The Mathieu twisted twining genera, i.e., the analogues of Norton's generalized Moonshine functions, are constructed for the elliptic genus of K3. It is shown that they satisfy the expected consistency conditions, and that their behaviour under modular transformations is controlled by a 3-cocycle in H 3 (M 24 , U(1)), just as for the case of holomorphic orbifolds. This suggests that a holomorphic VOA may be underlying Mathieu Moonshine.

Highly optimized complex transport networks serve crucial functions in many man-made and natural systems such as power grids and plant or animal vasculature. Often, the relevant optimization functional is nonconvex and characterized by many local extrema. In general, finding the global, or nearly global optimum is difficult. In biological systems, it is believed that such an optimal state is slowly achieved through natural selection. However, general coarse grained models for flow networks with local positive feedback rules for the vessel conductivity typically get trapped in low efficiency, local minima. In this work we show how the growth of the underlying tissue, coupled to the dynamical equations for network development, can drive the system to a dramatically improved optimal state. This general model provides a surprisingly simple explanation for the appearance of highly optimized transport networks in biology such as leaf and animal vasculature.

The Mathieu twisted twining genera, i.e. the analogues of Norton's generalised Moonshine functions, are constructed for the elliptic genus of K3. It is shown that they satisfy the expected consistency conditions, and that their behaviour under modular transformations is controlled by a 3-cocycle in H 3 (M24, U (1)), just as for the case of holomorphic orbifolds. This suggests that a holomorphic VOA may be underlying Mathieu Moonshine. Anexample 30 4.3. Computation of a twisted twining genus 32 5. Conclusions 34 5.1. Summary 34 5.2. Open problems and future work 35 Acknowledgements 37 Appendix A. Details on the unobstructed twisted twining genera 37 A.1. The characters φ 2B,4A 2 (group 13) φ 4B,4A 3 (group 23) and φ 4B,4A 4 (group 24) 38 A.2. The cases φ 3A,3A 3 (group 33) and φ 3A,3B 1 (group 34) 40 Appendix B. Some group cohomology 42 Appendix C. Projective representations of finite groups 43 C.1. Central extension and orbifolds 45 Appendix D. Centralisers C M 24 (g) and (projective) character tables 46 D.1. Character Tables 47

The failure of a single link can degrade the operation of a supply network up to the point of complete collapse. Yet, the interplay between network topology and locality of the response to such damage is poorly understood. Here, we study how topology affects the redistribution of flow after the failure of a single link in linear flow networks with a special focus on power grids. In particular, we analyze the decay of flow changes with distance after a link failure and map it to the field of an electrical dipole for lattice-like networks. The corresponding inverse-square law is shown to hold for all regular tilings. For sparse networks, a long-range response is found instead. In the case of more realistic topologies, we introduce a rerouting distance, which captures the decay of flow changes better than the traditional geodesic distance. Finally, we are able to derive rigorous bounds on the strength of the decay for arbitrary topologies that we verify through extensive numerical simulations. Our results show that it is possible to forecast flow rerouting after link failures to a large extent based on purely topological measures and that these effects generally decay with distance from the failing link. They might be used to predict links prone to failure in supply networks such as power grids and thus help to construct grids providing a more robust and reliable power supply.

SignificanceFunctionally optimized networks are ubiquitous in nature, e.g., in allosteric proteins that change conformation upon binding to a ligand or vascular networks that distribute oxygen and nutrients in animals or plants. Many of these networks are multifunctional, with proteins that can catalyze more than one substrate or vascular networks that can deliver enhanced flow to more than one localized region of the network. This work investigates the question of how many simultaneous functions a given network can be designed to fulfill, uncovering a phase transition that is related to other constraint–satisfaction transitions such as the jamming transition.