Generalized random sequential adsorption on Erd\H{o}s-R\'enyi random graphs

TL;DR

This paper studies a generalized greedy algorithm for random sequential adsorption on Erdős-Rényi graphs, analyzing the limiting behavior of the active vertex proportion in large graphs.

Contribution

It introduces and analyzes three generalizations of the classical RSA rule on Erdős-Rényi graphs, characterizing the asymptotic jamming constant.

Findings

Derived the limiting proportion of active vertices for the generalized rules.

Extended classical RSA analysis to broader local rules on random graphs.

Provided theoretical characterization of the jamming constant in the large-graph limit.

Abstract

We investigate Random Sequential Adsorption (RSA) on a random graph via the following greedy algorithm: Order the $n$ vertices at random, and sequentially declare each vertex either active or frozen, depending on some local rule in terms of the state of the neighboring vertices. The classical RSA rule declares a vertex active if none of its neighbors is, in which case the set of active nodes forms an independent set of the graph. We generalize this nearest-neighbor blocking rule in three ways and apply it to the Erd\H{o}s-R\'enyi random graph. We consider these generalizations in the large-graph limit $n\to\infty$ and characterize the jamming constant, the limiting proportion of active vertices in the maximal greedy set.