Expected Cover and Hitting Times of the Giant Component in Hyperbolic Random Graphs

Core Concepts

The expected cover time, maximum hitting time, and target time (average hitting time) of a simple random walk on the giant component of a hyperbolic random graph are Θ(n log^2 n), Θ(n log n), and Θ(n), respectively, asymptotically almost surely.

Abstract

The paper studies the behavior of simple random walks on the giant component of hyperbolic random graphs (HRGs), which are a model of "real-world" networks such as the Internet. The authors focus on determining the expected cover time, maximum hitting time, and target time (average hitting time) of the random walk on the giant component. Key highlights: The cover time, which is the expected time for the random walk to visit all vertices, is shown to be Θ(n log^2 n) asymptotically almost surely (a.a.s.). The maximum hitting time, which is the maximum expected time for the random walk to hit a given vertex, is Θ(n log n) a.a.s. The target time, which is the expected time for the random walk to travel between two vertices sampled from the stationary distribution, is Θ(n) a.a.s. The authors also provide sharp estimates for the commute time (sum of hitting times in both directions) between pairs of vertices added to the giant component under mild conditions. The proofs rely on carefully designed network flows to bound the effective resistances in the graph, which are then used to derive the results on random walk quantities. The results provide insights into the efficiency of random walks on the giant component of HRGs, which is an important model for complex networks.

Stats

The number of vertices in the giant component of the HRG is Θ(n) a.a.s. The number of edges in the giant component of the HRG is Θ(n) w.h.p.

Quotes

"The random walk is the quintessential random process, and studies of random walks have proven relevant for algorithm design and analysis; this coupled with the aforementioned appealing aspects of the HRG model motivates this research." "Quantities such as average hitting time and commute time are not meaningful for disconnected graphs (i.e., they are trivially equal to infinity). However, again for the range of parameters we are interested in, Bode, Fountoulakis and Müller [12] showed that it is very likely the graph has a component of linear size."

Key Insights Distilled From

Cover and Hitting Times of Hyperbolic Random Graphs

by Marcos Kiwi,... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2207.06956.pdf

Deeper Inquiries

How do the random walk properties on the giant component of HRGs compare to other random graph models, such as Erdős-Rényi or preferential attachment graphs

The random walk properties on the giant component of Hyperbolic Random Graphs (HRGs) exhibit some differences compared to other random graph models like Erdős-Rényi or preferential attachment graphs. In HRGs, the expected hitting time and cover time are determined to be of order n and n(log n)^2, respectively, on the giant component. This is in contrast to Erdős-Rényi graphs, where the hitting time is typically logarithmic in the number of vertices, and preferential attachment graphs, where the hitting time can be influenced by the degree distribution. Additionally, the commute time between vertices in HRGs is shown to be penalized by the lack of expansion in the graph, which is a distinguishing feature compared to other models that exhibit better expansion properties.

What are the implications of the obtained results on the design and analysis of algorithms that utilize random walks on complex networks modeled by HRGs

The results obtained on the behavior of random walks on the giant component of HRGs have significant implications for the design and analysis of algorithms in various applications. Understanding the hitting time, cover time, and commute time on HRGs can provide insights into network search efficiency, proximity of data, and cluster cohesion in complex networks modeled by HRGs. Algorithms that utilize random walks for tasks like load balancing, searching, resource location, property testing, and biological applications can benefit from the findings in this research. By leveraging the knowledge of these random walk properties, algorithms can be optimized for better performance and efficiency in navigating and exploring complex networks represented by HRGs.

Can the techniques developed in this work be extended to study other stochastic processes, such as diffusion or epidemic spreading, on hyperbolic random graphs

The techniques developed in this work for studying random walks on HRGs can be extended to investigate other stochastic processes on hyperbolic random graphs, such as diffusion or epidemic spreading. By adapting the framework used to analyze random walks, researchers can explore the behavior of diffusion processes, where particles spread through the network, or epidemic spreading, where diseases propagate through the graph. The understanding of effective resistances, energy dissipation in network flows, and geometric structures in HRGs can be leveraged to study the dynamics of these stochastic processes and their implications on the network connectivity and spread of information or diseases. This extension can provide valuable insights into the dynamics of complex systems modeled by HRGs.