Centrality is one of the most fundamental metrics in network science. Despite an abundance of methods for measuring centrality of individual vertices, there are by now only a few metrics to measure centrality of individual edges. We modify various, widely used centrality concepts for vertices to those for edges, in order to find which edges in a network are important between other pairs of vertices. Focusing on the importance of edges, we propose an edge-centrality-based network decomposition technique to identify a hierarchy of sets of edges, where each set is associated with a different level of importance. We evaluate the efficiency of our methods using various paradigmatic network models and apply the novel concepts to identify important edges and important sets of edges in a commonly used benchmark model in social network analysis, as well as in evolving epileptic brain networks.

Previous research has indicated that temporal changes of centrality of specific nodes in human evolving large-scale epileptic brain networks carry information predictive of impending seizures. Centrality is a fundamental network-theoretical concept that allows one to assess the role a node plays in a network. This concept allows for various interpretations, which is reflected in a number of centrality indices. Here we aim to achieve a more general understanding of local and global network reconfigurations during the pre-seizure period as indicated by changes of different node centrality indices. To this end, we investigate—in a time-resolved manner—evolving large-scale epileptic brain networks that we derived from multi-day, multi-electrode intracranial electroencephalograpic recordings from a large but inhomogeneous group of subjects with pharmacoresistant epilepsies with different anatomical origins. We estimate multiple centrality indices to assess the various roles the nodes play while the networks transit from the seizure-free to the pre-seizure period. Our findings allow us to formulate several major scenarios for the reconfiguration of an evolving epileptic brain network prior to seizures, which indicate that there is likely not a single network mechanism underlying seizure generation. Rather, local and global aspects of the pre-seizure network reconfiguration affect virtually all network constituents, from the various brain regions to the functional connections between them.

The field of Network Physiology aims to advance our understanding of how physiological systems and sub-systems interact to generate a variety of behaviors and distinct physiological states, to optimize the organism's functioning, and to maintain health. Within this framework, which considers the human organism as an integrated network, vertices are associated with organs while edges represent time-varying interactions between vertices. Likewise, vertices may represent networks on smaller spatial scales leading to a complex mixture of interacting homogeneous and inhomogeneous networks of networks. Lacking adequate analytic tools and a theoretical framework to probe interactions within and among diverse physiological systems, current approaches focus on inferring properties of time-varying interactions—namely strength, direction, and functional form—from time-locked recordings of physiological observables. To this end, a variety of bivariate or, in general, multivariate time-series-analysis techniques, which are derived from diverse mathematical and physical concepts, are employed and the resulting time-dependent networks can then be further characterized with methods from network theory. Despite the many promising new developments, there are still problems that evade from a satisfactory solution. Here we address several important challenges that could aid in finding new perspectives and inspire the development of theoretic and analytical concepts to deal with these challenges and in studying the complex interactions between physiological systems.

Transcutaneous auricular vagus nerve stimulation (taVNS) is a novel non-invasive brain stimulation technique considered as a potential supplementary treatment option for a wide range of diseases. Although first promising findings were obtained so far, the exact mode of action of taVNS is not fully understood yet. We recently developed an examination schedule to probe for immediate taVNS-induced modifications of large-scale epileptic brain networks. With this schedule, we observed short-term taVNS to have a topology-modifying, robustness- and stability-enhancing immediate effect on large-scale functional brain networks from subjects with focal epilepsies. We here expand on this study and investigate the impact of short-term taVNS on various local and global characteristics of large-scale evolving functional brain networks from a group of 30 subjects with and without central nervous system diseases. Our findings point to differential, at first glance counterintuitive, taVNS-mediated alterations of local and global topological network characteristics that result in a reconfiguration of networks and a modification of their stability and robustness properties. We propose a model of a stimulation-related stretching and compression of evolving functional brain networks that may help to better understand the mode of action of taVNS.

Many natural and man-made complex dynamical systems can be represented by networks with vertices representing system units and edges the coupling between vertices. If edges of such a structural network are inaccessible, a widely used approach is to identify them with interactions between vertices, thereby setting up a functional network. However, it is an unsolved issue if and to what extent important properties of a functional network on the global and the local scale match those of the corresponding structural network. We address this issue by deriving functional networks from characterizing interactions in paradigmatic oscillator networks with widely-used time-series-analysis techniques for various factors that alter the collective network dynamics. Surprisingly, we find that particularly key constituents of functional networks—as identified with betweenness and eigenvector centrality—coincide with ground truth to a high degree, while global topological and spectral properties—clustering coefficient, average shortest path length, assortativity, and synchronizability—clearly deviate. We obtain similar concurrences for an empirical network. Our findings are of relevance for various scientific fields and call for conceptual and methodological refinements to further our understanding of the relationship between structure and function of complex dynamical systems.

Electroencephalography (EEG) is a widely employed tool for exploring brain dynamics and is used extensively in various domains, ranging from clinical diagnosis via neuroscience, cognitive science, cognitive psychology, psychophysiology, neuromarketing, neurolinguistics, and pharmacology to research on brain computer interfaces. EEG is the only technique that enables the continuous recording of brain dynamics over periods of time that range from a few seconds to hours and days and beyond. When taking long-term recordings, various endogenous and exogenous biological rhythms may impinge on characteristics of EEG signals. While the impact of the circadian rhythm and of ultradian rhythms on spectral characteristics of EEG signals has been investigated for more than half a century, only little is known on how biological rhythms influence characteristics of brain dynamics assessed with modern EEG analysis techniques. At the example of multiday, multichannel non-invasive and invasive EEG recordings, we here discuss the impact of biological rhythms on temporal changes of various characteristics of human brain dynamics: higher-order statistical moments and interaction properties of multichannel EEG signals as well as local and global characteristics of EEG-derived evolving functional brain networks. Our findings emphasize the need to take into account the impact of biological rhythms in order to avoid erroneous statements about brain dynamics and about evolving functional brain networks.

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Vertex degree—the number of edges that are incident to a vertex—is a fundamental concept in network theory. It is the historically first and conceptually simplest centrality concept to rate the importance of a vertex for a network’s structure and dynamics. Unlike many other centrality concepts, for which joint metrics have been proposed for both vertices and edges, by now there is no concept for an edge centrality analogous to vertex degree. Here, we propose such a concept—termed nearest-neighbor edge centrality—and demonstrate its suitability for a non-redundant identification of central edges in paradigmatic network models as well as in real-world networks from various scientific domains.