The structural evolution of temporal hypergraphs through the lens of hyper-cores

TL;DR

A series of methods specifically designed to analyse the structural properties of temporal hypergraphs at multiple scales, distinguishing the higher-order structures and dynamics generated by different models from the empirical ones, and thus identifying the essential model mechanisms to reproduce the empirical hypergraph structure and evolution.

Abstract

The richness of many complex systems stems from the interactions among their components. The higher-order nature of these interactions, involving many units at once, and their temporal dynamics constitute crucial properties that shape the behaviour of the system itself. An adequate description of these systems is offered by temporal hypergraphs, that integrate these features within the same framework. However, tools for their temporal and topological characterization are still scarce. Here we develop a series of methods specifically designed to analyse the structural properties of temporal hypergraphs at multiple scales. Leveraging the hyper-core decomposition of hypergraphs, we follow the evolution of the hyper-cores through time, characterizing the hypergraph structure and its temporal dynamics at different topological scales, and quantifying the multi-scale structural stability of the system. We also define two static hypercoreness centrality measures that provide an overall description of the nodes aggregated structural behaviour. We apply the characterization methods to several data sets, establishing connections between structural properties and specific activities within the systems. Finally, we show how the proposed method can be used as a model-validation tool for synthetic temporal hypergraphs, distinguishing the higher-order structures and dynamics generated by different models from the empirical ones, and thus identifying the essential model mechanisms to reproduce the empirical hypergraph structure and evolution. Our work opens several research directions, from the understanding of dynamic processes on temporal higher-order networks to the design of new models of time-varying hypergraphs.

Figures (10)

• Figure 1: Evolution of the hyper-core structure in APS scientific collaborations.a: fraction of nodes $n_{(k,m)}$ in the $(k,m)$-core as a function of $k$ and $m$ for each 5-years time window. The numbers of active nodes $N_t$ and hyperedges $E_t$ are also reported and the insets show $n_{(k,m)}$ as a function of $k$ for $m=2$, $m=6$ and $m=10$. b: root-mean-square deviation similarity $\Sigma(t,t')$ between $n_{(k,m)}(t)$ and $n_{(k,m)}(t')$ (grey diagonal: $\Sigma(t,t)=1$). c: Jaccard similarity $J^*(t,t')$ between the sets of nodes belonging to the most central hyper-cores, i.e. to the $(k_{max}^m,m)$-cores $\forall m$, at time $t$ and $t'$ (grey diagonal: $J^*(t,t)=1$). d: Pearson correlation coefficient $\rho(t,t')$ between the nodes hypercoreness at times $t$ and $t'$, considering all the nodes that are active in at least one of the snapshots (grey diagonal: $\rho(t,t)=1$). e: similarity $\Sigma(t,t+1)$ vs. $t$. f: temporal evolution of $J^*(t,t+1)$ and Jaccard similarity $J_N(t,t+1)$ between the entire population in two consecutive time windows. g: temporal evolution of the correlation between the nodes hypercoreness in consecutive snapshots, considering all the nodes that are active in at least one of the snapshots, $\rho(t,t+1)$, or only those active in both, $\rho^*(t,t+1)$.
• Figure 2: Hypercoreness evolution for selected nodes in the APS scientific collaborations. We show the temporal evolution of the hypercoreness $r(i,t)$ for four authors and the mean $\langle r \rangle (t)$ value (average on active nodes): we show the authors I.Y. Lee ($\#_W1$) and R.V.F. Janssens ($\#_W2$), who occupy respectively the first and second position in the ranking produced by the aggregated hypercoreness $W$ over the period 1942-2021, and the authors Guang-Can Guo ($\#_{\overline{h}}1$) and Loren N. Pfeiffer ($\#_{\overline{h}}5$), who occupy respectively the first and fifth position in the ranking produced by the average number of interactions per active windows $\bar{h}$ over the period 1942-2021.
• Figure 3: Time-aggregated hypercoreness in APS scientific collaborations 1942-2021.a: scatter plot of the aggregated hypercoreness $W(i)$ as a function of the snapshot activity $a_w(i)$ for all nodes $i$, and average aggregated hypercoreness $\langle W \rangle$ as a function of $a_w$. b: aggregated hypercoreness $W(i)$ vs. average number of interactions per active window $\overline{h}(i)$ for all nodes $i$. c: aggregated hypercoreness $W(i)$ as a function of the activity-averaged hypercoreness $\overline{W}(i)$. In all panels the points are colored according to the activity $a_w$ of the corresponding node.
• Figure 4: Prevalent APS scientific communities in hyper-cores.a: temporal evolution over 5-years time windows of the prevalent journal within each $(k,m)$-hyper-core of the APS data set, defined as the most frequent hyperedge label in each core (we consider a journal dominant only if its frequency is larger than 0.5; white indicates hyper-cores which are empty or where a dominant journal cannot be defined). b: relative frequency $P$ of the various journals within the most central hyper-cores, i.e. $(k_{max}^m,m)$-cores $\forall m$, and its temporal evolution. c: same as b for the randomized data. We average the relative frequency over 50 randomized realizations of the hypergraph (see Methods). The error bars give the standard errors.
• Figure 5: Hyper-core structure evolution in daily interactions within a hospital (LH10). a: relative population $n_{(k,m)}$ of the $(k,m)$-core as a function of $k$ and $m$ for each time window. The number of active nodes $N_t$ and hyperedges $E_t$ is reported for each snapshot. b:$n_{(k,m)}$ as a function of $k$ for fixed values of $m$. c: root-mean-square deviation similarity $\Sigma(t,t')$ between $n_{(k,m)}(t)$ and $n_{(k,m)}(t')$ -- the grey diagonal corresponds to $\Sigma(t,t)=1$; d: Jaccard similarity $J^*(t,t')$ between the sets of nodes belonging to the most central hyper-cores, i.e. the $(k_{max}^m,m)$-cores $\forall m$, at time $t$ and $t'$ -- the grey diagonal corresponds to $J^*(t,t)=1$. e: Pearson correlation coefficient $\rho(t,t')$ between the nodes hypercoreness at time $t$ and $t'$, considering all the nodes that are active in at least one of the snapshots -- the grey diagonal corresponds to $\rho(t,t)=1$. f: similarity $\Sigma(t,t+1)$ as a function of $t$. g: temporal evolution of both the similarity $J^*(t,t+1)$ and the Jaccard similarity $J_N(t,t+1)$ between the entire population in consecutive time windows. h: temporal evolution of the correlation between the nodes hypercoreness in consecutive snapshots, considering all the nodes that are active in at least one of the snapshots, $\rho(t,t+1)$, or that are active in both, $\rho^*(t,t+1)$.