[Paper Review] The Jamming Constant of Random Graphs

Paola Bermolen, Matthieu Jonckheere+1 more|arXiv (Cornell University)|Oct 31, 2013

Stochastic processes and statistical mechanics19 references4 citations

TL;DR

This paper introduces a configuration-based approach to model the parking process on random graphs as a measure-valued Markov process, establishing a functional law of large numbers in the infinite-size limit. It characterizes the jamming constant—the limiting density of occupied vertices—for various random graph models, providing a rigorous asymptotic framework for understanding random sequential allocation dynamics.

ABSTRACT

Using a configuration approach, we define the dynamics of the “parking process” on random graphs as a measure-valued Markov process. We then establish a functional law of large numbers when the number of vertices grows to infinity. This allows us to characterize the jamming constant of various random graphs.

Motivation & Objective

• To model the random sequential parking process on random graphs as a stochastic dynamical system.
• To analyze the asymptotic behavior of the parking process as the number of vertices tends to infinity.
• To derive the jamming constant—the limiting fraction of vertices that can be occupied—across different random graph ensembles.
• To establish a functional law of large numbers for the empirical measure of occupied vertices in the large-graph limit.

Proposed method

• Formalizes the parking process on random graphs using a configuration model to define edge and vertex structures.
• Represents the state of the system as a measure-valued Markov process tracking the empirical distribution of occupied vertices.
• Applies a functional law of large numbers to the measure-valued process as the number of vertices grows to infinity.
• Uses the limit process to characterize the jamming constant as the equilibrium density of occupied vertices.
• Applies the framework to diverse random graph models, such as Erdős–Rényi and configuration model graphs.
• Derives the jamming constant through the solution of a system of equations derived from the limiting dynamics.

Experimental results

Research questions

• RQ1 What is the limiting density of occupied vertices in a random graph under the random sequential parking process?
• RQ2 How does the jamming constant depend on the underlying graph structure and degree distribution?
• RQ3 Can a functional law of large numbers be established for the empirical measure of occupied vertices in large random graphs?
• RQ4 What is the asymptotic behavior of the parking process on sparse random graphs with fixed degree sequences?

Key findings

• The jamming constant is characterized as the solution to a system of equations derived from the limiting dynamics of the measure-valued Markov process.
• The functional law of large numbers ensures that the empirical measure of occupied vertices converges to a deterministic limit in the infinite-size regime.
• The jamming constant is strictly less than 1 for all sparse random graphs, indicating that not all vertices can be occupied due to spatial constraints.
• The limiting behavior depends on the degree distribution, with higher variance in degrees leading to lower jamming constants.
• The framework applies uniformly across various random graph models, including Erdős–Rényi and configuration model graphs.
• The method provides a rigorous asymptotic characterization of the parking process, enabling precise prediction of jamming thresholds.