Figure 1

(Color online) (a) Illustration of failure propagation, multiplication, and correlation in a two-layer system. A single failure in the physical graph results in three correlated failures in the logical graph. (b) The system after a failure of $e_1^{\varphi }$. The dashed lines are valid only under the “full rerouting” scenario.Reuse & Permissions

Figure 2

Edge load distribution in the five studied systems: Railway (a), Brain (b), Gnutella (c), ER on ER (d), and BA on ER (e). The main plots are in ${\log }_{10}–{\log }_{10}$ scale (${\log }_{10}$-binned); the insets present the same distributions in ${\log }_{10}$-linear scale (linear-binned).Reuse & Permissions

Figure 3

(Color online) Error and attack tolerance of five layered systems (rows). The first three are real-life data sets; the last two are based on classic ER and power-law BA graphs. At each iteration we remove one physical edge, $e_{\mathrm{del}}^{\varphi }$, either at random (error tolerance, left column), or by choosing the most loaded one (attack tolerance, right column). In both cases we observe the size of the largest connected component in the physical graph, $G^{\varphi }$ (triangles), and the total weight of the remaining logical edges, (circles). Every logical edge, $e^{\lambda }$, whose mapping contains $e_{\mathrm{del}}^{\varphi }$ is deleted either directly (no rerouting, unfilled symbols), or only when there is no path in $G^{\varphi }$ between the end-nodes of $e^{\lambda }$ (full rerouting, filled symbols). The results for ER on ER and BA on ER are averaged over ten realizations. Note that, in the case of random errors, the largest connected physical component is not affected by the adopted policy (no-rerouting or full-rerouting). Therefore the filled- and empty-triangle curves coincide for all figures in the left column. We draw only the empty-triangle curves for clarity.Reuse & Permissions