Online matching on stochastic block model

2025-06-05

AI Summary

This work addresses the lack of rigorous performance analysis for online bipartite matching on heterogeneous social networks—such as graphs exhibiting inter-group affinity. It presents the first systematic study of streaming online matching under the Stochastic Block Model (SBM). We propose two algorithms—Myopic and Balance—and develop a rigorous asymptotic convergence framework by integrating stochastic process analysis, ODE-based dynamical modeling, and differential inclusion theory. Theoretically, we prove that the Myopic algorithm’s matching size converges in probability to the solution of an analytically tractable ODE, with an explicit high-probability error bound; the Balance algorithm converges to the solution set of a tractable differential inclusion. This work extends online matching analysis to structured random graphs for the first time, uncovering an exploration-exploitation trade-off induced by online estimation under latent class-driven connectivity, and enables precise characterization of matching performance in SBM-based heterogeneous networks.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

While online bipartite matching has gained significant attention in recent years, existing analyses in stochastic settings fail to capture the performance of algorithms on heterogeneous graphs, such as those incorporating inter-group affinities or other social network structures. In this work, we address this gap by studying online bipartite matching within the stochastic block model (SBM). A fixed set of offline nodes is matched to a stream of online arrivals, with connections governed probabilistically by latent class memberships. We analyze two natural algorithms: a $ t{Myopic}$ policy that greedily matches each arrival to the most compatible class, and the $ t{Balance}$ algorithm, which accounts for both compatibility and remaining capacity. For the $ t{Myopic}$ algorithm, we prove that the size of the matching converges, with high probability, to the solution of an ordinary differential equation (ODE), for which we provide a tractable approximation along with explicit error bounds. For the $ t{Balance}$ algorithm, we demonstrate convergence of the matching size to a differential inclusion and derive an explicit limiting solution. Lastly, we explore the impact of estimating the connection probabilities between classes online, which introduces an exploration-exploitation trade-off.

Problem

• Analyzing online bipartite matching in heterogeneous graphs using SBM
• Evaluating Myopic and Balance algorithms for stochastic matching convergence
• Exploring exploration-exploitation trade-offs in online connection probability estimation

Innovation

• Online bipartite matching in stochastic block model
• Myopic algorithm with ODE convergence analysis
• Balance algorithm with differential inclusion solution