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On Unique Neighborhoods in Bipartite and Expander Graphs

Stefan Rass

TL;DR

This paper studies the unique neighborhoods (UNN) property, where no neighborhood is a subset of another node's neighborhood, in two graph classes: random bipartite graphs and expander graphs. It proves that balanced random bipartite graphs with a fixed edge probability $p$ are almost surely in $\mathrm{UNN}$ as the graph grows, using a matrix formulation $B_{ij}=(\mathbf A\mathbf 1_{N\times N}-\mathbf A^2)_{ij}$. It also shows that UNN is logically independent of the Cheeger constant $h(G)$ for expanders, by constructing both UNN and non-UNN expanders across ranges of $h(G)$ and by demonstrating that 2-lifts preserve UNN. The results clarify how node-identity via neighborhoods interacts with graph sparsity and expansion, with implications for network design and identity schemes based on neighborhood structure.

Abstract

An undirected graph is said to have \emph{unique neighborhoods} if any two distinct nodes have also distinct sets of neighbors. In this way, the connections of a node to other nodes can characterize a node like an "identity", irrespectively of how nodes are named, as long as two nodes are distinguishable. We study the uniqueness of neighborhoods in (random) bipartite graphs, and expander graphs.

On Unique Neighborhoods in Bipartite and Expander Graphs

TL;DR

This paper studies the unique neighborhoods (UNN) property, where no neighborhood is a subset of another node's neighborhood, in two graph classes: random bipartite graphs and expander graphs. It proves that balanced random bipartite graphs with a fixed edge probability are almost surely in as the graph grows, using a matrix formulation . It also shows that UNN is logically independent of the Cheeger constant for expanders, by constructing both UNN and non-UNN expanders across ranges of and by demonstrating that 2-lifts preserve UNN. The results clarify how node-identity via neighborhoods interacts with graph sparsity and expansion, with implications for network design and identity schemes based on neighborhood structure.

Abstract

An undirected graph is said to have \emph{unique neighborhoods} if any two distinct nodes have also distinct sets of neighbors. In this way, the connections of a node to other nodes can characterize a node like an "identity", irrespectively of how nodes are named, as long as two nodes are distinguishable. We study the uniqueness of neighborhoods in (random) bipartite graphs, and expander graphs.
Paper Structure (3 sections, 8 theorems, 4 equations, 2 figures, 1 table)

This paper contains 3 sections, 8 theorems, 4 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. UNN in Random Bipartite Graphs
  3. UNN in Expander Graphs

Key Result

Theorem 1

Let a random bipartite graph $G_{n,m}$ be given with $n=\left|V_1\right|, m=\left|V_2\right|$ and $n\in\Theta(m)$. Let the edges of $G$ appear with a probability $p$ that does not depend on $n$ or $m$. Then, as $n+m\to\infty$, $G_{m,n}$ will almost surely be a UNN.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (15)

  • Definition 1: UNN rassTellMeWho2024
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 5 more