Stable and efficient networks with neighborhood-influenced externalities

This paper analyzes the incentives of individuals to add or sever links that imply the formation of stable and efficient networks when a society is partitioned into groups. In terms of group partitioning, we determine cost topology by arranging a model in which a pair of players pays equally for the link connecting them and in which such a cost depends on the neighborhood composition of the pair when they belong to different groups. To be more precise, the cost of a link between players can be reduced if at least one of these players has neighbors from the group the other player belongs to. We examine specific network structures (i.e., minimal networks, minimally connected networks, complete networks, majority complete networks, and complete bipartite networks) when they are stable and efficient. Our analysis demonstrates how players' distribution among groups modifies the conditions of stability and efficiency. More significantly, we identify some fascinating phenomena which sharply contrast with most literature dealing with stable and efficient networks: (i) the nonminimal network can be stable in the absence of a benefit decay through the path; (ii) a player may prefer to link with players in other groups with a higher average link cost abandoning connection with the partners from her own group; (iii) it is impossible to ensure that the complete network will be efficient for partition with certain characteristics irrespective of the decay factor and the value of costs. The numerical examples are provided to illustrate our theoretical findings.