Cover and Hitting Times of Hyperbolic Random Graphs

TL;DR

The paper develops a rigorous electrical-network framework for random walks on the giant component of Hyperbolic Random Graphs in the power-law regime with $\tfrac{1}{2}<\alpha<1$, showing that key walk-related times scale as $t_{\odot}(\mathcal{C})=\Theta(n)$, $t_{hit}(\mathcal{C})=\Theta(n\log n)$, and $t_{cov}(\mathcal{C})=\Theta(n\log^{2}n)$, while the Kirchhoff index satisfies $\mathcal{K}(\mathcal{C})=\Theta(n^{2})$ and the average resistance is $\Theta(1)$. Central to the approach is bounding resistances via carefully designed low-energy flows on a tiling of the hyperbolic plane, overlaid with a forest-like structure, which also yields concentration results. The paper further provides sharp polylogarithmic-bound commute-time estimates for pairs of vertices added at prescribed radii, highlighting how the HRG’s geometry yields non-expansion that nevertheless permits tight control of random-walk quantities. Together, these results advance the understanding of diffusion-like processes on HRGs and have implications for network exploration and routing in complex, heavy-tailed networks.

Abstract

We study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range $(2,3)$. In particular, we first focus on the expected time for a random walk to hit a given vertex or visit, i.e. cover, all vertices. We show that, a.a.s. (with respect to the HRG), and up to multiplicative constants: the cover time is $n(\log n)^2$, the maximum hitting time is $n\log n$, and the average hitting time is $n$. We then determine the expected time to commute between two given vertices a.a.s., up to a small factor polylogarithmic in $n$, and under some mild hypothesis on the pair of vertices involved. Our results are proved by controlling effective resistances using the energy dissipated by carefully designed network flows associated to a tiling of the hyperbolic plane, on which we overlay a forest-like structure.

Figures (4)

• Figure 1: Instances of $\mathrm{\mathcal{G}}_{\alpha,\nu}(n)$ for $n=100$, $\nu = 1.832$, $\alpha=0.6$ (left) and $\alpha=0.9$ (right).
• Figure 2: (a) Partial illustration of a tiling of $B_O(R)$ (not at scale). (b) Illustration of flow between vertices $s$ and $t$ with no common ancestor tile. Coloured regions contain vertices through which flow from $s$ to $t$ is routed. Flow is pushed radially towards the origin $O$ from a yellow shaded tile to its parent half-tile. Flow is pushed in an angular direction from dark to light yellow shaded half-tiles. The direction of flow is reversed in the green shaded region.
• Figure 3: (a) Shaded region corresponds to $\mathcal{R}_a:=(\mathcal{R}_a^{\mathrm{ct}}\cup\mathcal{R}_a^{\mathrm{pt}}\cup\mathcal{R}_a^{\mathrm{bf}})\setminus\mathcal{R}_O$, and (b) hatched region corresponds to band $\mathcal{B}_a$. (Picture not to scale.)
• Figure 4: Picture not to scale. For simplicity, we have illustrated the case where $h_{\ell}=\rho$ and $h_{\ell'}=\rho'$. (a) The lightly shaded region corresponds to band $\mathcal{B}_a$. Each division of the shaded region corresponds to a $\mathcal{B}'_j$. (b) The darker shaded region correspond to $\mathcal{H}_a$, i.e., the union of all half-tiles $H_s^{(i)}$ with $s\in\{\ell,...,\ell'\}$ and $i\in [6]$. (c) Vertex $v_a$ is represented as a star. (d) Vertices in $\mathcal{P}_a$ are represented as a curved segment. The existence of the path represented as a piecewise linear segment guarantees that the path $\mathcal{P}_a$ belongs to the center component.

Theorems & Definitions (109)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 5
• Lemma 6
• proof
• Lemma 7: BerTam
• Lemma 8: FriedrichK18
• Remark 9