  The yielding of disordered materials is a complex transition involving
significant changes of the material's microstructure and dynamics. After
yielding, many soft materials recover their quiescent properties over time as
they age. There remains, however, a lack of understanding of the nature of this
recovery. Here, we elucidate the mechanisms by which fibrillar networks restore
their ability to support stress after yielding. Crucially, we observe that the
aging response bifurcates around a critical stress $\sigma_\mathrm{c}$, which
is equivalent to the material yield stress. After an initial yielding event,
fibrillar networks subsequently yield faster and at lower magnitudes of stress.
For stresses $\sigma<\sigma_\mathrm{c}$, the time to yielding increases with
waiting time $t_\mathrm{w}$ and diverges once the network has restored
sufficient entanglement density to support the stress. When $\sigma >
\sigma_\mathrm{c}$, the yield time instead plateaus at a finite value because
the developed network density is insufficient to support the applied stress. We
quantitatively relate the yielding and aging behavior of the network to the
competition between stress-induced disentanglement and dynamic fluctuations of
the fibrils rebuilding the network. The bifurcation in the material response
around $\sigma_c$ provides a new possibility to more rigorously localize the
yield stress in disordered materials with time-dependent behavior.
