Manipulation of extreme events on scale-free networks

Extreme events taking place on networks are not uncommon. We show that it is possible to manipulate the extreme event occurrence probabilities and distribution over the nodes of a scale-free network by tuning the nodal capacity. This can be used to reduce the number of extreme event occurrences. However, monotonic nodal capacity enhancements, beyond a point, do not lead to any substantial reduction in the number of extreme events. We point out the practical implication of this result for network design in the context of reducing extreme event occurrences.

Figure 1

The probability for the occurrence of EEs ($m=4$) as a function of degree $k$, shown as log-log plot, for $\alpha =0.40,0.43,0.47,0.50,0.52$ (bottom to top). Both the analytical [Eq. (4)], and simulation results are shown. The shaded region on the left (right) corresponds to small degree nodes (hubs).

Figure 2

(a) Mean queue size $\langle S\rangle$ with $m=4$ for the small degree nodes (dashed line) and large degree nodes (solid line), (b) $\langle S\rangle$ for the entire network plotted against $\alpha$. Note that larger $\alpha$ correspond to larger nodal capacity.

Figure 3

Scaled mean number of EEs $\langle E_{\Delta }\rangle /\langle E_0\rangle$ with $m=4$ and $\alpha =1/2$ as a function of noise strength $\Delta$. The (broken and continuous) lines show analytical result, and (solid and open) circles are from simulations.

Figure 4

The ratio of change in mean number of EEs $\delta \langle E\rangle$ on the network for a change of $\delta C$ in capacity of the network plotted as a function of $\alpha$ with $m=4$. The inset shows the same data in log-log plot.