Fast Decoding for Non-Adaptive Learning of Erdős--Rényi Random Graphs

2025-11-21

AI Summary

This paper studies non-adaptive edge detection in Erdős–Rényi random graphs $ G sim mathcal{ER}(n,q) $, where each query selects a subset of vertices and returns whether at least one edge exists within it. For sparse graphs with expected edge count $ ar{k} = qinom{n}{2} $, we propose a test-and-decode framework based on generalized binary splitting. We prove that $ O(ar{k} log n) $ queries suffice to recover the edge set exactly with asymptotically vanishing error probability. Moreover, decoding time is reduced to $ O(ar{k}^{1+delta} log n) $ for arbitrarily small $ delta > 0 $, achieving— for the first time—optimal query complexity alongside sublinear decoding time. Unlike deterministic graph learning, which faces fundamental lower bounds, our approach leverages the random graph assumption to break those barriers. It is the first non-adaptive scheme that simultaneously attains information-theoretically optimal query count and efficient decoding for Erdős–Rényi graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 (Ωig(min{k^2log n,,n^2}ig)) tests even when a small error probability is allowed. We focus on learning Erdős--Rényi (ER) graphs (GsimER(n,q)) in the non-adaptive setting, where the expected number of edges is (ar{k}=qinom{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(ar{k}log n)) but incurs (Ω(n^2)) decoding time, whereas their proposed sublinear-time algorithm incurs an extra ((log ar{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(ar{k}log n)) tests while attaining decoding time (O(ar{k}^{1+δ}log n)) for any fixed (δ>0).

Problem

• Learning Erdős-Rényi graphs via group queries
• Reducing decoding time while maintaining optimal tests
• Achieving sublinear decoding with vanishing error probability

Innovation

• Fast decoding for learning Erdős-Rényi graphs
• Uses binary splitting approach from group testing
• Achieves near-linear tests and subquadratic decoding time