Aktuelles & Events

Higher-order networks have become a popular tool in the network science community to model dynamics such as synchronization and diffusion. The linearized system often depends on a Laplacian operator and its spectral properties. We introduce a Laplacian operator for uniform hypergraphs and study the limiting operator for an increasing sequence of dense uniform hypergraphs using the theory of graph limits. Although a theory of dense hypergraph limits has been developed by Elek and Szegedy, and independently Zhao, not much of its implications to spectral properties is known. We show that a weaker notion of convergence for the sequence of hypergraphs is sufficient to obtain pointwise convergence of the spectrum of the Laplacians.

We present a framework for determining effectively the spectrum and stability of traveling waves on networks with symmetries, such as rings and lattices, by computing master stability curves (MSCs). Unlike traditional methods, MSCs are independent of system size and can be readily used to assess wave destabilization and multistability in small and large networks.

Joint work with Alexandra Blessing (Universität Konstanz). The dynamics of rough differential equations (RDEs) has recently received a lot of interest. For example, the existence of local random center manifolds for RDEs has been established. In this work, we present an approximation for local random center manifolds for RDEs driven by geometric rough paths. To this aim, we combine tools from rough path and deterministic center manifold theory to derive Taylor-like approximations of local random center manifolds. The coefficients of this approximation are stationary solutions of RDEs driven by the same geometric rough path as the original equation. We illustrate our approach for stochastic differential equations (SDEs) with linear and nonlinear multiplicative noise. If necessary here is the link for the pre-print on arxiv: https://arxiv.org/abs/2510.00971

The Cluster-cluster model was defined by Meakin in 1984. Consider a stochastic process on the graph Z^d. Each x in Z^d starts with a cluster of size 1 with probability p in (0,1] independently. Each cluster C performs a continuous time SRW with rate |C|^{-\alpha}. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. In this talk we will present results about explosion, non-explosion and cluster growth rates, as a function of the dimension - d, percolation density - p and diffusion rate parameter - alpha. Joint work with Noam Berger (TUM), Eviatar B. Procaccia (Technion) and Dominik Schmid (Augsburg University).