A bounded-degree network formation game

TL;DR

This paper studies a network formation game with bounded out-degree, analyzing the existence, construction, and properties of pure Nash equilibria, especially in uniform weight scenarios, and explores convergence behaviors and special graph structures.

Contribution

It introduces the Bounded Degree Network Formation game, proves NP-hardness of equilibrium existence, provides constructions for pure Nash equilibria in uniform cases, and analyzes convergence and structural properties.

Findings

Pure Nash equilibria may not exist with arbitrary weights.

Constructed pure Nash equilibria for uniform weight cases.

Diameter of equilibria is bounded by $O(\sqrt{n \log_k n})$.

Abstract

Motivated by applications in peer-to-peer and overlay networks we define and study the \emph{Bounded Degree Network Formation} (BDNF) game. In an $(n,k)$-BDNF game, we are given $n$ nodes, a bound $k$ on the out-degree of each node, and a weight $w_{vu}$ for each ordered pair $(v,u)$ representing the traffic rate from node $v$ to node $u$. Each node $v$ uses up to $k$ directed links to connect to other nodes with an objective to minimize its average distance, using weights $w_{vu}$, to all other destinations. We study the existence of pure Nash equilibria for $(n,k)$-BDNF games. We show that if the weights are arbitrary, then a pure Nash wiring may not exist. Furthermore, it is NP-hard to determine whether a pure Nash wiring exists for a given $(n,k)$-BDNF instance. A major focus of this paper is on uniform $(n,k)$-BDNF games, in which all weights are 1. We describe how to construct a pure Nash equilibrium wiring given any $n$ and $k$, and establish that in all pure Nash wirings the cost of individual nodes cannot differ by more than a factor of nearly 2, whereas the diameter cannot exceed $O(\sqrt{n \log_k n})$. We also analyze best-response walks on the configuration space defined by the uniform game, and show that starting from any initial configuration, strong connectivity is reached within $\Theta(n^2)$ rounds. Convergence to a pure Nash equilibrium, however, is not guaranteed. We present simulation results that suggest that loop-free best-response walks always exist, but may not be polynomially bounded. We also study a special family of \emph{regular} wirings, the class of Abelian Cayley graphs, in which all nodes imitate the same wiring pattern, and show that if $n$ is sufficiently large no such regular wiring can be a pure Nash equilibrium.