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.