  We propose a simple model of the evolution of a social network which involves
local search and volatility (random decay of links). The model captures the
crucial role the network plays for information diffusion. This is responsible
for a feedback loop which results in a first-order phase transition between a
very sparse network regime and a highly-connected phase. Phase coexistence and
hysteresis take place for intermediate value of parameters. We derive a
mean-field theory which correctly reproduces this behavior, including the
distribution of degree connectivity and the non-trivial clustering properties.
