Swap Equilibria under Link and Vertex Destruction

We initiate the study of the \emph{destruction model} (\aka \emph{adversary model}) introduced by Kliemann (2010), using the stability concept of \emph{swap equilibrium} introduced by Alon et. al (2010). The destruction model is a network formation game incorporating the robustness of a network under a more or less targeted attack. In addition to bringing in the swap equilibrium (SE) concept, we extend the model from an attack on the edges of the network to an attack on its vertices. Vertex destruction can generally cause more harm and tends to be more difficult to analyze. We prove structural results and linear upper bounds or super-linear lower bounds on the social cost of SE under different attack scenarios. The most complex case is when the vertex to be destroyed is chosen uniformly at random from the set of those vertices where each causes a maximum number of player pairs to be separated (called a max-sep vertex). We prove a lower bound on the social cost of $\Omega(n^{3/2})$ for this case and initiate an understanding of the structural properties of SE in this scenario. Namely, we prove that there is no SE that is a tree and has only one max-sep vertex. We conjecture that this result can be generalized, in particular we conjecture that there is no SE that is a tree. On the other hand, we prove that if the vertex to be destroyed is chosen uniformly at random from the set of \emph{all} vertices, then each SE is a tree (unless it is two-connected). Our conjecture would imply that moving from the uniform probability measure to a measure concentrated on the max-sep vertices, means moving from no SE having a cycle (unless two-connected) to each SE having a cycle. This would ask for a more detailed study of this transition in future work.