The Strategic Formation of Multi-Layer Networks

TL;DR

This paper investigates how multiple layers of networks form strategically, analyzing the complexity of optimal design and showing that certain common network structures like hub-and-spoke emerge as Nash equilibria.

Contribution

It generalizes distance-based network formation to multi-layer settings, characterizes properties of optimal networks, and models multi-layer formation as a game with strategic substitutes.

Findings

Optimal multi-layer networks are NP-hard to design.

Low-cost players tend to dominate high-cost players in the game.

Hub-and-spoke networks naturally arise as Nash equilibria.

Abstract

We study the strategic formation of multi-layer networks, where each layer represents a different type of relationship between the nodes in the network and is designed to maximize some utility that depends on the topology of that layer and those of the other layers. We start by generalizing distance-based network formation to the two-layer setting, where edges are constructed in one layer (with fixed cost per edge) to minimize distances between nodes that are neighbors in another layer. We show that designing an optimal network in this setting is NP-hard. Despite the underlying complexity of the problem, we characterize certain properties of the optimal networks. We then formulate a multi-layer network formation game where each layer corresponds to a player that is optimally choosing its edge set in response to the edge sets of the other players. We consider utility functions that view the different layers as strategic substitutes. By applying our results about optimal networks, we show that players with low edge costs drive players with high edge costs out of the game, and that hub-and-spoke networks that are commonly observed in transportation systems arise as Nash equilibria in this game.