Activity-driven model

In network science, the activity-driven model is a temporal network model in which each node has a randomly-assigned "activity potential",^[1] which governs how it links to other nodes over time.

Each node ${\displaystyle j}$ (out of ${\displaystyle N}$ total) has its activity potential ${\displaystyle x_{i}}$ drawn from a given distribution ${\displaystyle F(x)}$. A sequence of timesteps unfolds, and in each timestep each node ${\displaystyle j}$ forms ties to ${\displaystyle m}$ random other nodes at rate ${\displaystyle a_{i}=\eta x_{i}}$ (more precisely, it does so with probability ${\displaystyle a_{i}\,\Delta t}$ per timestep). All links are then deleted after each timestep.

Properties of time-aggregated network snapshots are able to be studied in terms of ${\displaystyle F(x)}$. For example, since each node ${\displaystyle j}$ after ${\displaystyle T}$ timesteps will have on average ${\displaystyle m\eta x_{i}T}$ outgoing links, the degree distribution after ${\displaystyle T}$ timesteps in the time-aggregated network will be related to the activity-potential distribution by

${\displaystyle P_{T}(k)\propto F\left({\frac {k}{m\eta T}}\right).}$

Spreading behavior according to the SIS epidemic model was investigated on activity-driven networks, and the following condition was derived for large-scale outbreaks to be possible:

${\displaystyle {\frac {\beta }{\lambda }}>{\frac {2\langle a\rangle }{\langle a\rangle +{\sqrt {\langle a^{2}\rangle }}}},}$

where ${\displaystyle \beta }$ is the per-contact transmission probability, ${\displaystyle \lambda }$ is the per-timestep recovery probability, and (${\displaystyle \langle a\rangle }$, ${\displaystyle \langle a^{2}\rangle }$) are the first and second moments of the random activity-rate ${\displaystyle a_{j}}$.

Extensions

A variety of extensions to the activity-driven model have been studied. One example is activity-driven networks with attractiveness,^[2] in which the links that a given node forms do not attach to other nodes at random, but rather with a probability proportional to a variable encoding nodewise attractiveness. Another example is activity-driven networks with memory,^[3] in which activity-levels change according to a self-excitation mechanism.

References

1. Perra, Nicola; B. Gonçalves; R. Pastor-Satorras; A. Vespignani (2012-06-25). "Activity driven modeling of time varying networks".
2. Pozzana, Iacopo; K. Sun; N. Perra (2017-10-26). "Epidemic spreading on activity-driven networks with attractiveness". Physical Review E. Vol. 96, no. 4. doi:10.1103/PhysRevE.96.042310.
3. Zino, Lorenzo; A. Rizzo; M. Porfiri (2018-12-11). "Modeling Memory Effects in Activity-Driven Networks". SIAM Journal on Applied Dynamical Systems. 17 (4): 2830–2854. doi:10.1137/18M1171485. S2CID 102354985.