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", which governs how it links to other nodes over time.

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

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

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

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:

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

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

Extensions

A variety of extensions to the activity-driven model have been studied. One example is activity-driven networks with attractiveness, 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, in which activity-levels change according to a self-excitation mechanism.