Mathematical principles behind the transmission of energy and synchronisation in complex networks

Understanding how the transmission of energy between the providers (such as nuclear power stations, renewable resources, or any type of supplying entity) and the consumers (such as factories, homes, or any type of demanding entity) depends on the structure of the inter-connections between them and on their dynamical be- haviour, is of paramount importance for the design of power-grid systems that are resilient to failures, e.g., failures due to structural modifications or energy fluctu- ations. In this thesis, we derive the implicit relationship between structure and behaviour that flow and power networks have, namely, the mathematical principles behind the transmission of energy in complex networks. From our novel derivations, we determine exact and approximate strategies to create self-controlled and stable systems (i.e., resilient to failures without the need for external controllers) that have an optimal (i.e., with less cost and power dissipation) and smart (i.e., allowing the decentralisation of large power-stations to smaller fluctuating renewable resources) energy distribution. Moreover, not only we achieve analytical solutions for problems that usually require a numerical analysis, but we also propose a change in the analy- sis view-point of complex systems, namely, systems composed of many dynamically interacting units forming a network. We show that in order to explain the emer- gent behaviour in these systems, instead of focusing on the network structure of the interactions, we should focus on the functional form of the interactions. In particu- lar, we derive a general framework to study the existence and stability of emergent collective behaviour in networks of interacting phase-oscillators, namely, the math- ematical principles behind the synchronisation in complex networks