Stochastische Prozesse auf evolvierenden Netzwerken
Zusammenfassung der Projektergebnisse
The project “Stochastic processes on evolving networks” addressed the critical need for a rigorous mathematical framework to understand complex systems that are characterized by their states evolving by network interactions and evolution of interactions. Imagine a social network where friendships form and dissolve over time, or a biological network where interactions between molecules change as an organism develops. These are some examples of evolving networks. The project focused on developing mathematical tools to analyze how stochastic processes – models of phenomena subject to random inputs and uncertainty unfolding over time – behave on such evolving networks. The most important scientific advances include: • New mathematical models for evolving networks: The network developed novel models that go beyond static network representations, capturing the dynamic changes in network structure. These models provide a more realistic and nuanced description of complex systems. • Probabilistic techniques for dynamic network processes: The project advanced the mathematical toolkit for analyzing stochastic processes on these evolving networks. This includes adapting existing techniques and developing new methods to understand the behavior of random phenomena, like the spread of information or diseases, on networks that are themselves changing. • Foundation for statistical analysis of dynamic network data: The network laid the groundwork for developing statistical methods to infer the structure and dynamics of realworld evolving networks from data. This is essential for applying these mathematical models to practical problems and extracting meaningful insights from complex datasets. These advances have broad application potential in various fields. For instance, in epidemiology, improved models can help predict and control the spread of diseases in populations with evolving contact patterns. In social sciences, they can provide a deeper understanding of opinion dynamics and information diffusion in dynamic social networks. In the long term, these fundamental mathematical advances can contribute to more accurate predictions, better risk assessments, and optimized design of complex systems across diverse domains. While the project progressed largely as planned, a positive “surprise” was the high level of engagement and productivity within the network. The workshops consistently attracted strong participation from both network members and invited experts, fostering a highly collaborative and stimulating research environment. The number of publications and ongoing projects resulting from the network’s activities exceeded initial expectations, demonstrating the effectiveness of the network structure and the enthusiasm of the participants. The adaptability of the network to the challenges posed by the COVID-19 pandemic, maintaining momentum and productivity despite disruptions, was also a noteworthy positive outcome.
Projektbezogene Publikationen (Auswahl)
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Markovian Dynamics of Exchangeable Arrays. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, 209-228.
Černý, Jiří & Klimovsky, Anton
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Quenched invariance principle for random walks on dynamically averaging random conductances. Electronic Communications in Probability, 26.
Bethuelsen, Stein Andreas; Hirsch, Christian & Mönch, Christian
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Recurrence versus transience for weight-dependent random connection models. Electronic Journal of Probability, 27.
Gracar, Peter; Heydenreich, Markus; Mönch, Christian & Mörters, Peter
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An Invariance Principle for a Random Walk Among Moving Traps via Thermodynamic Formalism
Athreya, S.; Drewitz, A. & Sun, R.
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Cumulants asymptotics for the zeros counting measure of real Gaussian processes. Electronic Journal of Probability, 28.
Gass, Louis
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Lipschitz cutset for fractal graphs and applications to the spread of infections
Drewitz, A.; Gallo, G. & Gracar, P.
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Meeting times of Markov chains via singular value decomposition
van Belle, T. & Klimovsky, A.
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The grapheme-valued Wright–Fisher diffusion with mutation. Theoretical Population Biology, 158, 76-88.
Greven, Andreas; den Hollander, Frank; Klimovsky, Anton & Winter, Anita
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Voronoi cells in random split trees. Annales Henri Lebesgue, 7, 123-159.
Drewitz, Alexander; Heydenreich, Markus & Mailler, Cécile
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Continuum graph dynamics via population dynamics: Well-posedness, duality and equilibria. Stochastic Processes and their Applications, 188, 104670.
Greven, Andreas; den Hollander, Frank; Klimovsky, Anton & Winter, Anita
