Non-Adaptive Learning of Sparse Erd\H{o}s--R\'enyi Graphs via Affine Splitting

2026-09-27

AI Summary

This study addresses the problem of efficiently and exactly reconstructing the edge structure of sparse Erdős-Rényi graphs under non-adaptive queries. To this end, it proposes a non-adaptive learning scheme based on affine splitting, which integrates random affine hashing over finite fields with binary splitting techniques. Specifically, queries are designed via a direct whole-graph splitting strategy, and the overall decoding complexity is rigorously bounded. The proposed method achieves exact recovery with high probability using O(k log n) queries and matching time complexity, demonstrating significant improvements over existing approaches in both query efficiency and computational overhead.

Abstract

Graph learning from edge-detecting queries concerns the reconstruction of an unknown edge set on a known vertex set. Each query reports whether a specified vertex subset contains at least one edge. We study non-adaptive schemes, in which all queries are fixed before any outcomes are observed, with the goal of achieving exact recovery using few queries and fast decoding. For general graphs on $n$ vertices with at most $k$ edges, non-adaptive recovery requires $\Omega(\min\{k^2\log n,n^2\})$ queries in the worst case, even when a small error probability is allowed. In this paper, we consider Erd\H{o}s--R\'enyi ($\mathrm{ER}$) graphs $G\sim \mathrm{ER}(n,q)$, with expected edge count $\bar{k}=q\binom{n}{2}$. Our scheme uses $O(\bar{k}\log n)$ queries and achieves exact recovery in $O(\bar{k}\log n)$ decoding time with probability tending to one throughout the regime $\bar{k}\to\infty$ and $\bar{k}=o(n^2)$. This improves the previous $O(\bar{k}^{1+\delta}\log n)$ decoding guarantee for any fixed $\delta>0$, while maintaining the same query order. The guarantee also extends beyond the previously studied regime $\bar{k}=\Theta(n^{2\theta})$ with fixed $\theta\in(0,1)$. Our approach builds on the binary splitting method used in prior work, which organizes vertices into a hierarchy of successively smaller groups. We introduce three main changes: (i) we use random affine hash functions over a finite field to process each candidate pair in constant time; (ii) we apply the splitting procedure directly to the full graph, avoiding the need to combine solutions to multiple smaller graph-learning subproblems; and (iii) we bound the total decoding workload directly rather than deriving separate high-probability bounds on candidate counts at each level.

Problem

• Graph learning
• Non-adaptive recovery
• Erdős–Rényi graphs
• Edge-detecting queries
• Sparse graphs

Innovation

• Non-adaptive graph learning
• Sparse Erdős–Rényi graphs
• Affine splitting
• Random affine hashing
• Exact recovery