Fig. 2: Effectiveness and robustness of shielding network structures.
From: Network isolators inhibit failure spreading in complex networks
a, b Adjacency matrices for the graphs shown in Fig. 1a, b. Two random graphs G(30,0.4) are inter-connected via a fraction c = 0.2 of their nodes chosen at random, and links are added with probability μ, interpolating between weak (a) or strong (b) interconnectivity (see Methods for details). c The average ratio of flow changes R(ℓ) in the two components (Eq. (8)) is strongly suppressed for both high and low interconnectivity μ. The blue line represents the median value over all distances and the shaded region indicates the 0.25- and 0.75-quantiles. d Adjacency matrix for the six-regular graph shown in Fig. 1c and containing a network isolator. Note that all nodes in the graph including those in the network isolator have degree equal to six, which allows us to exclude any potential impact of heterogeneity in the degree on failure spreading in this case. e The ratio of flow changes R, now averaged over all possible trigger links ℓ and distances d, vanishes for a network isolator described by the condition ξ(A12) = 0 and increases algebraically with the coherence parameter ξ (cf. Eq. (10)) when perturbed (see Methods for details on the simulation). Again, median and 0.25- and 0.75-quantiles are shown resulting from averaging over all distances and then trigger links.
