Social, biological and physical systems are widely studied through the modeling of dynamical processes over networks, and one commonly investigates the interplay between structure and dynamics. I will discuss how cyclic patterns in networks can influence models for collective and diffusive processes, including generalized models in which dynamics are defined over simplicial complexes and multiplex networks. Our approach relies on homology theory, which is the subfield of mathematics that formally studies cycles (and more generally, k-dimensional holes). We will make use of techniques including persistent homology and Hodge theory to examine the role of cycles in helping organize dynamics onto low-dimensional manifolds. This pursuit represents an emerging interface between the fields of network-coupled dynamical systems and topological data analysis.

Consider the injection of a fluid onto an impermeable surface for an infinite length of time... Does the injected fluid reach a finite height, or does it keep on growing forever? The classical theory of gravity currents suggests that the height remains finite, causing the radius to grow outwards like the square root of time. When the fluid resides within a porous medium, the same is thought to be true. However, recently I used some small scale experiments and numerical simulations, spanning 12 orders of magnitude in dimensionless time, to demonstrate that the height actually grows very slowly, at a rate ~t^(1/7)*(log(t))^(1/2). This strange behaviour can be explained by analysing the flow in a narrow "inner region" close to the source, in which there are significant vertical velocities and non-hydrostatic pressures. Analytical scalings are derived which match closely with both numerics and experiments, suggesting that the blob of fluid is in fact ever-growing, and therefore becomes unbounded with time.