In this talk I will first recall the short history of dynamical- system-theoretically defined Riemannian metrics on deformation spaces -- pressure metrics. Then I will focus on a moduli space of metric graphs and discuss two pressure metrics on it. These two pressure metrics could be thought of as analogues of Weil Petersson metrics (or Thurston's Riemannian metrics) for spaces of metric graphs. In particular, I will discuss and compare Riemannian geometric features of these two metrics with the "classic" Weil-Petersson metric in Teichmueller theory. This work is motivated by an earlier work of Pollicott and Sharp. Speaker(s): Kao Lien-Yung (Notre Dame)