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Achievability of Heterogeneous Hypergraph Recovery from its Graph Projection

Alexander Morgan, Chenghao Guo

TL;DR

A heterogeneous random hypergraph model is formulated, and an achieveability result for recovery of hyperedges from the observed projected graph is provided, based on the idea of selecting maximal cliques of size $d_j$ in the projected graph.

Abstract

We formulate and analyze a heterogeneous random hypergraph model, and we provide an achieveability result for recovery of hyperedges from the observed projected graph. We observe a projected graph which combines random hyperedges across all degrees, where a projected edge appears if and only if both vertices appear in at least one hyperedge. Our goal is to reconstruct the original set of hyperedges of degree $d_j$ for some $j$. Our achievability result is based on the idea of selecting maximal cliques of size $d_j$ in the projected graph, and we show that this algorithm succeeds under a natural condition on the densities. This achievability condition generalizes a known threshold for $d$-uniform hypergraphs with noiseless and noisy projections. We conjecture the threshold to be optimal for recovering hyperedges with the largest degree.

Achievability of Heterogeneous Hypergraph Recovery from its Graph Projection

TL;DR

A heterogeneous random hypergraph model is formulated, and an achieveability result for recovery of hyperedges from the observed projected graph is provided, based on the idea of selecting maximal cliques of size in the projected graph.

Abstract

We formulate and analyze a heterogeneous random hypergraph model, and we provide an achieveability result for recovery of hyperedges from the observed projected graph. We observe a projected graph which combines random hyperedges across all degrees, where a projected edge appears if and only if both vertices appear in at least one hyperedge. Our goal is to reconstruct the original set of hyperedges of degree for some . Our achievability result is based on the idea of selecting maximal cliques of size in the projected graph, and we show that this algorithm succeeds under a natural condition on the densities. This achievability condition generalizes a known threshold for -uniform hypergraphs with noiseless and noisy projections. We conjecture the threshold to be optimal for recovering hyperedges with the largest degree.
Paper Structure (14 sections, 9 theorems, 62 equations, 1 algorithm)

This paper contains 14 sections, 9 theorems, 62 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Results
  3. Notation
  4. Preliminiaries
  5. Outline
  6. Probabilities of Events on $\mathcal{\psi}$
  7. Combinatorial Properties of $g(E, \mathbf{\Delta})$
  8. Proof of the Achievability Result, Theorem \ref{['theorem:maximal_clique_recovery']}
  9. Conclusion
  10. Postponed Proofs
  11. Proof of Lemma \ref{['lemma:binom_union_bound_tight_general']}
  12. Proof of Lemma \ref{['lemma:max_rate_dominates']}
  13. Proof of Lemma \ref{['lemma:optimal_cover_without_entire_clique']}
  14. Proof of Lemma \ref{['lemma:optimal_star_cover']}

Key Result

Theorem 1.1

Consider the model specified in Definiton def:heterogeneous_random_hypergraph. Fix a $j$ with $d_j \ge 3$, and suppose Equation eqn:key_inequality holds for this $j$. Let $\mathcal{H}_j$ be the set of true hyperedges of degree $d_j$. Estimating $\mathcal{H}_j$ by applying Algorithm alg:maximal_cliqu

Theorems & Definitions (19)

  • Definition 1: Heterogeneous Random Hypergraph
  • Theorem 1.1
  • Lemma 3.1
  • Corollary 3.2
  • Lemma 3.3: Max Rate Dominates
  • Lemma 5.1: Implied Hyperedge Probability
  • proof
  • Lemma 5.2: General Implied Hyperedge Probability
  • proof
  • Theorem 5.3
  • ...and 9 more