Abstract
Strategic network formation arises in settings where agents receive some benefit from their connectedness to other agents, but also incur costs for forming these links. We consider a new network formation game that incorporates an adversarial attack, as well as immunization or protection against the attack. An agent’s network benefit is the expected size of her connected component post-attack, and agents may also choose to immunize themselves from attack at some additional cost. Our framework can be viewed as a stylized model of settings where reachability rather than centrality is the primary interest (as in many technological networks such as the Internet), and vertices may be vulnerable to attacks (such as viruses), but may also reduce risk via potentially costly measures (such as an anti-virus software).
Our main theoretical contributions include a strong bound on the edge density at equilibrium. In particular, we show that under a very mild assumption on the adversary’s attack model, every equilibrium network contains at most only \(2n-4\) edges for \(n \ge 4\), where n denotes the number of agents and this upper bound is tight. We also show that social welfare does not significantly erode: every non-trivial equilibrium with respect to several adversarial attack models asymptotically has social welfare at least as that of any equilibrium in the original attack-free model.
We complement our sharp theoretical results by a behavioral experiment on our game with over 100 participants, where despite the complexity of the game, the resulting network was surprisingly close to equilibrium.
The full version of this paper with all the omitted details is available at https://arxiv.org/abs/1511.05196.
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Notes
- 1.
The spread of the initial attack to reachable non-immunized vertices is deterministic in our model, and the protection of immunized vertices is absolute. It is also natural to consider relaxations such as probabilistic attack spreading and imperfect immunization, as well as generalizations such as multiple initial attack vertices. However, as we shall see, even the basic model we study here exhibits substantial complexity. We refer the reader to the full version for a discussion on possible extensions/relaxations.
- 2.
The index \(k'\) in the definition of \(\mathcal {T}\) satisfies \(k'\le k\) (see k in the definition of \(\mathcal {V}\)).
- 3.
If a vertex is killed, the size of her connected component is defined to be 0.
- 4.
Lenzner [17] introduced this equilibrium concept under the name greedy equilibrium.
- 5.
Vertex v is k-critical in a connected network if the size of the largest connected component after removing v is k.
- 6.
We represent immunized and vulnerable vertices as blue and red, respectively. Although we treat the networks as undirected (the benefits and risks are bilateral), we use directed edges in some figures to denote which player purchased the edge.
- 7.
- 8.
We view this condition as the most interesting regime, since in natural circumstances we do not expect the edge or immunization costs to grow with the population size.
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Acknowledgments
We thank Chandra Chekuri, Yang Li and Aaron Roth for useful suggestions. We also thank anonymous reviewers for detailed suggestions regarding some of the proofs. Sanjeev Khanna is supported in part by National Science Foundation grants CCF-1552909, CCF-1617851, and IIS-1447470.
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Goyal, S., Jabbari, S., Kearns, M., Khanna, S., Morgenstern, J. (2016). Strategic Network Formation with Attack and Immunization. In: Cai, Y., Vetta, A. (eds) Web and Internet Economics. WINE 2016. Lecture Notes in Computer Science(), vol 10123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54110-4_30
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