  We develop a topology data analysis-based method to detect early signs for
critical transitions in financial data. From the time-series of multiple stock
prices, we build time-dependent correlation networks, which exhibit topological
structures. We compute the persistent homology associated to these structures
in order to track the changes in topology when approaching a critical
transition. As a case study, we investigate a portfolio of stocks during a
period prior to the US financial crisis of 2007-2008, and show the presence of
early signs of the critical transition.
