  We study network formation with n players and link cost \alpha > 0. After the
network is built, an adversary randomly deletes one link according to a certain
probability distribution. Cost for player v incorporates the expected number of
players to which v will become disconnected. We show existence of equilibria
and a price of stability of 1+o(1) under moderate assumptions on the adversary
and n \geq 9.
  As the main result, we prove bounds on the price of anarchy for two special
adversaries: one removes a link chosen uniformly at random, while the other
removes a link that causes a maximum number of player pairs to be separated.
For unilateral link formation we show a bound of O(1) on the price of anarchy
for both adversaries, the constant being bounded by 10+o(1) and 8+o(1),
respectively. For bilateral link formation we show O(1+\sqrt{n/\alpha}) for one
adversary (if \alpha > 1/2), and \Theta(n) for the other (if \alpha > 2
considered constant and n \geq 9). The latter is the worst that can happen for
any adversary in this model (if \alpha = \Omega(1)). This points out
substantial differences between unilateral and bilateral link formation.
