Discrete and Continuous Models in the Theory of Networks

This book contains contributions from the participants of the research group hosted by the ZiF - Center for Interdisciplinary Research at the University of Bielefeld during the period 2013-2017 as well as from the conclusive conference organized at Bielefeld in December 2017. The contributions consist of original research papers: they mirror the scientific developments fostered by this research program or the state-of-the-art results presented during the conclusive conference.

The volume covers current research in the areas of operator theory and dynamical systems on networks and their applications, indicating possible future directions. The book will be interesting to researchers focusing on the mathematical theory of networks; it is unique as, for the first time, continuous network models - a subject that has been blooming in the last twenty years - are studied alongside more classical and discrete ones. Thus, instead of two different worlds often growing independently without much intercommunication, a new path is set, breaking with the tradition. The fruitful and beneficial exchange of ideas and results of both communities is reflected in this book.

Contenu

Stability matters for reaction diffusion-equations on metric graphs under the anti-Kirchhoff vertex condition.- Many-particle quantum graphs: A review.- Deterministic and stochastic mean-field SIRS models on heterogeneous networks.- Krein formula and convergence of Hamiltonians with scald potentials in dimension one.- Ground states of the L2-critical NLS equation with localized nonlinearity on a tadpole graph.- An asymptotic expansion of the trace of the heat kernel of a singular two-particle contact interaction in one-dimension.- Modeling dynamic coupling in social interactions.- A note on Cheeger inequalities for piecewise at surfaces.- Unravelling topological determinants of excitable dynamics on graphs using analytical mean-field approaches.- Spectral zeta functions.- A family of diameter-based eigenvalue bounds for quantum graphs.- Missing-level statistics in chaotic microwave networks versus level statistics of partially chaotic systems.- Signatures, lifts, and eigenvalues of graphs.- Strong Isoperimetric Inequality for Tessellating Quantum Graphs.- Spectral monotonicity for Schrödinger operators on metric graphs.- Random graphs and their subgraphs.- Massive Modes for Quantum Graphs.