  We investigate weak localization in metallic networks etched in a two
dimensional electron gas between $25\:$mK and $750\:$mK when electron-electron
(e-e) interaction is the dominant phase breaking mechanism. We show that, at
the highest temperatures, the contributions arising from trajectories that wind
around the rings and trajectories that do not are governed by two different
length scales. This is achieved by analyzing separately the envelope and the
oscillating part of the magnetoconductance. For $T\gtrsim0.3\:$K we find
$\Lphi^\mathrm{env}\propto{T}^{-1/3}$ for the envelope, and
$\Lphi^\mathrm{osc}\propto{T}^{-1/2}$ for the oscillations, in agreement with
the prediction for a single ring \cite{LudMir04,TexMon05}. This is the first
experimental confirmation of the geometry dependence of decoherence due to e-e
interaction.
