Vulnerability and Cosusceptibility Determine the Size of Network Cascades

In a network, a local disturbance can propagate and eventually cause a substantial part of the system to fail in cascade events that are easy to conceptualize but extraordinarily difficult to predict. Here, we develop a statistical framework that can predict cascade size distributions by incorporating two ingredients only: the vulnerability of individual components and the cosusceptibility of groups of components (i.e., their tendency to fail together). Using cascades in power grids as a representative example, we show that correlations between component failures define structured and often surprisingly large groups of cosusceptible components. Aside from their implications for blackout studies, these results provide insights and a new modeling framework for understanding cascades in financial systems, food webs, and complex networks in general.

Figure 1

Example of cosusceptibility of cascading failures in a power grid. (a) Color-coded network of positive correlations between failures of transmission lines (blue dots) in the Texas network. In the gray background network, the nodes represent (all) transmission lines, while each link represents a direct physical connection through a common substation. (b) Network in (a) after a correlation-based repositioning of the nodes.

Figure 2

Two-stage algorithm for identifying sets of cosusceptible components. (a) First stage, in which we partition the reduced unweighted correlation graph into multiple cliques $\{Q_k\}$ and index the cliques in descending order of size. (b) Second stage, in which we find the cosusceptible groups $\{R_k\}$ by recursively agglomerating cliques that have enough connections.

Figure 3

Correlations between transmission line failures in the Texas power grid. (a)–(d) Estimated correlation matrix ${\widehat{C}}^{(\mathrm{sig})}$ [(a), (b)] and vulnerabilities $p_{\ell }$ [(c), (d)] under the 2011 summer on-peak condition. In (a) and (c), the transmission lines are indexed so that $p_{\ell }$ is decreasing in $\ell$. In (b) and (d), they are indexed so that the sets of cosusceptible lines appear as diagonal blocks. (e) Sizes of the groups of cosusceptible transmission lines, indicated by the lengths of the individual segments of the red portion of the top bar (on-peak snapshot) and of the blue portion of the bottom bar (off-peak snapshot) for each season. The gray portion of each bar corresponds to groups of fewer than three lines among groups $R_k$. (f) Relative difference ${\overline{\sigma }}^2$ between the variance of the cascade size and its counterpart under the no-correlation assumption.

Figure 4

Validation of the cosusceptibility model. (a) Cumulative distributions $S_N(x)$ and $S_{\mathcal{N}}(x)$ of cascade sizes $N$ and $\mathcal{N}$ from the cascade-event data and from the cosusceptibility model, respectively, under the 2011 summer on-peak condition. (Inset) Binned probabilities from the two distributions plotted against each other. The shaded area indicates the 95% confidence interval. (b) Distance between two distributions as a function of the total load in megawatts (MW) for all 24 snapshots.