The Unit Acquisition Number of Binomial Random Graphs
Konstantinos Georgiou, Somnath Kundu, Pawel Pralat

TL;DR
This paper studies the unit acquisition number in Erdős-Rényi random graphs, showing that it almost surely becomes 1 exactly when the graph becomes connected, revealing a sharp phase transition.
Contribution
It proves that the unit acquisition number of Erdős-Rényi graphs drops to 1 precisely at the connectivity threshold, establishing a strong link between graph connectivity and acquisition number.
Findings
a_u(G) = 1 at the connectivity threshold
The acquisition number is at least 2 if the graph is disconnected
The result holds asymptotically almost surely
Abstract
Let be a graph in which each vertex initially has weight 1. In each step, the unit weight from a vertex to a neighbouring vertex can be moved, provided that the weight on is at least as large as the weight on . The unit acquisition number of , denoted by , is the minimum cardinality of the set of vertices with positive weight at the end of the process (over all acquisition protocols). In this paper, we investigate the Erd\H{o}s-R\'{e}nyi random graph process , where . We show that asymptotically almost surely right at the time step the random graph process creates a connected graph. Since trivially if the graphs is disconnected, the result holds in the strongest possible sense.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Graph theory and applications · Markov Chains and Monte Carlo Methods
