A local mean field analysis of security investments in networks

ABSTRACT

Getting agents in the Internet, and in networks in general, to invest in and deploy security features and protocols is a challenge, in particular because of economic reasons arising from the presence of network externalities. Our goal in this paper is to model and investigate the impact of such externalities on security investments in a network.

Specifically, we study a network of interconnected agents subject to epidemic risks such as viruses and worms where agents can decide whether or not to invest some amount to deploy security solutions. We consider both cases when the security solutions are strong (they perfectly protect the agents deploying them) and when they are weak. We make three contributions in the paper. First, we introduce a general model which combines an epidemic propagation model with an economic model for agents which captures network effects and externalities. Second, borrowing ideas and techniques used in statistical physics, we introduce a Local Mean Field (LMF) model, which extends the standard mean-field approximation to take into account the correlation structure on local neighborhoods. Third, we solve the LMF model in a network with externalities, and we derive analytic solutions for sparse random graphs of agents, for which we obtain asymptotic results. We find known phenomena such as free riders and tipping points. We also observe counter-intuitive phenomena, such as increasing the quality of the security technology can result in a decreased adoption of that technology in the network. In general, we find that both situations with strong and weak protection exhibit externalities and that the equilibrium is not socially optimal - therefore there is a market failure. Insurance is one mechanism to address this market failure. In related work, we have shown that insurance is a very effective mechanism [3,4], and argue that using insurance would increase the security in a network such as the Internet.

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INDEX TERMS

Primary Classification:
  J. Computer Applications
  J.4 SOCIAL AND BEHAVIORAL SCIENCES
      Subjects: Economics

Additional Classification:
  C. Computer Systems Organization
  C.4 PERFORMANCE OF SYSTEMS
      Subjects: Modeling techniques

  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.2 Graph Theory
          Subjects: Network problems

General Terms:
Design, Economics, Theory

Keywords:
economics, epidemics, free-rider problem, game theory, price of anarchy, security, tipping