Percolation and Topological Properties of Temporal Higher-Order Networks

Many complex systems that exhibit temporal nonpairwise interactions can be represented by means of generative higher-order network models. Here, we propose a hidden variable formalism to analytically characterize a general class of higher-order network models. We apply our framework to a temporal higher-order activity-driven model, providing analytical expressions for the main topological properties of the time-integrated hypergraphs, depending on the integration time and the activity distributions characterizing the model. Furthermore, we provide analytical estimates for the percolation times of general classes of uncorrelated and correlated hypergraphs. Finally, we quantify the extent to which the percolation time of empirical social interactions is underestimated when their higher-order nature is neglected.

Figure 1

Topological properties of HOAD networks. (a) Hyperdegree distribution $P_T(k^{(m)})$; Eq. (8) shown as a dashed line. (b) Hyperdegree correlations ${\overline{k}}_{nn,T}^{(m)}(k)$; Eq. (9) shown as a dashed line. Network size $N={10}^6$, orders $m=1$, 2, 5, integration time $T={10}^3$. Different values of $T$ and $m$ are shown in the Supplemental Material [56]. The activity distributions $\rho (a)$ of order $m$ have a power-law form for every order with exponent $\gamma =2.25$.

Figure 2

Percolation time of HOAD networks. Orders $m=1$, 2, 5. (a) Giant component size $S/N$ (continuous line) and the peak of its variance $\sigma (S{)}^2$ (dashed line) over time. The theoretical prediction given by Eq. (10) is indicated as a vertical line. (b) Finite-size scaling analysis of the relative difference $[T^{(m)}(N)-T_c^{(m)}]/T_c^{(m)}$ (circles) and corresponding scaling law $N^{-\nu }$ (dashed line). Results are averaged over ${10}^2$ runs.

Figure 3

Percolation times in empirical data. Geology (stars) and history (circles) scientific collaboration networks. Blue points, ratios between the first-order ($T^{(1)}$) and $m$-order ($T^{(m)}$) percolation times of networks informed by empirical activities, estimated from numerical simulations; yellow points, ratios between the theoretical prediction from Eq. (10) $T_c^{(m)}$ and the percolation times of networks informed by empirical activities estimated from numerical simulations $T^{(m)}$.