A Fast Binary Splitting Approach for Non-Adaptive Learning of Erdős--Rényi Graphs

TL;DR

This work tackles non-adaptive learning of Erdős–Rényi graphs from edge-detecting queries, where the goal is to recover the edge set with vanishing error while using a near-optimal number of tests. It extends a fast binary splitting approach to graph learning by organizing vertices into a hierarchical block structure and performing randomized tests that preserve high information gain per test. A key innovation is permutation amplification via partitioned subgraphs and shared random permutations, which reduces decoding time from previous $O(\bar{k}^{1.5})$-type bounds while keeping $O(\bar{k}\log n)$ tests. The main results show that one can recover the entire edge set with probability $1-o(1)$ using $O(\bar{k}\log n)$ tests and a decoding time of $O(\bar{k}^{1+\delta}\log n)$ for any fixed $\delta>0$, approaching the best trade-off between test efficiency and computational efficiency in this setting. This has practical impact for efficiently learning sparse networks from indirect queries in large-scale graphs, with potential extensions to hypergraphs and noisy observations.

Abstract

We study the problem of learning an unknown graph via group queries on node subsets, where each query reports whether at least one edge is present among the queried nodes. In general, learning arbitrary graphs with $n$ nodes and $k$ edges is hard in the non-adaptive setting, requiring $Ω\big(\min\{k^2\log n,\,n^2\}\big)$ tests even when a small error probability is allowed. We focus on learning Erdős--Rényi (ER) graphs $G\sim\mathrm{ER}(n,q)$ in the non-adaptive setting, where the expected number of edges is $\bar{k}=q\binom{n}{2}$, and we aim to design an efficient testing--decoding scheme achieving asymptotically vanishing error probability. Prior work (Li--Fresacher--Scarlett, NeurIPS 2019) presents a testing--decoding scheme that attains an order-optimal number of tests $O(\bar{k}\log n)$ but incurs $Ω(n^2)$ decoding time, whereas their proposed sublinear-time algorithm incurs an extra $(\log \bar{k})(\log n)$ factor in the number of tests. We extend the binary splitting approach, recently developed for non-adaptive group testing, to the ER graph learning setting, and prove that the edge set can be recovered with high probability using $O(\bar{k}\log n)$ tests while attaining decoding time $O(\bar{k}^{1+δ}\log n)$ for any fixed $δ>0$.

Figures (1)

• Figure 1: Example tree structures at levels $\ell\in\{0,1,2,3,4\}$ for graph learning with $n=16$ and $k=4$. Defective blocks are colored red, and edges between blocks are indicated with a solid line.

Theorems & Definitions (29)

• Definition 1
• Lemma 1
• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['thm:main']}