Simplicial Activity Driven Model

Many complex systems find a convenient representation in terms of networks: structures made by pairwise interactions (links) of elements (nodes). For many biological and social systems, elementary interactions involve, however, more than two elements, and simplicial complexes are more adequate to describe such phenomena. Moreover, these interactions often change over time. Here, we propose a framework to model such an evolution: the simplicial activity driven model, in which the building block is a simplex of nodes representing a multiagent interaction. We show analytically and numerically that the use of simplicial structures leads to crucial structural differences with respect to the activity driven model, a paradigmatic temporal network model involving only binary interactions. It also impacts the outcome of paradigmatic processes modeling disease propagation or social contagion. In particular, fluctuations in the number of nodes involved in the interactions can affect the outcome of models of simple contagion processes, contrarily to what happens in the activity driven model.

Figure 1

SAD model. (a) At each time step, a node $i$ activates with probability $a_i\mathrm{\Delta }t$. Upon activation it creates a coherent unit of $s$ nodes [an ($s-1$)-simplex], with links between all pairs. (b) In contrast, in the standard AD model (nAD), only the $s-1$ edges stemming from the activated node are added. (c) In the eAD model, $(\genfrac{}{}{0ex}{}{s}{2})$ links stem from the activated node, conserving at each interaction the number of links of the SAD model.

Figure 2

Structural properties of SAD model. In all plots we use $N=2000$ nodes and activities sampled from $F(a)=(a/a_0{)}^{-\alpha }$, where $\alpha =2.1$ and $a_0=5\times {10}^{-3}$. (a) Average aggregated degree $⟨k_T⟩$ for the SAD model and corresponding nAD and eAD models versus simplex size $s$, with $T=10$. Symbols, numerical values; lines, theoretical predictions [for the SAD model, Eq. (3) averaged over all nodes]. (b) Temporal growth of the aggregated GCC size $S$ for the SAD (solid lines), nAD (crosses), and eAD (dashed lines) models for various fixed simplex size $s$. (c) Empty symbols are the average $⟨k_2(i,T)⟩$ over all nodes $i$ of the number of 2-simplices to which $i$ belongs, in the SAD model aggregated until time $T$, for various values of $s$ and of $T$. Continuous lines are the prediction [Eq. (4)] averaged over nodes. The filled symbols give instead the average number of triangles to which a node $i$ belongs in the 1-skeleton of the aggregated SAD model. (d) Eigenspectrum of the simplicial Laplacian ${\mathcal{L}}_1$ computed on the aggregated SAD 1-skeleton (here with $s=4$), on the actual aggregated SAD simplicial complex, and on the clique complex of the 1-skeleton of the aggregated SAD simplicial complex [in which each ($k+1$) clique of the 1-skeleton is promoted to a $k$-simplex]. Aggregation time $T=100$.

Figure 3

SIS epidemic threshold for AD and SAD models. (a) Epidemic prevalence versus $\lambda =\beta /\mu$: the epidemic transition in the SAD model is delayed as compared with an SIS model on the corresponding eAD model (here, $N=1000$, $s=4$, $T=20000$). Vertical lines correspond to the theoretical values of the epidemic thresholds [Eq. (6) for the SAD model]. (b) Increasing the average connectivity of the underlying network lowers the epidemic threshold in all models; for the $s$-regular SAD model ${\lambda }_c$ [Eq. (6)] is always larger than for the corresponding eAD model. In both panels, node activities were sampled from $F(a)=(a/a_0{)}^{-\alpha }$, where $\alpha =2.1$ and $a_0=5\times {10}^{-3}$.