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Colour ratio in Prim's ranking of bipartite graphs

Félix Kahane, Minmin Wang

TL;DR

This work analyzes Prim's algorithm on the random weighted complete bipartite graph $K_{n_b,n_w}$ with i.i.d. edge weights, focusing on the black-vertex proportion $\rho^{(n)}_k$ within the Prim ranking. It identifies two asymptotic regimes as $n,k\to\infty$: a sublinear regime where $k=o(n)$ and a linear regime where $k\sim sn$, deriving distinct limiting laws that typically differ from the global black fraction $\theta$ unless $\theta=1/2$. The analysis hinges on a bond-percolation coupling and a two-type branching process that characterise the giant-component structure, enabling precise limit formulas via invasion-percolation intuition. The results illuminate how local growth dynamics in Prim’s order diverge from global colour proportions and provide a rigorous framework for understanding colour-ratio evolution in growing random graphs with bipartite structure.

Abstract

We consider a complete bipartite graph of size $n$ endowed with i.i.d. uniform edge weights and run Prim's Algorithm to obtain a ranking of its vertices. Let $ρ^{(n)}_k$ be the proportion of black vertices among the first $k$ vertices in this ranking. We characterise the limit behaviour of $ρ^{(n)}_k$ as both $n$ and $k$ tend to infinity. Our results show that in general the limit of $ρ^{(n)}_k$, when existing, differs from the overall proportion of the black vertices in the graph.

Colour ratio in Prim's ranking of bipartite graphs

TL;DR

This work analyzes Prim's algorithm on the random weighted complete bipartite graph with i.i.d. edge weights, focusing on the black-vertex proportion within the Prim ranking. It identifies two asymptotic regimes as : a sublinear regime where and a linear regime where , deriving distinct limiting laws that typically differ from the global black fraction unless . The analysis hinges on a bond-percolation coupling and a two-type branching process that characterise the giant-component structure, enabling precise limit formulas via invasion-percolation intuition. The results illuminate how local growth dynamics in Prim’s order diverge from global colour proportions and provide a rigorous framework for understanding colour-ratio evolution in growing random graphs with bipartite structure.

Abstract

We consider a complete bipartite graph of size endowed with i.i.d. uniform edge weights and run Prim's Algorithm to obtain a ranking of its vertices. Let be the proportion of black vertices among the first vertices in this ranking. We characterise the limit behaviour of as both and tend to infinity. Our results show that in general the limit of , when existing, differs from the overall proportion of the black vertices in the graph.
Paper Structure (18 sections, 31 theorems, 189 equations, 1 figure)

This paper contains 18 sections, 31 theorems, 189 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Discussion.
  3. Organisation of the paper.
  4. Notation.
  5. Graph exploration in Prim's ranking
  6. Vertex ordering and graph exploration order
  7. A 2-neighbourhood exploration of $G(n_b, n_w, p)$
  8. Bond percolation of the complete bipartite graph
  9. A two-type branching process
  10. Size of the giant component
  11. Proof in the linear regime
  12. Distribution of the neighbourhoods
  13. Proof in the sublinear regime
  14. An overview of the proof in the sublinear regime
  15. The dual problem.
  16. ...and 3 more sections

Key Result

Theorem 1

Assume that hyp: theta holds. Denote $\gamma_\theta = \sqrt{\frac{1-\theta}{\theta}}$. Let $(\kappa_n)_{n\ge 1}$ be a sequence of positive integers that tends to infinity and satisfies $\kappa_n/n\to 0$ as $n\to\infty$. Then

Figures (1)

  • Figure 1: Plot of the function $s\mapsto \rho\circ \ell^{-1}(s)$ in the cases (a) $\theta=0.1$ and (b) $\theta=0.7$.

Theorems & Definitions (63)

  • Theorem 1: Sublinear regime
  • Theorem 2: Linear regime
  • Remark 1: Influence of the starting vertex
  • Remark 2: Interpretation in invasion percolation
  • Definition 1: Graph exploration order (GEO)
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 53 more