The importance of edges in complex networks

Abstract

The sun, the climate, world politics, the stock market, or the human brain, complex networked systems are deeply intertwined in the world we live in. Their dynamics, which include unexpected and catastrophic extreme events, can have tremendous impact on a single human or man-kind as a whole. Hence, studying these systems' dynamical phenomena as well as their properties is essential to improve our knowledge about them. Under the key premise that a complex system can be divided into interacting elementary units, the network ansatz poses a very useful and decisive approach to characterise the system. Associating network vertices with elementary units and network edges with interactions between them, this ansatz yields vast applicability to various natural or man-made systems. Even for those cases, where interactions have no structural correlate or cannot be inferred directly, utilizing time-series-analysis techniques to investigate the units' dynamics allows to characterize properties of interactions, like their strength, direction or even coupling functions, ultimately constituting a time-evolving functional network. Graph theory assesses networks as mathematical structures and provides a multitude of concepts and metrics to assess network characteristics from a global scale, viewing the network as a whole, over an intermediate scale, focusing on substructures in it, to a local scale, inspecting properties of single vertices and edges. Knowledge gained in this way about the properties of the network can then be related to properties of the investigated system and aid to understand its complex emergent global dynamics. While in many ways it is the intricate interplay of interactions between the systems' elements that dictates its properties and dynamics, the edges of networks and their properties have been vastly overlooked. Therefore in this thesis, we embarked on a more edge-centric approach to investigate complex systems utilizing the network ansatz. We developed novel concepts, advanced local network metrics, proposed novel edge-centric metrics and introduced network decomposition algorithms, set out to improve our understanding of real-world systems and their complex dynamics. We demonstrated the applicability and added value of these concepts and metrics, and gained vital insights about archetypical network topologies, spreading phenomena, as well as critical transitions and their entailed extreme events. On the prime example of a complex dynamical system, able to self-generate extreme events, the human epileptic brain, we elucidated vital aspects of network mechanisms involved in the generation of epileptic seizures, e.g. in revealing specific tipping elements and tipping subnetworks. This can ultimately aid in developing more refined approaches to characterize, predict and possibly even mitigate extreme events, such as epileptic seizures. We further revealed limitations of the network ansatz, and how this approach can aid in tackling fundamental challenges encountered especially when studying such real-world systems as the brain, included sampling issues and influences of endogenous and external driving forces. Employing the network ansatz and focusing on the intricate interplay of a complex networked system's interactions, provided considerable advances in understanding these systems and their dynamical phenomena, while also paving the way for future research by displaying the immense potential the network ansatz -- and especially the study of important edges -- can hold.