Network Connectivity: Structure vs Dynamics

=======
Abstract
=======
Many questions about complex systems are fundamentally questions of connectivity. How does information flow through a network? What hidden structures govern these flows? And can large networks be simplified without losing their essential dynamical behaviour? In this talk, I present a research programme centred on the interplay between network structure and diffusion. Using random walks as a unifying framework, I explore how network dynamics reveal hidden structure, and how structural organisation constrains diffusion. I begin by discussing recent work on first-return time statistics, showing that return-time distributions are governed primarily by local neighbourhood structure and short cycles, challenging the common assumption that diffusion observables necessarily reflect community structure. I then introduce a coarse-graining framework for directed networks based on ergodic sets and transport flows. By identifying dynamically relevant modes and compressing core mixing behaviour large directed networks can be reduced while preserving their dominant diffusion pathways. Finally, I present a statistical-mechanics-inspired framework that unifies several classical notions of connectivity—including shortest paths, diffusion, effective resistance, and percolation—within a common probabilistic model. Together, these results suggest that connectivity provides a common language linking structure, dynamics, and inference in complex networks, while raising new questions about the relationship between diffusion, transport, and multiscale organisation.

===============
About the Speaker
===============
Erik Hormann is a Lecturer in Mathematics at James Cook University Singapore. He completed a DPhil in Mathematics at the University of Oxford, where he worked on network science under the supervision of Renaud Lambiotte. Prior to joining JCU Singapore, he held a teaching position at the University of Edinburgh. His research focuses on the interplay between network structure and dynamics, with particular interests in random walks, directed networks, statistical mechanics, community detection, and multi scale network modelling. His work has appeared in journals including PNAS, Physical Review E, and Nature Communications.