Measuring network reliability and repairability against cascading failures

Abstract

Cascading failures on techno-socio-economic systems can have dramatic and catastrophic implications in society. A damage caused by a cascading failure, such as a power blackout, is complex to predict, understand, prevent and mitigate as such complex phenomena are usually a result of an interplay between structural and functional non-linear dynamics. Therefore, systematic and generic measurements of network reliability and repairability against cascading failures is of a paramount importance to build a more sustainable and resilient society. This paper contributes a probabilistic framework for measuring network reliability and repairability against cascading failures. In contrast to related work, the framework is designed on the basis that network reliability is multifaceted and therefore a single metric cannot adequately characterize it. The concept of ‘repairability envelope’ is introduced that illustrates trajectories of performance improvement and trade-offs for countermeasures against cascading failures. The framework is illustrated via four model-independent and application-independent metrics that characterize the topological damage, the network spread of the cascading failure, the evolution of its propagation, the correlation of different cascading failure outbreaks and other aspects by using probability density functions and cumulative distribution functions. The applicability of the framework is experimentally evaluated in a theoretical model of damage spread and an empirical one of power cascading failures. It is shown that the reliability and repairability in two systems of a totally different nature undergoing cascading failures can be better understood by the same generic measurements of the proposed framework.

Appendices

Appendix A: Theoretical model: damage correlation

To better understand the spreading dynamics, the evolution of damage correlation for the two artificial networks is shown in Figs. 12 and 13. The two figures show how correlated the spreading of network damages are in all possible pairs of perturbations in the network and how the correlations evolve during the cascading failure.

Fig. 12

Damage correlation in the Barabási-Albert network at different cascade iterations. (a)-(d) Baseline, (e)-(h) strategy A and (i)-(l) strategy B

Fig. 13

Damage correlation in the small world network at different cascade iterations. (a)-(d) Baseline, (e)-(h) strategy A and (i)-(l) strategy B

Figure 12 confirms that the hubs of the Barabási-Albert network result in highly positively correlated processes of damage spread, which are actually due to the high levels of damage diffused in the network. When the recovery process takes place, there is a low increase in the average correlations for strategy B compared to baseline, for instance, 3.95% at the 30th cascade iteration compared to 0.5% at the 20th iteration.

Figure 13 confirms that the damage spread in small world networks has a strong locality influence. The damage of neighboring nodes results in correlated spread of damages as it can be clearer seen in Fig. 13a, e and i. As the spread of the damage increases according the trajectories of Fig. 1b, the correlation of the damages becomes more polarized and vary between highly positive and negative correlation in different parts of the network. Strategies decrease correlations compared to the baseline, for instance, 25.44% at the 30th cascade iteration and 41.5% at the 70th iteration for strategy B. However, damage correlations increase on average during the cascading failure, for example, 87% and 83.6% for baseline and strategy B respectively and from the 30th to the 70th iteration of the cascading failure.

B: Theoretical model: network visualizations

Figures 14 and 15 show the two artificial networks in the respective cascade iterations that are also shown for the damage correlations. The initial damaged node is node 7.

Fig. 14

Visualization of cascading failure in the Barabási-Albert network. (a), (d), (g), (j): Baseline, (b), (e), (h), (k) strategy A and (c), (f), (i), (l) strategy B. The pallete indicates the damage level of the nodes in the range [0,4]

Fig. 15

Visualization of cascading failure in the small world network. (a), (d), (g), (j): Baseline, (b), (e), (h), (k) strategy A and (c), (f), (i), (l) strategy B. The pallete indicates the damage level of the nodes in the range [0,4]

The Barabási-Albert network in Fig. 14 clearly shows that the damage of the peripheral node influences the neighboring nodes (Fig. 14d, f), however, as the central hub is only two hops away from the damaged node, the damage spreads and eventually affects severely the whole network already at the 20th iteration (Fig. 14g, i). The repairability strategies can only alleviate the damage level of individual nodes and therefore, they do not restrict the cascading disaster.

In contrast, the small world network of Fig. 15 shows that the damage spread is in general localized at a higher level than the Barabási-Albert network. Strategy B maintains the lowest damage level in the nodes as also confirmed by Fig. 1b.