Optimal Welfare in Noncooperative Network Formation under Attack
📝 Abstract
Communication networks are essential for our economy and our everyday lives. This makes them lucrative targets for attacks. Today, we see an ongoing battle between criminals that try to disrupt our key communication networks and security professionals that try to mitigate these attacks. However, today’s networks, like the Internet or peer-to-peer networks among smart devices, are not controlled by a single authority, but instead consist of many independently administrated entities that are interconnected. Thus, both the decisions of how to interconnect and how to secure against potential attacks are taken in a decentralized way by selfish agents. This strategic setting, with agents that want to interconnect and potential attackers that want to disrupt the network, was captured via an influential game-theoretic model by Goyal, Jabbari, Kearns, Khanna, and Morgenstern (WINE 2016). We revisit this model and show improved tight bounds on the achieved robustness of networks created by selfish agents. As our main result, we show that such networks can resist attacks of a large class of potential attackers, i.e., these networks maintain asymptotically optimal welfare post attack. This improves several bounds and resolves an open problem. Along the way, we show the counter-intuitive result, that attackers that aim at minimizing the social welfare post attack do not actually inflict the greatest possible damage.
💡 Analysis
Communication networks are essential for our economy and our everyday lives. This makes them lucrative targets for attacks. Today, we see an ongoing battle between criminals that try to disrupt our key communication networks and security professionals that try to mitigate these attacks. However, today’s networks, like the Internet or peer-to-peer networks among smart devices, are not controlled by a single authority, but instead consist of many independently administrated entities that are interconnected. Thus, both the decisions of how to interconnect and how to secure against potential attacks are taken in a decentralized way by selfish agents. This strategic setting, with agents that want to interconnect and potential attackers that want to disrupt the network, was captured via an influential game-theoretic model by Goyal, Jabbari, Kearns, Khanna, and Morgenstern (WINE 2016). We revisit this model and show improved tight bounds on the achieved robustness of networks created by selfish agents. As our main result, we show that such networks can resist attacks of a large class of potential attackers, i.e., these networks maintain asymptotically optimal welfare post attack. This improves several bounds and resolves an open problem. Along the way, we show the counter-intuitive result, that attackers that aim at minimizing the social welfare post attack do not actually inflict the greatest possible damage.
📄 Content
We rely on various networks: for communication, for transportation, and also for our energy infrastructure. Given their importance, networks have always been prone to attack and network operators have always tried to secure their networks against possible threats. A prominent example is the use of firewalls in routers to prevent a computer virus spreading in some part of the network from infecting other parts. Also, social distancing and vaccination measures in the past COVID pandemic can be understood as security measures in a (social) network to prevent the spread of a virus.
However, both examples show that in today’s networks there is no central authority that could enforce certain security measures or a certain structure of the network. The use of a firewall or to vaccinate is an individual decision of the participants of the networks. The same holds for the decision of which connections to establish: while a direct connection yields benefit in terms of low latency, it also poses a risk since a possible attack could spread along the created link.
Thus, today’s networks can be better modeled and understood as a complex multi-agent system consisting of strategic smart agents. These agents can be people, routers, smart devices, or simply interacting components of an AI system. Each agent strategically decides on its security measures and on its links it wants to establish towards other agents.
This viewpoint of networks as multi-agent systems of strategic agents has sparked the multifaceted research on game-theoretic network formation models in Economics, Computer Science, and Artificial Intelligence in the last three decades, see e.g. (Papadimitriou 2001;Jackson et al. 2008). In these models, agents that correspond to nodes of a network strategically decide which connections to other nodes to establish. The agents are selfish and act according to some given utility function that encodes the agents’ objectives. In this paper, we focus on the objective of network robustness. Modern communication and infrastructure networks have to cope with hardware failures or even deliberate attacks while still providing a reliable service. For incorporating this, researchers have studied agent-based network formation models where the agents prepare for single-link failures (Bala and Goyal 2003;Meirom, Mannor, and Orda 2015;Chauhan et al. 2016) or try to maximize the obtained min-cut in the created network (Echzell et al. 2020). Also the scenario of fighting a virus that spreads in the created network was considered and this is our main reference point.
In this work we revisit the elegant strategic network formation model with attack and immunization by Goyal, Jabbari, Kearns, Khanna, and Morgenstern (Goyal et al. 2016) that features a smart adversary. Each agent selfishly decides which costly links to form and if to acquire protection from attack. In such an attack, a single node of the network is targeted for infection and then this infection spreads along the subgraph consisting of unprotected nodes. The authors consider three natural types of attack:
• maximum carnage: the attacker targets a node to infect as many nodes as possible, • random attack: the attacker targets an unprotected node uniformly at random, and • maximum disruption: the attacker targets a node to minimize the social welfare post attack.
The utility of uninfected agents is the (expected) number of reachable uninfected nodes post attack, while infected agents have utility zero. The social welfare is the total utility. Goyal et al. (2016) give non-trivial bounds on the social welfare of equilibrium networks for the maximum carnage and the random adversary and they pose the analysis of the maximum disruption attacker as open problem. In this paper, we completely resolve the question of the social welfare by providing tight optimal bounds for all three attackers. Even more, we show that the optimal bound holds for more general class of attackers that subsumes the maximum disruption attacker. In fact, we show that the created equilibrium networks have asymptotically the same social welfare post attack as in a setting without adversary. This highlights that networks that are created in a decentralized way by strategic agents are highly robust against various types of attackers.
Sets & Functions: We use N to denote the set of natural numbers and R + to denote the set of non-negative real numbers. We will refer to the derivative of a discrete function f : N → R + as the function f ′ : n → f (n + 1) -f (n).
The Game: For our network formation game, we mostly use the original notation by Goyal et al. (2016). A game instance is defined by the tuple (n, C E , C I , A), where n is the number of agents (or nodes of the network), the value C E > 1 is the cost at which agents can buy an edge to any other node, the cost for a player to immunize itself is C I > 0, and A describes the attacker (or opponent) targeting the created network. We use [n] := {1, . .
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