Topologies of strategically formed social networks based on a generic value function - Allocation rule model

Abstract A wide variety of game theoretic models have been proposed in the literature to explain social network formation. Topologies of networks formed under these models have been investigated, keeping in view two key properties, namely efficiency and stability. Our objective in this paper is to investigate the topologies of networks formed with a more generic model of network formation. Our model is based on a well known model, the value function – allocation rule model. We choose a specific value function and a generic allocation rule and derive several interesting topological results in the network formation context. A unique feature of our model is that it simultaneously captures several key determinants of network formation such as (i) benefits from immediate neighbors through links, (ii) costs of maintaining the links, (iii) benefits from non-neighboring nodes and decay of these benefits with distance, and (iv) intermediary benefits that arise from multi-step paths. Based on this versatile model of network formation, our study explores the structure of the networks that satisfy one or both of the properties, efficiency and pairwise stability. The following are our specific results: (1) we first show that the complete graph and the star graph are the only topologies possible for non-empty efficient networks; this result is independent of the allocation rule and corroborates the findings of more specific models in the literature. (2) We then derive the structure of pairwise stable networks and come up with topologies that are richer than what have been derived for extant models in the literature. (3) Next, under the proposed model, we state and prove a necessary and sufficient condition for any efficient network to be pairwise stable. (4) Finally, we study topologies of pairwise stable networks in some specific settings, leading to unravelling of more specific topological possibilities.