  We study network formation with the bilateral link formation rule (Jackson
and Wolinsky 1996) with $n$ players and link cost $\alpha>0$. After the network
is built, an adversary randomly destroys one link according to a certain
probability distribution. Cost for player $v$ incorporates the expected number
of players to which $v$ will become disconnected. This model was previously
studied for unilateral link formation (K. 2011).
  We prove existence of pairwise Nash equilibria 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 destroys a link chosen uniformly at
random, while the other destroys a link that causes a maximum number of player
pairs to be separated. We prove bounds tight up to constants, namely $O(1)$ 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 $\alpha=\Omega(1)$).
