Abstract
The recent availability of huge high resolution datasets on human activities has revealed the heavy-tailed nature of the interevent time distributions. In social simulations of interacting agents the standard approach has been to use Poisson processes to update the state of the agents, which gives rise to very homogeneous activity patterns with a well defined characteristic interevent time. As a paradigmatic opinion model we investigate the voter model and review the standard update rules and propose two new update rules which are able to account for heterogeneous activity patterns. For the new update rules each node gets updated with a probability that depends on the time since the last event of the node, where an event can be an update attempt (exogenous update) or a change of state (endogenous update). We find that both update rules can give rise to power law interevent time distributions, although the endogenous one more robustly. Apart from that for the exogenous update rule and the standard update rules the voter model does not reach consensus in the infinite size limit, while for the endogenous update there exist a coarsening process that drives the system toward consensus configurations.
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Notes
- 1.
S(t) is the probability of being in an active configuration at simulation time t. Then the probability of reaching an absorbing state at time t is\(\frac{d} {\mathit{dt}}(1 - S(t)) = -\frac{d} {\mathit{dt}}S(t)\). The average time to reach consensus is then\(\langle T\rangle = -\int _{0}^{\infty }(t \frac{d} {\mathit{dt}}S(t))\mathit{dt}\) and, integrating by parts one finds that ⟨T⟩ = ∫ 0 ∞ S(t).
- 2.
In the remainder we will refer to the complementary cumulative distribution just as cumulative distribution.
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Acknowledgements
We acknowledge financial support from the MINECO (Spain) and FEDER (EU) through projects FISICOS (FIS2007-60327) and MODASS (FIS2011-24785). JFG acknowledges support from the Government of the Balearic Islands through the Conselleria d’Educaci, Cultura i Universitats and the ESF.
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Fernández-Gracia, J., Eguíluz, V.M., Miguel, M.S. (2013). Timing Interactions in Social Simulations: The Voter Model. In: Holme, P., Saramäki, J. (eds) Temporal Networks. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36461-7_17
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