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Topological Data Analysis, and in particular the method of persistent homology, allow to summarise higher order patterns in complex networks. In this talk I will review some functional summaries of persistence, present their stability properties, discuss their computation and opportunity for data analysis. I will then focus on the case of biological detailed reconstructions of brain networks and on going work with J. Hallgren Kotaleski's group to identify specific network alterations in Parkinson disease progression.