[Paper Review] Random Regular Graphs are not Asymptotically Gromov Hyperbolic

Gabriel H. Tucci|arXiv (Cornell University)|Mar 22, 2012

Geometric and Algebraic Topology7 references3 citations

TL;DR

This paper demonstrates that random $d$-regular graphs with $d \geq 3$ are not asymptotically Gromov hyperbolic, as they fail the four-point condition for any fixed $\delta \geq 0$ almost surely as $n \to \infty$. The authors establish that traffic congestion under geodesic routing scales as $O(n \log_{d-1}^3 n)$, significantly below the $\Theta(n^2)$ scaling expected in hyperbolic graphs, indicating a non-hyperbolic structure despite high expansion properties.

ABSTRACT

In this paper we prove that random $d$--regular graphs with $d\geq 3$ have traffic congestion of the order $O(n\log_{d-1}^{3}(n))$ where $n$ is the number of nodes and geodesic routing is used. We also show that these graphs are not asymptotically $δ$--hyperbolic for any non--negative $δ$ almost surely as $n o\infty$.

Motivation & Objective

• To investigate whether random $d$-regular graphs with $d \geq 3$ are asymptotically Gromov hyperbolic.
• To analyze the maximum vertex congestion under geodesic routing in such graphs.
• To determine whether the observed congestion scaling aligns with the $\Theta(n^2)$ behavior expected in hyperbolic networks.
• To establish that these graphs violate the four-point condition for any fixed $\delta \geq 0$ asymptotically almost surely.

Proposed method

• Use of the four-point condition to test Gromov hyperbolicity, where a graph is $\delta$-hyperbolic if $d(x_1,x_3) + d(x_2,x_4) \leq \max\{d(x_1,x_2)+d(x_3,x_4), d(x_1,x_4)+d(x_2,x_3)\} + 2\delta$ for all quadruples of vertices.
• Leverage the existence of almost geodesic cycles in random $d$-regular graphs, with cycle length $|C| = 2\log_{d-1}n + O(\omega(n))$ and error $e(n) = \log_{d-1}\log_{d-1}n + \omega(n)$.
• Construct a quadruple of vertices on such a cycle with distances approximately $|C|/4$ apart to test the four-point condition.
• Bound the geodesic distances in the graph using the cycle distance and the error $e(n)$, showing that $d(x_1,x_3) + d(x_2,x_4)$ exceeds the other sum by $\log_{d-1}n - 2\log_{d-1}\log_{d-1}n + O(\omega(n))$.
• Apply concentration bounds and diameter estimates from [5] to show that the diameter $D(G) \leq \log_{d-1}n + \log_{d-1}\log_{d-1}n + C$ a.s.
• Use Lemma 2.3 to bound vertex congestion $T(v) \leq d^D D^2$, leading to the congestion scaling $M_n = O(n \log_{d-1}^3 n)$.

Experimental results

Research questions

• RQ1 Do random $d$-regular graphs with $d \geq 3$ satisfy the four-point condition for any fixed $\delta \geq 0$ as $n \to \infty$?
• RQ2 What is the asymptotic scaling of maximum vertex congestion under geodesic routing in random $d$-regular graphs?
• RQ3 Is the observed congestion scaling consistent with the $\Theta(n^2)$ behavior expected in Gromov hyperbolic graphs?
• RQ4 Do random $d$-regular graphs contain almost geodesic cycles of length $\Theta(\log n)$ with small distortion?

Key findings

• Random $d$-regular graphs with $d \geq 3$ are not asymptotically $\delta$-hyperbolic for any $\delta \geq 0$, almost surely as $n \to \infty$.
• The maximum vertex congestion under geodesic routing scales as $O(n \log_{d-1}^3 n)$, which is sub-quadratic and significantly less than $\Theta(n^2)$.
• The congestion scaling is derived from a diameter bound $D(G) \leq \log_{d-1}n + \log_{d-1}\log_{d-1}n + C$ holding with high probability.
• The violation of the four-point condition is demonstrated by constructing a quadruple of vertices where $d(x_1,x_3) + d(x_2,x_4)$ exceeds the other sum by $\log_{d-1}n - 2\log_{d-1}\log_{d-1}n + O(\omega(n))$, which tends to infinity.
• The existence of almost geodesic cycles with $|C| = 2\log_{d-1}n + O(\omega(n))$ and $e(n) = \log_{d-1}\log_{d-1}n + \omega(n)$ is used to construct the counterexample to hyperbolicity.
• Despite having a large spectral gap and being good expanders, random $d$-regular graphs do not exhibit the high congestion typical of hyperbolic networks.