On Optimization of Complete Social Networks

A balanced social network is a social network where, for any member of the social network, the following two statements are true; a friend of my friend is my friend and an enemy of my enemy is my friend. In this paper we demonstrate a polynomial time greedy algorithm that balances any complete social network with n members by changing at most ⌈n ² /4 − n/2⌉ of the initial relationships between the members of the network. We also demonstrate that the problem of determining the minimum number of relationships that needs to change so that a complete social network, where each member has at least as many friends as enemies, becomes balanced is still NP-Complete.