Fast Geographic Routing in Fixed-Growth Graphs

TL;DR

Fixed-growth graphs with dimensionality $α$ generalize Kleinberg-like small-worlds to nonlattice networks by embedding a randomized highway layer with long-range edges scaled by $d(u,v)^{-α}$. The authors extend the randomized highway model to FG graphs, establishing tight greedy-routing and diameter bounds, with both expectation and high-probability guarantees, and validate the framework on U.S. road networks showing $α$ better predicts optimal clustering than network size. They demonstrate that for $k ∈ Θ(\log n)$, routing between nodes at distance $d(s,t)=Θ(\sqrt[α]{n})$ can be achieved in $Θ(\log n)$ hops in expectation, with matching high-probability results and corresponding diameter bounds. The work provides a rigorous, scalable theory for fast geographic routing in realistic, nonlattice networks, informing routing strategies and network-design considerations for infrastructure graphs like road networks.

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{á}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 $n \times n$ 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 $α$, 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 $α$ than by the network's size, as was conjectured in a previous work.

Figures (5)

• Figure 1: In these figures we show balls of various widths surrounding some central node $u$ in a lattice graph (left) and a fixed-growth graph (right). Note how balls grow far more predictably in lattices than they do in general fixed-growth graphs. We refer to the shaded regions in between two balls as shells, and the difference in radius between two balls as the shell width $w$.
• Figure 2: A visualization of the two sets $S$ and $T$. $S$ is the set of highway nodes that can be efficiently reached from the source $s$, while $T$ is the set of highway nodes that can efficiently reach the destination $t$. We show that these two sets intersect with high probability, and thus $s$ can route to $t$ efficiently.
• Figure 3: Here we compare the average greedy routing performance of randomized highway graphs constructed from the road networks of all 50 U.S. states and DC using different clustering coefficients $s$. Specifically, we compare the performance for when $s$ is set to 2 (blue), which is the value used by previous papers who assume that the road networks behave like a 2-dimensional lattice, to when $s$ is set to $\alpha$ (orange), the estimated dimensionality of the graph assuming that the road networks behave like fixed-growth graphs. Modeling them as fixed-growth graphs always provides better greedy routing performance, as evident by their difference (green) always being positive.
• Figure 4: Here we compare our estimated dimensionality $\alpha$, as used in \ref{['fig:greedy_routing_comp']}, to the empirically determined optimal clustering exponent $s$ for each state. There is a clear correlation between our values of $s$ and $\alpha$, and importantly, none of the optimal clustering coefficients are close to two, the assumed dimensionality in previous work gila2023highwaygoodrich2022modelingkleinberg2000small. This suggests that our fixed-growth model is able to take advantage of the intrinsic growth rate of these graphs in a way that abstracting them to a lattice does not.
• Figure 5: A comparison between the dimensionality $\alpha$ (left) and the number of nodes $n$ (right) as indicators for the optimal clustering coefficient $s$. It is clear that the dimensionality $\alpha$ is a much better indicator than the network size.

Theorems & Definitions (25)

• lemma thmcounterlemma
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• lemma thmcounterlemma
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• lemma thmcounterlemma
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• lemma thmcounterlemma
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• theorem thmcountertheorem
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