Topological properties of a time-integrated activity-driven network

Here we consider the topological properties of the integrated networks emerging from the activity-driven model [N. Perra et al., Sci. Rep. 2, 469 (2012)], a temporal network model recently proposed to explain the power-law degree distribution empirically observed in many real social networks. By means of a mapping to a hidden-variable network model, we provide analytical expressions for the main topological properties of the integrated network, depending on the integration time and the distribution of activity potential characterizing the model. The expressions obtained, exacts in some cases, the results of controlled asymptotic expansions in others, are confirmed by means of extensive numerical simulations. Our analytical approach, which highlights the differences of the model with respect to the empirical observations made in real social networks, can be easily extended to deal with improved, more realistic modifications of the activity-driven network paradigm.

Figure 1

Rescaled degree distribution $P_T\left(k\right)$ for integrated networks corresponding to different values of $T$, with power-law activity distribution with exponents $\gamma =3$ and $2.25$. Network size $N={10}^6$. The behavior predicted by Eq. (27) is represented as dashed lines.

Figure 2

Degree distribution $P_T\left(k\right)$ for integrated networks corresponding to different values of $T$, with power-law activity distribution with exponent $\gamma =2.25$. Network size $N={10}^3$. The result of a numerical integration of Eq. (22) is showed as continuous lines. The inset shows the rescaled $P_T\left(k\right)$ against Eq. (27) as a blue dashed line.

Figure 3

Rescaled average degree of the nearest neighbors of the vertices of degree $k$, ${\overline{k}}_T^{nn}\left(k\right)$, for the integrated network with size $N={10}^4$ and power-law activity distribution with $\gamma =2.5$, for different values of $T$. The prediction of Eq. (33) is shown as a dashed blue line. The inset shows the average degree of the neighbors of the vertices with activity $a$, ${\overline{k}}_T^{nn}\left(a\right)$, for the same integrated network. The predictions from Eq. (32) are shown as continuous lines.

Figure 4

Rescaled clustering coefficient of the nodes of degree $k$, ${\overline{c}}_T\left(k\right)$, of the integrated network with size $N={10}^4$ and power-law activity distribution with $\gamma =2.5$, for different values of $T$. The prediction of Eq. (39) is shown as a dashed blue line. The inset shows the clustering coefficient of the nodes with activity $a$, ${\overline{c}}_T\left(a\right)$, of the same integrated network. The predictions from Eq. (37) are shown as continuous lines.