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Highway Preferential Attachment Models for Geographic Routing

Ofek Gila, Evrim Ozel, Michael T. Goodrich

TL;DR

This work advances theoretical understanding of geographic routing in small-world networks by integrating highway-inspired structures with Kleinberg-like and preferential-attachment mechanisms. It introduces Kleinberg Highway, Randomized Highway, and Windowed NPA models, and provides rigorous, high-probability routing bounds that surpass Kleinberg’s classic $O( obreak )$ results on grid networks while maintaining a constant average degree. The windowed NPA model, in particular, enforces a power-law degree distribution and achieves $O( obreak ^{1+\epsilon} n)$ greedy routing on grids for any $\epsilon>0$, supported by a parallelizable construction and empirical validation on road networks. These results offer a closer theoretical bridge between social-network-inspired routing and real-world geographic networks, with practical implications for efficient decentralized routing in road-like topologies.

Abstract

In the 1960s, the world-renowned social psychologist Stanley Milgram conducted experiments that showed that not only do there exist ``short chains'' of acquaintances between any two arbitrary people, but that these arbitrary strangers are able to find these short chains. This phenomenon, known as the \emph{small-world phenomenon}, is explained in part by any model that has a low diameter, such as the Barabási and Albert's \emph{preferential attachment} model, but these models do not display the same efficient routing that Milgram's experiments showed. In the year 2000, Kleinberg proposed a model with an efficient $\mathcal{O}(\log^2{n})$ greedy routing algorithm. In 2004, Martel and Nguyen showed that Kleinberg's analysis was tight, while also showing that Kleinberg's model had an expected diameter of only $Θ(\log{n})$ -- a much smaller value than the greedy routing algorithm's path lengths. In 2022, Goodrich and Ozel proposed the \emph{neighborhood preferential attachment} model (NPA), combining elements from Barabási and Albert's model with Kleinberg's model, and experimentally showed that the resulting model outperformed Kleinberg's greedy routing performance on U.S. road networks. While they displayed impressive empirical results, they did not provide any theoretical analysis of their model. In this paper, we first provide a theoretical analysis of a generalization of Kleinberg's original model and show that it can achieve expected $\mathcal{O}(\log{n})$ routing, a much better result than Kleinberg's model. We then propose a new model, \emph{windowed NPA}, that is similar to the neighborhood preferential attachment model but has provable theoretical guarantees w.h.p. We show that this model is able to achieve $\mathcal{O}(\log^{1 + ε}{n})$ greedy routing for any $ε> 0$.

Highway Preferential Attachment Models for Geographic Routing

TL;DR

This work advances theoretical understanding of geographic routing in small-world networks by integrating highway-inspired structures with Kleinberg-like and preferential-attachment mechanisms. It introduces Kleinberg Highway, Randomized Highway, and Windowed NPA models, and provides rigorous, high-probability routing bounds that surpass Kleinberg’s classic results on grid networks while maintaining a constant average degree. The windowed NPA model, in particular, enforces a power-law degree distribution and achieves greedy routing on grids for any , supported by a parallelizable construction and empirical validation on road networks. These results offer a closer theoretical bridge between social-network-inspired routing and real-world geographic networks, with practical implications for efficient decentralized routing in road-like topologies.

Abstract

In the 1960s, the world-renowned social psychologist Stanley Milgram conducted experiments that showed that not only do there exist ``short chains'' of acquaintances between any two arbitrary people, but that these arbitrary strangers are able to find these short chains. This phenomenon, known as the \emph{small-world phenomenon}, is explained in part by any model that has a low diameter, such as the Barabási and Albert's \emph{preferential attachment} model, but these models do not display the same efficient routing that Milgram's experiments showed. In the year 2000, Kleinberg proposed a model with an efficient greedy routing algorithm. In 2004, Martel and Nguyen showed that Kleinberg's analysis was tight, while also showing that Kleinberg's model had an expected diameter of only -- a much smaller value than the greedy routing algorithm's path lengths. In 2022, Goodrich and Ozel proposed the \emph{neighborhood preferential attachment} model (NPA), combining elements from Barabási and Albert's model with Kleinberg's model, and experimentally showed that the resulting model outperformed Kleinberg's greedy routing performance on U.S. road networks. While they displayed impressive empirical results, they did not provide any theoretical analysis of their model. In this paper, we first provide a theoretical analysis of a generalization of Kleinberg's original model and show that it can achieve expected routing, a much better result than Kleinberg's model. We then propose a new model, \emph{windowed NPA}, that is similar to the neighborhood preferential attachment model but has provable theoretical guarantees w.h.p. We show that this model is able to achieve greedy routing for any .
Paper Structure (28 sections, 16 theorems, 19 equations, 6 figures)

This paper contains 28 sections, 16 theorems, 19 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Kleinberg's Model
  3. The Neighborhood Preferential Attachment Model
  4. Our Results
  5. Preliminaries
  6. Kleinberg Highway
  7. Results
  8. Randomized Highway
  9. Results
  10. Greedy Routing Sketch
  11. Windowed Neighborhood Preferential Attachment
  12. Results
  13. Efficient Construction
  14. Future Work
  15. Appendix
  16. ...and 13 more sections

Key Result

theorem 1

The expected decentralized routing time in a Kleinberg highway network is $\mathcal{O}(\sqrt{k} + \log^2(n)/k + \log{n})$ for $1 \leq k \leq n^2$ when each node knows the positioning of the highway grid, and $\mathcal{O}(k + \log^2(n)/k)$ otherwise.

Figures (6)

  • Figure 1: The average greedy routing path length of the Kleinberg highway model for different values of parameter $k$.
  • Figure 2: An example of the Kleinberg highway model with $n = 9$, $k = 9$, and $Q = 1/9$. The solid black and curved solid blue lines represent local connections for the entire grid and for the highway grid, respectively. The value of $Q$ was picked such that each highway node has only one long-range connection (represented by the dashed light green directed lines) to make the graph less cluttered. If $Q$ were 1, each highway node would have 9 long-range connections.
  • Figure 3: An example of the randomized highway model with $n = 9$, $k = 9$, and $Q = 1/9$. The solid black and curved solid blue lines represent local connections for the entire grid. In this model, there are no local connections for the highway subgraph. The value of $Q$ was picked such that each highway node has only one long-range connection (represented by the dashed light green directed lines) to make the graph less cluttered. If $Q$ were 1, each highway node would have 9 long-range connections.
  • Figure 4: Comparison of greedy routing times for Kleinberg's model and the windowed NPA model when $Q = 1, \epsilon = 0.5, A = 1.01$. The right plot is in log scale.
  • Figure 5: The greedy routing times for the windowed NPA model on the 50 US states when $Q = 32$, $\epsilon = 0.5$, and $A = 1.01$.
  • ...and 1 more figures

Theorems & Definitions (30)

  • theorem 1
  • theorem 2
  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4
  • lemma 5
  • theorem 3
  • lemma 6
  • proof
  • ...and 20 more