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Author: Hellmann, Frank × End date: 2023 × Start date: 2020 × Publisher: [London] : IOP ×

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    Ensemble analysis of complex network properties—an MCMC approach
    ([London] : IOP, 2022) Pfeffer, Oskar; Molkenthin, Nora; Hellmann, Frank
    What do generic networks that have certain properties look like? We use relative canonical network ensembles as the ensembles that realize a property R while being as indistinguishable as possible from a background network ensemble. This allows us to study the most generic features of the networks giving rise to the property under investigation. To test the approach we apply it to study properties thought to characterize ‘small-world networks’. We consider two different defining properties, the ‘small-world-ness’ of Humphries and Gurney, as well as a geometric variant. Studying them in the context of Erdős-Rényi and Watts-Strogatz ensembles we find that all ensembles studied exhibit phase transitions to systems with large hubs and in some cases cliques. Such features are not present in common examples of small-world networks, indicating that these properties do not robustly capture the notion of small-world networks. We expect the overall approach to have wide applicability for understanding network properties of real world interest, such as optimal ride-sharing designs, the vulnerability of networks to cascades, the performance of communication topologies in coordinating fluctuation response or the ability of social distancing measures to suppress disease spreading.
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    Monte Carlo basin bifurcation analysis
    ([London] : IOP, 2020) Gelbrecht, Maximilian; Kurths, Jürgen; Hellmann, Frank
    Many high-dimensional complex systems exhibit an enormously complex landscape of possible asymptotic states. Here, we present a numerical approach geared towards analyzing such systems. It is situated between the classical analysis with macroscopic order parameters and a more thorough, detailed bifurcation analysis. With our machine learning method, based on random sampling and clustering methods, we are able to characterize the different asymptotic states or classes thereof and even their basins of attraction. In order to do this, suitable, easy to compute, statistics of trajectories with randomly generated initial conditions and parameters are clustered by an algorithm such as DBSCAN. Due to its modular and flexible nature, our method has a wide range of possible applications in many disciplines. While typical applications are oscillator networks, it is not limited only to ordinary differential equation systems, every complex system yielding trajectories, such as maps or agent-based models, can be analyzed, as we show by applying it the Dodds-Watts model, a generalized SIRS-model, modeling social and biological contagion. A second order Kuramoto model, used, e.g. to investigate power grid dynamics, and a Stuart-Landau oscillator network, each exhibiting a complex multistable regime, are shown as well. The method is available to use as a package for the Julia language. © 2020 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.