Burstiness and fractional diffusion on complex networks

@article{deNigris2016BurstinessAF,
  title={Burstiness and fractional diffusion on complex networks},
  author={Sarah de Nigris and Anthony Hastir and Renaud Lambiotte},
  journal={The European Physical Journal B},
  year={2016},
  volume={89},
  pages={1-7}
}
Abstract Many dynamical processes on real world networks display complex temporal patterns as, for instance, a fat-tailed distribution of inter-events times, leading to heterogeneous waiting times between events. In this work, we focus on distributions whose average inter-event time diverges, and study its impact on the dynamics of random walkers on networks. The process can naturally be described, in the long time limit, in terms of Riemann-Liouville fractional derivatives. We show that all… 
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11 Citations

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References

SHOWING 1-10 OF 36 REFERENCES

Analytically Solvable Model of Spreading Dynamics with Non-Poissonian Processes

An analytically solvable model of susceptible-infected spreading dynamics in infinite systems for arbitrary inter-event time distributions and for the whole time range shows that for early and intermediate times, the burstiness accelerates the spreading as compared to a Poisson-like process with the same mean and same lower bound of inter- event times.

Diffusion on networked systems is a question of time or structure.

A generalized formalism for linear dynamics on complex networks, able to incorporate statistical properties of the timings at which events occur, is proposed, showing that the diffusion dynamics is affected by the network community structure and by the temporal properties of waiting times between events.

Spreading dynamics on networks: the role of burstiness, topology and non-stationarity

It is demonstrated that finite, so called ‘locally tree-like’ networks, like the Barabási–Albert networks behave very differently from real tree graphs if the IETD is strongly fat-tailed, as the lack or presence of rare alternative paths modifies the spreading.

Spreading dynamics following bursty human activity patterns.

The susceptible-infected model with power-law waiting time distributions P(τ)~τ^{-α), as a model of spreading dynamics under heterogeneous human activity patterns, is studied to unify individual activity patterns with macroscopic collective dynamics at the network level.

Fractional dynamics on networks: emergence of anomalous diffusion and Lévy flights.

The general fractional diffusion formalism applies to regular, random, and complex networks and can be implemented from the spectral properties of the Laplacian matrix, providing an important tool to analyze anomalous diffusion on networks.

Generalized Master Equations for Non-Poisson Dynamics on Networks

The effects of non-Poisson inter-event statistics on the dynamics of edges are examined, and the concept of a generalized master equation is applied to the study of continuous-time random walks on networks.

Random walks on temporal networks.

It is shown that the random walk exploration is slower on temporal networks than it is on the aggregate projected network, even when the time is properly rescaled, and a fundamental role is played by the temporal correlations between consecutive contacts present in the data.

Impact of non-Poissonian activity patterns on spreading processes.

It is shown that the non-Poisson nature of the contact dynamics results in prevalence decay times significantly larger than predicted by the standard Poisson process based models.

Solvable non-Markovian dynamic network.

The analytically solvable Mittag-Leffler model provides an excellent approximation to the case when the network dynamics is characterized by power-law-distributed interevent times.

Bursts of Vertex Activation and Epidemics in Evolving Networks

A stochastic model to generate temporal networks where vertices make instantaneous contacts following heterogeneous inter-event intervals, and may leave and enter the system is proposed, finding that prevalence is generally higher for heterogeneous patterns, except for sufficiently large infection duration and transmission probability.