  With increasing connectivity among comprising agents or (sub-)systems in
large, complex systems, there is a growing interest in understanding
interdependent security and dealing with inefficiency in security investments.
Making use of a population game model and the well-known Chung-Lu random graph
model, we study how one could encourage selfish agents to invest more in
security by internalizing the externalities produced by their security
investments.
  To this end, we first establish an interesting relation between the local
minimizers of social cost and the Nash equilibria of a population game with
slightly altered costs. Secondly, under a mild technical assumption, we
demonstrate that there exists a unique minimizer of social cost and it
coincides with the unique Nash equilibrium of the population game. This finding
tells us how to modify the private cost functions of selfish agents in order to
enhance the overall security and reduce social cost. In addition, it reveals
how the sensitivity of overall security to security investments of agents
influences their externalities and, consequently, penalties or taxes that
should be imposed for internalization of externalities. Finally, we illustrate
how the degree distribution of agents influences their security investments and
overall security at both the NEs of population games and social optima.
