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Functionality of Random Graphs

John Sylvester, Viktor Zamaraev, Maksim Zhukovskii

TL;DR

This work determines the functionality of random graphs G(n,p) up to a constant factor across the entire range of p. By combining three technical approaches—degeneracy-based bounds, small distinguishing sets, and dominating-set arguments in random bipartite graphs—the authors derive tight upper bounds that peak near p* = √(ln n / n) and match lower bounds in both sparse and dense regimes. The main findings show that fun(G(n,p)) scales as Θ( np p ln(ew)/ln n ) around p = √( ln n /(n w) ) and as Θ( ln(ew)/p ) around p = √( w ln n / n ), with the maximum value Θ(√(n/ln n)) achieved near p*. Functionality thus grows more slowly than many classical width parameters, and the results substantially tighten previous general bounds, revealing a nuanced phase transition in the locality-aware encoding power of random graphs. The paper introduces novel domination bounds in random bipartite graphs and leverages them to bridge subgraph-correctness with global structure, advancing our understanding of adjacency-labeling schemes and graph encodings.

Abstract

The functionality of a graph $G$ is the minimum number $k$ such that in every induced subgraph of $G$ there exists a vertex whose neighbourhood is uniquely determined by the neighborhoods of at most $k$ other vertices in the subgraph. The functionality parameter was introduced in the context of adjacency labeling schemes, and it generalises a number of classical and recent graph parameters including degeneracy, twin-width, and symmetric difference. We establish the functionality of a random graph $G(n,p)$ up to a constant factor for every value of $p$.

Functionality of Random Graphs

TL;DR

This work determines the functionality of random graphs G(n,p) up to a constant factor across the entire range of p. By combining three technical approaches—degeneracy-based bounds, small distinguishing sets, and dominating-set arguments in random bipartite graphs—the authors derive tight upper bounds that peak near p* = √(ln n / n) and match lower bounds in both sparse and dense regimes. The main findings show that fun(G(n,p)) scales as Θ( np p ln(ew)/ln n ) around p = √( ln n /(n w) ) and as Θ( ln(ew)/p ) around p = √( w ln n / n ), with the maximum value Θ(√(n/ln n)) achieved near p*. Functionality thus grows more slowly than many classical width parameters, and the results substantially tighten previous general bounds, revealing a nuanced phase transition in the locality-aware encoding power of random graphs. The paper introduces novel domination bounds in random bipartite graphs and leverages them to bridge subgraph-correctness with global structure, advancing our understanding of adjacency-labeling schemes and graph encodings.

Abstract

The functionality of a graph is the minimum number such that in every induced subgraph of there exists a vertex whose neighbourhood is uniquely determined by the neighborhoods of at most other vertices in the subgraph. The functionality parameter was introduced in the context of adjacency labeling schemes, and it generalises a number of classical and recent graph parameters including degeneracy, twin-width, and symmetric difference. We establish the functionality of a random graph up to a constant factor for every value of .
Paper Structure (31 sections, 18 theorems, 115 equations, 3 figures)

This paper contains 31 sections, 18 theorems, 115 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our Contribution & Discussion
  3. Proof Strategies
  4. General upper bounds (small and large values of $p$).
  5. Upper bounds via dominating sets ($p$ around $p^*$).
  6. Lower bounds.
  7. Related Graph Parameters
  8. Preliminaries
  9. Graphs.
  10. Functionality.
  11. Probability.
  12. Properties of random graphs.
  13. General Upper Bounds
  14. Upper bound via degeneracy.
  15. Upper bound via distinguishing set.
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $G_n \sim G(n,p)$, where $\Omega(1/n)= p:=p(n)\leqslant 1/2$. Then for any $w := w(n)\geqslant 1$, w.h.p.

Figures (3)

  • Figure 1: Qualitative behaviour of the functionality of $G(n,p)$
  • Figure 2: Upper and lower bounds on the functionality of $G(n,p)$. The brown segments show the intervals on which the corresponding bounds hold. The thick brown segments show the intervals where the bounds are order optimal.
  • Figure 3: Illustration of the proof of \ref{['lem:clm1']}. The vertex $y\in [s]$ has a $3$-dominating set for two of the vertices in its neighbourhood.

Theorems & Definitions (48)

  • Theorem 1.1: Main
  • Theorem 1.2
  • Lemma 2.4: JansonBook
  • Lemma 2.5: AlonSpencer
  • Lemma 2.6: JansonTail
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • proof
  • Theorem 3.1
  • ...and 38 more