  It is believed that, much like a cat's cradle, the cytoskeleton can be
thought of as a network of strings under tension. We show that both regular and
random bond-disordered networks having bonds that buckle upon compression
exhibit a variety of phase transitions as a function of temperature and
extension. The results of self-consistent phonon calculations for the regular
networks agree very well with computer simulations at finite temperature. The
analytic theory also yields a rigidity onset (mechanical percolation) and the
fraction of extended bonds for random networks. There is very good agreement
with the simulations by Delaney et al. (Europhys. Lett. 2005). The mean field
theory reveals a nontranslationally invariant phase with self-generated
heterogeneity of tautness, representing ``antiferroelasticity''.
