Greedy Routing Reachability Games

TL;DR

The paper analyzes a decentralized network formation game where autonomous agents in a metric space connect to enable greedy routing. It distinguishes directed and undirected edge variants, proving that directed equilibria exist and are socially optimal, with polynomial-time construction in Euclidean spaces, while best-response computations remain NP-hard. For undirected edges, it establishes a tight price of anarchy around 1.75–1.8 in 2D and sub-2 bounds in higher dimensions, and provides a polynomial-time procedure to compute approximate Nash equilibria that outperform the Delaunay triangulation in 2D. Overall, the work links game theory and computational geometry to illuminate how navigable networks form under selfish behavior and offers practical algorithms for near-optimal navigable networks.

Abstract

Today's networks consist of many autonomous entities that follow their own objectives, i.e., smart devices or parts of large AI systems, that are interconnected. Given the size and complexity of most communication networks, each entity typically only has a local view and thus must rely on a local routing protocol for sending and forwarding packets. A common solution for this is greedy routing, where packets are locally forwarded to a neighbor in the network that is closer to the packet's destination. In this paper we investigate a game-theoretic model with autonomous agents that aim at forming a network where greedy routing is enabled. The agents are positioned in a metric space and each agent tries to establish as few links as possible, while maintaining that it can reach every other agent via greedy routing. Thus, this model captures how greedy routing networks are formed without any assumption on the distribution of the agents or the specific employed greedy routing protocol. Hence, it distills the essence that makes greedy routing work. We study two variants of the model: with directed edges or with undirected edges. For the former, we show that equilibria exist, have optimal total cost, and that in Euclidean metrics they can be found efficiently. However, even for this simple setting computing optimal strategies is NP-hard. For the much more challenging setting with undirected edges, we show for the realistic setting with agents in 2D Euclidean space that the price of anarchy is between 1.75 and 1.8 and for higher dimensions it is less than 2. Also, we show that best response dynamics may cycle, but that in Euclidean space almost optimal approximate equilibria can be computed in polynomial time. Moreover, for 2D Euclidean space, these approximate equilibria outperform the well-known Delaunay triangulation.

Figures (5)

• Figure 1: Illustration of greedy routing paths in $\mathbb{R}^2$. The pink and blue $s$-$t$-paths are greedy routing paths, while the purple $s$-$t$-path is not, since $15 = d(s,t) < d(u,t) = \sqrt{12^2+15^2}$. Greedy routing is enabled only for node $s$.
• Figure 2: (right) The nearest neighbor graph of 20 points $\mathcal{P}$ in $\mathbb{R}^2$, (left) The Delaunay triangulation of $\mathcal{P}$.
• Figure 3: The kissing number in $\mathbb{R}^3$ is 12.
• Figure 4: (left) A greedy routing set of agent $u$, (right) the minimum greedy routing set with $\phi(u)=3$ and $\phi^+(u)=1$.
• Figure 11: Illustration of the function $\alpha(u)$.

Theorems & Definitions (28)

• Lemma 1
• Lemma 2
• Remark 1
• Lemma 3
• Lemma 4
• Remark 2
• Theorem 1
• Corollary 1
• Lemma 5
• Theorem 2