It has long been known that the magnetic field structures on the Sun are highly complex. By determining the skeleton of global magnetic fields extrapolated from high-resolution observed magnetograms it is now possible to characterise the complexity of these fields. Here, we will discuss the evolution of the global complexity on both short (days) and long (solar cycle) time-scales. In particular, highlighting how changes in the numbers/strengths of sunspots or the strength of the polar fields influences the distribution of null points, separators and the location of streamers and pseudo-streamers.