Prize Talk: Irwin Oppenheim Award: Spectra of Large Networks: Theory and Applications

The interactions between the constituents of complex systems, such as, neural networks, ecosystems, or financial networks, can be described with large, directed networks. In recent years, mathematical methods have been developed to determine the spectral properties of large, directed graphs as a function of their topological properties. These approaches extend beyond the extensively studied case of densely connected networks, and instead consider networks of finite connectivity, as they occur in real-world systems. In this talk I will summarize some key results on the spectral analysis of large graphs, specifically focusing on the role of network topology and the properties of the edge weights. Subsequently, we discuss the implications of these results for the dynamics of complex systems defined on large networks.