Fig. 5: Robust design of network isolators in the Kuramoto model.

a To study the effect of non-linearity on network isolators, we simulate the failure of a single link (red) in a network consisting of two modules that are connected via a network isolator. b We consider the median absolute non-linear flow changes \(| {{\Delta }}\tilde{F}(\ell )|\) (Eq. (11)) on a link ℓ after the removal of the link shown in (a). We analyse the effect of edge distance to the failing link (x-axis) and increasing degree of non-linearity (colour code from light to dark). We compare the flow changes in the lower module that contains the failing link (curves on the upper left) and the isolated module (curves on the lower right) by averaging the flow changes over all links in the given module at a fixed distance. As expected, flow changes in the upper module are lowest for a weakly non-linear system (bright line) and increase with the non-linearity, but a strong isolation effect persists even for a high degree of non-linearity (dark purple line). Shaded region indicates the 0.25- and 0.75-quantiles evaluated over the given distance. c We fix the overall available edge weight of the four edges forming the isolator to ∑_ia_i = 4 and systematically scan over the remaining degrees of freedom, measuring the isolator performance in terms of the mean logarithmic flow changes \(\langle {\log }_{10}(|\varDelta \tilde{F}|)\rangle\) for a fixed degree of (intermediate) non-linearity. We observe that a heterogeneous isolator where the weights differ strongly provides the best shielding. d We evaluate the available worst-case N − 1 weight, i.e. the overall edge weight connecting the two modules after the failure of a single link in the isolator, for the same set of edge weights as in (c). Here, isolators with homogeneous weights perform best. Edge weights of all non-isolator edges are set to unity, A_ij = 1, \(\forall(i, j) \in\) E(G) in all panels.