Brain connectivity, particularly based on resting-state fMRI data has been the subject of intense study for over a decade, stimulated by growing evidence of network involvement in brain diseases. But such study needs methods of analysis that can compare networks with differing numbers of nodes and links and provide results that are stable over different spatial and temporal data resolutions. Current graph analysis methods cannot do this. Enter topological data analysis; in particular persistent homology. By focussing on network 'shape' these new methods can attack those questions. Using our recent development of frequency domain persistent homology, we discuss: network interpretation and comparison of sparsely and densely connected brain networks; multi-scale (spatial and temporal) network analyses; data quality diagnostics, and defining of a core brain architecture. We illustrate results with fMRI data.