  We study a network formation game where agents receive benefits by forming
connections to other agents but also incur both direct and indirect costs from
the formed connections. Specifically, once the agents have purchased their
connections, an attack starts at a randomly chosen vertex in the network and
spreads according to the independent cascade model with a fixed probability,
destroying any infected agents. The utility or welfare of an agent in our game
is defined to be the expected size of the agent's connected component
post-attack minus her expenditure in forming connections.
  Our goal is to understand the properties of the equilibrium networks formed
in this game. Our first result concerns the edge density of equilibrium
networks. A network connection increases both the likelihood of remaining
connected to other agents after an attack as well the likelihood of getting
infected by a cascading spread of infection. We show that the latter concern
primarily prevails and any equilibrium network in our game contains only
$O(n\log n)$ edges where $n$ denotes the number of agents. On the other hand,
there are equilibrium networks that contain $\Omega(n)$ edges showing that our
edge density bound is tight up to a logarithmic factor. Our second result shows
that the presence of attack and its spread through a cascade does not
significantly lower social welfare as long as the network is not too dense. We
show that any non-trivial equilibrium network with $O(n)$ edges has
$\Theta(n^2)$ social welfare, asymptotically similar to the social welfare
guarantee in the game without any attacks.
