Burstiness and Aging in Social Temporal Networks

The presence of burstiness in temporal social networks, revealed by a power-law form of the waiting time distribution of consecutive interactions, is expected to produce aging effects in the corresponding time-integrated network. Here, we propose an analytically tractable model, in which interactions among the agents are ruled by a renewal process, that is able to reproduce this aging behavior. We develop an analytic solution for the topological properties of the integrated network produced by the model, finding that the time translation invariance of the degree distribution is broken. We validate our predictions against numerical simulations, and we check for the presence of aging effects in a empirical temporal network, ruled by bursty social interactions.

Figure 1

Top: degree distribution $P_t(k)$ at time $t={10}^4$ as a function of the rescaled degree $k-⟨r{⟩}_t$ from numerical simulation of the NoPAD model with a network of size $N={10}^6$, $c_0=1$, and different values of $\alpha$ and $\beta$. The exponent $\gamma$, as given by Eq. (8), is plotted as a dashed line. Bottom: rescaled degree distribution $P_t(k)/(c_{0}t{)}^{\beta }$ for different times $t$, in network size $N={10}^6$, and $c_0=1$, $\alpha =0.6$, and $\beta =1.2$. The theoretical decay exponent $\gamma =3$ is plotted as a dashed line.

Figure 2

Rescaled average degree $⟨k{⟩}_{t_a,t}/t^{\alpha }$ as a function of the aging time $t_a$, obtained from numerical simulations of the NoPAD model with a network of size $N={10}^6$, and parameters $\alpha =0.6$, $c_0=1$, $\beta =1.2$, integrated for different time intervals $t$. The behavior predicted for $t\ll t_a$ is shown as a dashed line. Inset: degree distribution $P_{t_a,t}(k)$ of the NoPAD networks of the main plot, for integration time $t=100$ and different aging time $t_a$. The behavior predicted by Eq. (8) is shown as a dashed line.

Figure 3

Waiting time distribution $\psi (\tau )$ for two consecutive publications in PRL by the same author, in unit of weeks. A decay of the form given by Eq. (6), with $\alpha =0.3$, is plotted as a dashed line. Inset: rescaled average degree $⟨k{⟩}_{t_a,t}/t^{\alpha }$ as a function of the aging time $t_a$ of the PRL temporal network, integrated for different time intervals $t=1$, 2, or 3 yr. The behavior predicted by the NoPAD model for $t\ll t_a$, $⟨k{⟩}_{t,t_a}\sim t_a^{\alpha -1}$, is shown as a dashed line.