Cascading failures in interdependent systems under a flow redistribution model

Robustness and cascading failures in interdependent systems has been an active research field in the past decade. However, most existing works use percolation-based models where only the largest component of each network remains functional throughout the cascade. Although suitable for communication networks, this assumption fails to capture the dependencies in systems carrying a flow (e.g., power systems, road transportation networks), where cascading failures are often triggered by redistribution of flows leading to overloading of lines. Here, we consider a model consisting of systems $A$ and $B$ with initial line loads and capacities given by ${\{L_{A,i},C_{A,i}\}}_{i=1}^n$ and ${\{L_{B,i},C_{B,i}\}}_{i=1}^n$, respectively. When a line fails in system $A$, $a$ fraction of its load is redistributed to alive lines in $B$, while remaining $(1-a)$ fraction is redistributed equally among all functional lines in $A$; a line failure in $B$ is treated similarly with $b$ giving the fraction to be redistributed to $A$. We give a thorough analysis of cascading failures of this model initiated by a random attack targeting $p_1$ fraction of lines in $A$ and $p_2$ fraction in $B$. We show that (i) the model captures the real-world phenomenon of unexpected large scale cascades and exhibits interesting transition behavior: the final collapse is always first order, but it can be preceded by a sequence of first- and second-order transitions; (ii) network robustness tightly depends on the coupling coefficients $a$ and $b$, and robustness is maximized at non-trivial $a,b$ values in general; (iii) unlike most existing models, interdependence has a multifaceted impact on system robustness in that interdependency can lead to an improved robustness for each individual network.

Figure 1

Possible transition behaviors under the load redistribution based cascade model. We see that final system collapse is always first order, which may be preceded with one or more first- or second-order transitions.

Figure 2

Illustration of a two-network system. When failures happen in network $B$, $b$ portion of the failed loads goes to network $A$ and $(1-b)$ portion stays in $B$. Similarly in network $A$, $(1-a)$ portion stays and $a$ portion goes to $B$. Failed loads will be redistributed equally and globally among the remaining lines in each network.

Figure 3

Final system size under different load-free space distributions and coupling coefficients. We observe interesting transition behaviors under different load-free space distributions and coupling level, and the simulation represented in symbol matches with the analytical results represented in lines.

Figure 4

Effect of coupling on the robustness of a single system. We see that contrary to percolation-based models, robustness can indeed be improved by having nonzero coupling between the constituent networks. (Inset) The critical point $p_{★}$ defined as the smallest $p_1$ at which $n_{\infty ,A}\left(p_1\right)$ deviates from $1-p_1$. The optimal (i.e., largest) $p_{★}$ is attained at a nontrivial coupling level $a=b=\simeq 0.53$.

Figure 5

Number of steps needed to reach steady state for identical networks ($a=b$), for various $a$ values. For the case when $a=0.37$, we observe an unforeseen transition behavior.

Figure 6

Final system size in two networks when only network $A$ has been attacked initially. The two networks are statistically identical with $a=b=0.37$. Their loads follow a Weibull distribution with $k=0.4$, $\lambda =100$, $L_{\text{min}}=10$, and $S=0.6L$

Figure 7

Survival regions of the coupled system under load-redistribution based model. When coupling is introduced, regions where both networks survive or collapse ($S_{12}$ and $S_0$, respectively) get larger, while regions where only one network survives ($S_1$ and $S_2$) shrink significantly.

Figure 8

Color map of the critical attack size under different coupling coefficients $a$ and $b$. Darker colors indicate larger $p_{\text{sys}}^{★}$ values, meaning that the interdependent system is more robust.

Figure 9

Extra load per alive line $Q_{t,A}$ is shown (at different attack sizes $p_1$ on Network $A$) as a function of cascade step $t=0,1,...$, for the setting considered in Fig. 6. The jumps in the transitions divide the final system curve into four regions (marked with circled numbers), which correspond to four clusters in the $Q_{t,A}$ plots (distinguished by four colors).

Figure 10

Multiple transitions in a single network and the corresponding function $g\left(x\right)$ [defined at Eq. (8)] is plotted when $L$ follows Weibull distribution with $k=0.4$, $\lambda =100$, $L_{\text{min}}=10$, and $S=\alpha L$, where $\alpha =1.74$. The inset zooms in to the region where $g\left(x\right)$ has a local maximum.

Figure 11

Effect of parameter $\gamma$, which controls the fraction of failed load that will be redistributed locally according to network topology, on the robustness of interdependent systems.