Slow, bursty dynamics as a consequence of quenched network topologies

Bursty dynamics of agents is shown to appear at criticality or in extended Griffiths phases, even in case of Poisson processes. I provide numerical evidence for a power-law type of intercommunication time distributions by simulating the contact process and the susceptible-infected-susceptible model. This observation suggests that in the case of nonstationary bursty systems, the observed non-Poissonian behavior can emerge as a consequence of an underlying hidden Poissonian network process, which is either critical or exhibits strong rare-region effects. On the contrary, in time-varying networks, rare-region effects do not cause deviation from the mean-field behavior, and heterogeneity-induced burstyness is absent.

Figure 1

Intercommunication time distribution in the 1D critical CP of size $N={10}^5$. The full line denote histogramming from all times, the dashed line from high times, and long dashes from low times. The dotted line shows a power-law fit for $t>200$ resulting in $\propto t^{-1.85\left(5\right)}$. The solid thin line corresponds to runs from seed initial conditions. Inset: the same data multiplied by the $t^{1.85}$ corresponding to the tail decay.

Figure 2

Intercommunication time distribution in the GP of a GSW network with $\beta =0.1$ of size $N={10}^6$ and $\lambda =2.97$, 3.02, 3.07 (bottom to top solid curves). The solid thin line corresponds to runs from seed initial conditions measuring all activation attempts. Inset: effective exponents defined as (4) of the same data.

Figure 3

Intercommunication time distribution in the GP of a GSW network with $\beta =0.2$ of size $N={10}^6$ and $\lambda =2.75$, 2.8, 2.85 (bottom to top curves). Inset: effective exponents defined as (4) of the same data.

Figure 4

Intercommunication time distribution in the GP of an aging BA network of size $N={10}^5$ for $\lambda =2.47$, 2.5, 2.55, 2.59, 2.6, 2.65 (top to bottom curves) and measured at different ($i=1$ and 100) sites. Power-law fitting exponents for the tail behavior are shown in the text. Inset: $P\left(\Delta \right){\Delta }^4$ at the ${\lambda }_c$ of the CP defined on the pure BA graph.

Figure 5

Density decay of the contact process in time-varying networks of sizes $N={10}^7$ for $\lambda =2.936$, 2.938, 2.940, 2.942 (bottom to top). The dashed line shows a power-law fit to the $\lambda =2.94$ (critical) curve. The activity potential decays with $\gamma =3$.

Figure 6

Density decay in the ARW in time-varying networks of sizes $N={10}^7$ for $\gamma =0.6$, 0.8, 0.9, 1, 3.8 (bottom to top). The curves are the average of ${10}^3$ independent runs.