau})$ and $P(\Delta B_{T})$, respectively for nodes and hyperedges, for the SFHH dataset. We consider only the nodes and hyperedges with at least $10$ activation events.
• Figure 3: Participation of nodes at different interaction orders. For each dataset, at each order $m$ we rank the nodes based on the time they spend interacting at size $m$, and we show the Pearson's correlation coefficient $\rho(m,m')$ between the rankings obtained at order $m$ and $m'$ (panels a,c,e,g,i,k,m,o). We also compute the fraction $n_f^m(t)$ of nodes active at time $t$ (in hyperedges of any size) among the nodes occupying the top $f N$ positions of the nodes ranking at order $m$, and show the temporal evolution of $n_f^m(t)$ for different $m$ (see legend), fixing $f=0.1$ (panels b,d,f,h,j,l,n,p); we also plot the total fraction $N_t/N$ of active nodes in the population at time $t$ in each panel.
• Figure 4: Sketch of the Emerging Activity Temporal Hypergraph (EATH) model. The EATH model consists in: (i) generation of the activity dynamics of individual nodes, which transition between low- and high-activity states; (ii) the overall activity of the system emerges from the activity dynamics of individual nodes and from the size distribution $\Psi(m)$; (iii) the nodes that are part of active interactions are selected with memory and activity mechanisms, hence producing the (iv) groups dynamics. The panel System dynamics shows the evolution of the number of active hyperedges $E_t$, also divided in sizes for $m \in [2,3,4]$, and the hyperedge size distribution $\Psi(m)$ (inset) for the SFHH dataset.
• Figure 5: Generated system activity and topology.a: evolution of the number of active hyperedges $E_t$ and distributions of the hyperedges' sizes $\Psi(m)$ (inset), for the empirical dataset and for the EATH model. b: evolution of the number of active hyperedges of size $m \in [2,3,4]$ for the EATH model. c, d: respectively, distribution $P(D)$ of the total hyperdegree $D$ and distribution $P(S)$ of the total hyperstrength $S$, for the aggregated hypergraphs of the empirical dataset and of the surrogate hypergraphs obtained in the model with (EATH) and without memory (EATHw). e, f: same as c,d for the degree and the strength in the aggregated projected pairwise graph. In all panels we consider the SFHH dataset, with model parameters extracted from the empirical hypergraph as described in the main text. In panels c-f the dashed vertical lines indicate the average values of the corresponding distributions.