Week 11.11.2024 – 17.11.2024

Complex networks have emerged as the primary framework for modeling the dynamics of interacting systems. However, networks inherently describe pairwise interactions, while real-world systems often involve interactions among groups of three or more units. In this talk, I will explore social systems as a natural testing ground for higher-order network approaches. I will briefly demonstrate how incorporating higher-order mechanisms can lead to the emergence of novel phenomena, presenting recent results on the influence of structural features and seeding strategies on emergent dynamics. Finally, I will delve into the microscopic dynamics of empirical higher-order structures, examining the mechanisms governing their temporal dynamics at both the individual and group levels. This will involve characterizing how individuals navigate groups and how groups form and dissolve. I will conclude by proposing a dynamical hypergraph model that closely reproduces empirical observations.

Percolation is the study of long-range connectivity in randomly connected systems, and has been studied extensively for the last 60+ years, in fact quite intensely at King’s College (Cyril Domb, Michael Fisher, John Essam, M. F. Sykes) in the early days. In this talk I survey some of important results, such as crossing (John Cardy, Gerard Watts), scaling, universality, and exact results. Determination of exact thresholds over a wide range of lattices, using a generalization of the Sykes-Essam result based upon the star-triangle transformation, will be discussed (work with Chris Scullard). Numerical methods (work with Chris Lorenz, Stephen Mertens, Youjin Deng, Mark Newman) will also be discussed. Some recent work on Simply-Connected Compoennts (work with Peter Grassberger) will also be presented.