View source for Activity-driven model

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{{Network science}}

In [[Network Science]], the activity-driven model is a [[temporal network]] model in which each node has a randomly-assigned "activity potential" <ref name="perra">{{cite news
|title=Activity driven modeling of time varying networks
|last=Perra
|first=Nicola
|author2=B. Gonçalves
|author3=R. Pastor-Satorras
|author4=A. Vespignani 
|date=2012-06-25
|url = https://www.nature.com/articles/srep00469}}</ref>, which governs how it links to other nodes over time. 

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

Properties of time-aggregated network snapshots are able to be studied in terms of <math> F(x) </math>. For example, since each node <math>j</math> after <math>T</math> timesteps will have on average <math>m\eta x_i T</math> outgoing links, the [[degree distribution]] after <math>T</math> timesteps in the time-aggregated network will be related to the activity-potential distribution by

: <math> P_T(k) \propto F\left(\frac{k}{m\eta T}\right). </math>

Spreading behavior according to the [[Compartmental models in epidemiology#The SIS model|SIS epidemic model]] was investigated on activity-driven networks, and the following condition was derived for large-scale outbreaks to be possible:

: <math> \frac{\beta}{\lambda} > \frac{2\langle a\rangle}{\langle a\rangle + \sqrt{\langle a^2\rangle}}, </math>

where <math>\beta</math> is the per-contact transmission probability, <math>\lambda</math> is the per-timestep recovery probability, and (<math> \langle a\rangle </math>, <math> \langle a^2\rangle</math>) are the first and second moments of the random activity-rate <math> a_j</math>.


== Extensions ==

A variety of extensions to the activity-driven model have been studied. One example is activity-driven networks with attractiveness<ref name="pozzana">{{cite news
|title=Epidemic spreading on activity-driven networks with attractiveness
|last=Pozzana
|first=Iacopo
|author2=K. Sun
|author3=N. Perra
|date=2017-10-26
|url = https://journals.aps.org/pre/abstract/10.1103/PhysRevE.96.042310}}</ref>, 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<ref name="zino">{{cite news
|title=Modeling Memory Effects in Activity-Driven Networks
|last=Zino
|first=Lorenzo
|author2=A. Rizzo 
|author3=M. Porfiri
|date=2018-12-11
|url=https://epubs.siam.org/doi/abs/10.1137/18M1171485}}</ref>, in which activity-levels change according to a self-excitation mechanism.




== References ==
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[[Category:Random graphs]]

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