Fig. 1: Different structural patterns separate networks and increase network robustness.
a Topology of the Scandinavian power grid, with weak connectivity between different geographic units, in particular Finland. b Venation network of the leaf Schizolobium amazonicum, with a strong central vein separating the leaf into left and right. The width of the lines in a, b encode the edge weights. c, d Flow changes ∣ΔF∣ after the failure of a single edge (colored in red) for the two networks shown in a, b. The impact of the failure is strongly suppressed in another part of the network, i.e., in Finland and the right half of the leaf, respectively. This highlights the existence and impact of boundaries that separate the network into communities. e Simulation of a classic model of global cascades24,25 in a lattice with inhomogenous edge weights. Edges in the middle have weight wℓ (indicated by thin/thick lines), while all other edges have a weight of wij = 1 (see “Methods” for details). Infected/faulty nodes are shown as yellow triangles, healthy/operational nodes as green circles. A global cascade occurs for a homogeneous lattice (wℓ = 1), while both weak (wℓ = 10−2) and strong connections (wℓ = 102) stop the cascade from propagating to the right part of the network. f Final fraction ρ∞ of nodes that become infected/faulty during the cascade as a function of the weight parameter w. The line represents the median and the shaded region the 25–75% quantile for 1000 random initial conditions. For homogeneous lattices (w = 1), the cascade reaches all nodes (ρ∞ = 1). For both weak (w ≪ 1) and strong (w ≫ 1) conditions, the cascade stops at the boundary such that ρ∞ ≈ 0.5.
