Network formation with multigraphs and strategic complementarities

This paper examines the formation of one network G when connections in a second network H are inherited under two scenarios: (i) H is asymmetric allowing for a wide range of networks called nested split graphs, and (ii) H is a connected regular graph. The bulk of our paper assumes that both G and H are interdependent because the respective actions in each are (weak) strategic complements. This complementarity creates a “silver spoon” effect whereby those who inherit high Katz-Bonacich centrality in H will continue to have high Katz-Bonacich centrality in G. There is, however, a “silver lining”: depending on the costs of link formation, the formed network G may allow for an improvement in centrality. As an application, we introduce an overlapping generations models to analyze intergenerational social mobility. We show that the silver spoon effect persists across generations with descendants of agents who are highly connected continuing to remain highly connected. Finally, we explore the implications of actions being strategic substitutes across networks. This can lead to an outcome where well-connected agents in H establish no links in G, and those with no connections in H form all the links in G. Our analysis provides insight into preferential attachment, how asymmetries in one network may be magnified or diminished in another, and why players with links in one network may form no links in another network.