C3

Real complex systems exhibit rich collective behavior, yet identifying which components of an interaction network drive such dynamics remains a central challenge. Here, we show that complexity itself can resolve this problem. In large random and empirical networks, structural disorder and heterogeneity induce spectral localization, causing Laplacian modes to concentrate on small subsets of nodes. This converts global modes into identifiable dynamical units tied to specific structural components. Exploiting this principle, we develop a node-resolved stability framework that predicts instability onsets, identifies the nodes responsible for collective transitions, and restores interpretability in systems where classical modal theories fail. In heterogeneous reaction networks, the same mechanism enables collective states beyond those usually associated with homogeneous assumptions. More broadly, our results show that complexity can be revealed, rather than obscure, the microscopic drivers of macroscopic dynamics.

We will introduce a class of countable homeomorphism groups that share many properties with Thompson's group V, known as FSS* groups. This talk from Patrick Henry DeBonis will focus on some of the group constructions and deformation/rigidity arguments needed to prove FSS* group von Neumann algebras are prime - and have potential for wider applications.

Physical networks are spatially embedded complex networks composed of nodes and links that are tangible objects which cannot overlap. Examples of physical networks range from neural networks and networks of bio-molecules to computer chips and disordered meta-materials. It is hypothesized that the unique features of physical networks, such as the non-trivial shape of nodes and links and volume exclusion affect their network structure and function. However, the traditional tool set of network science cannot capture these properties, calling for a suitable generalization of network theory. Here, I present recent efforts to understand the impact of physicality through tractable models of network formation.

We introduce a graph renormalization procedure based on the coarse-grained Laplacian, which generates reduced-complexity representations across scales. This method retains both dynamics and large-scale topological structures, while reducing redundant information, facilitating the analysis of large graphs by decreasing the number of vertices. Applied to graphs derived from electroencephalogram recordings of human brain activity, our approach reveals collective behavior emerging from neuronal interactions, such as coordinated neuronal activity. Additionally, it shows dynamic reorganization of brain activity across scales, with more generalized patterns during rest and more specialized and scale-invariant activity in the occipital lobe during attention.