Efficient networks in two-way flow model with player heterogeneity and small decay

This paper examines the trade-off between minimizing link formation costs and minimizing the distance over which benefits must travel in an efficient network. In such networks, a central objective is to keep the total cost of forming links as low as possible. However, efforts to reduce costs can lead to sparser structures in which information or other benefits must traverse longer paths. This increases the likelihood of decay or degradation of value as it flows through the network. In other words, reducing link costs may come at the expense of increased distances between agents, which can lower overall efficiency due to the attenuation of benefits over longer paths. This paper investigates the structure of efficient networks under this trade-off, particularly in the context of two-way information flow and small decay. I show that even when a tension exists between cost and distance, the structure of an efficient network remains relatively compact. Specifically, I prove that the diameter of any efficient network is at most four. Moreover, the network takes the form of a generalized center-sponsored star, where a central group of nodes facilitates indirect connections among others, balancing the need to minimize link costs while maintaining short communication paths to preserve the value of transmitted benefits.