Networks in which the formation of connections is governed by a random
process often undergo a percolation transition, wherein around a critical
point, the addition of a small number of connections causes a sizable
fraction of the network to suddenly become linked together. Typically such
transitions are continuous, so that the percentage of the network linked
together tends to zero right above the transition point. I will discuss
how incorporating a limited amount of choice in the classic Erdos-Renyi
network formation model allows us to enhance or delay the onset of
large-scale connectivity and can even cause the percolation transition to
become discontinuous.  In the latter case, despite the discontinuity, the
transition is accompanied by diverging correlation lengths, similar to
phase transitions observed in NP-complete problems in computer science and
most recently observed for models of jamming in granular materials.