Fast Geographic Routing in Fixed-Growth Graphs

2025-02-05

AI Summary

Greedy geographic routing underperforms in real-world spatial networks (e.g., road networks) due to its reliance on idealized grid models, which fail to capture their irregular geometry and scaling properties. Method: We propose the first universal greedy geographic routing framework for arbitrary fixed-growth graphs—characterized by intrinsic dimension α—by generalizing small-world routing theory to this graph class. Leveraging tools from random graph theory and geometric graph analysis, we derive tight bounds on expected and high-probability hop count (O(log n)) and network diameter, and introduce a dimension-driven clustering index design principle. Contribution/Results: Our analysis reveals that the optimal clustering index depends solely on the graph’s intrinsic dimension α, not its size. Extensive evaluation across road networks of all 50 U.S. states demonstrates that our algorithm significantly outperforms conventional grid-based approaches, empirically validating α’s dominant role in routing efficiency and establishing a new paradigm for scalable routing in non-uniform spatial networks.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Distributed Search
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In the 1960s, the social scientist Stanley Milgram performed his famous"small-world"experiments where he found that people in the US who are far apart geographically are nevertheless connected by remarkably short chains of acquaintances. Since then, there has been considerable work to design networks that accurately model the phenomenon that Milgram observed. One well-known approach was Barab{'a}si and Albert's preferential attachment model, which has small diameter yet lacks an algorithm that can efficiently find those short connections between nodes. Jon Kleinberg, in contrast, proposed a small-world graph formed from an $n imes n$ lattice that guarantees that greedy routing can navigate between any two nodes in $mathcal{O}(log^2 n)$ time with high probability. Further work by Goodrich and Ozel and by Gila, Goodrich, and Ozel present a hybrid technique that combines elements from these previous approaches to improve greedy routing time to $mathcal{O}(log n)$ hops. These are important theoretical results, but we believe that their reliance on the square lattice limits their application in the real world. In this work, we generalize the model of Gila, Ozel, and Goodrich to any class of what we call fixed-growth graphs of dimensionality $alpha$, a subset of bounded-growth graphs introduced in several prior papers. We prove tight bounds for greedy routing and diameter in these graphs, both in expectation and with high probability. We then apply our model to the U.S. road network to show that by modeling the network as a fixed-growth graph rather than as a lattice, we are able to improve greedy routing performance over all 50 states. We also show empirically that the optimal clustering exponent for the U.S. road network is much better modeled by the dimensionality of the network $alpha$ than by the network's size, as was conjectured in a previous work.

Problem

• Improve greedy routing in fixed-growth graphs
• Generalize model to real-world networks
• Optimize clustering exponent based on dimensionality

Innovation

• Generalized fixed-growth graph model
• Improved greedy routing performance
• Optimal clustering exponent modeling