We study a fourth order geometric evolution problem on a network of curves
in a bounded domain
. The flow decreases a weighted total length of the curves and
preserves the enclosed volumes. Stationary solutions of the flow are critical points of
a partition problem in
. In this paper we study the linearized stability of stationary
solutions using the H−1-gradient flow structure of the problem. Important issues are the
development of an appropriate PDE formulation of the geometric problem and Poincar´e
type estimate on a network of curves.