\textbf{Quantifying Stability in Complex Networks: From Linear to Basin Stability }

The human brain, power grids, arrays of coupled lasers and the Amazon rainforest are all characterized by multistability. The likelihood that these systems will remain in the most desirable of their many stable states depends on their stability against significant perturbations, particularly in a state space populated by undesirable states. Here we claim that the traditional linearization-based approach to stability is in several cases too local to adequately assess how stable a state is. Instead, we quantify it in terms of basin stability, a new measure related to the volume of the basin of attraction. Basin stability is non-local, nonlinear and easily applicable, even to high-dimensional systems. It provides a long-sought-after explanation for the surprisingly regular topologies of neural networks and power grids, which have eluded theoretical description based solely on linear stability. Specifically, we employ a component-wise version of basin stability, a nonlinear inspection scheme, to investigate how a grid's degree of stability is influenced by certain patterns in the wiring topology. Various statistics from our ensemble simulations all support one main finding: The widespread and cheapest of all connection schemes, namely dead ends and dead trees, strongly diminish stability. For the Northern European power system we demonstrate that the inverse is also true: `Healing' dead ends by addition of transmission lines substantially enhances stability. This indicates a crucial smart-design principle for tomorrow's sustainable power grids: add just a few more lines to avoid dead ends. Further, we analyse the particular function of certain network motifs to promote the stability of the system. Here we uncover the impact of so-called detour motifs on the appearance of nodes with a poor stability score and discuss the implications for power grid design. Moreover, it will be shown that basin stability enables uncovering the mechanism for explosive synchronization and understanding of evolving networks. \\ \\Reference: P. Menck, J. Heitzig, N. Marwan, and J. Kurths, Nature Physics 9, 89 (2013) P. Menck, J. Heitzig, J. Kurths, and H. Schellnhuber, Nature Communication 5, 3969 (2014) P. Schultz, J. Heitzig, and J. Kurths, New Journal Physics 16, 125001 (2014) V. Kohar, P. Ji, A. Choudhary, S. Sinha, and J. Kurths, Phys. Rev. E 90, 022812 (2014) Y. Zou, T. Pereira,~M. Small,~ Z. Liu, and J. Kurths, Phys. Rev. Lett. 112, 114102 (2014)