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Correlated Stochastic Block Models and Graph Matching

Kazemi, Hadi | 2024

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 57403 (05)
  4. University: Sharif University of Technology
  5. Department: Electrical Engineering
  6. Advisor(s): Yassaee, Mohammad Hossein
  7. Abstract:
  8. Stochastic block models are the most common statistical models for simulating graphs with block structures. These models have been studied extensively for community detection problem and evaluating the performance of algorithms for community detection. Commu- nity detection has various applications in the study of social networks, protein networks, image processing, and natural language processing. In the classical setting, community detection is studied when only one graph is available. In 2021, a model for correlated stochastic block graphs was introduced, in which multiple edge-correlated graphs with the same block structure are observed. In this model, node labels are not available, making it suitable for simulating scenarios where user privacy is important or direct access to correspondence between nodes is not feasible. The exact community detection problem for this novel model has been fully explored in two recent papers. In this work, we investigate the weak community detection problem for correlated stochastic block models. We focus on the constant average degree regime. Given that most natural graphs exhibit constant average degree, studying this regime is crucial. We introduce a new model called “broadcasting on correlated trees” and use it to demonstrate that in the sense of weak recovery, the presence of two correlated graphs does not change the statistical threshold compared to the single-graph case. Furthermore, based on ideas from “broadcasting on correlated trees,” we define a hypothesis testing problem related to the graph matching of correlated stochastic block models. While we establish an infeasibility result for a restricted range of parameters, this hypothesis testing problem remains widely open and requires further research. We also provide an information-theoretic impossibility result for the graph matching problem in correlated stochastic block models
  9. Keywords:
  10. Graph Matching ; Broadcasting on Correlated Trees (BOCT) ; Broadcasting on Trees (BOT) ; Stochastic Block Models (SBM) ; Correlated Stochastic Block Models (CSBM) ; Exact Community Detection ; Weak Community Detection

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  • مقدمه
    • مدل‌های بلوکی تصادفی
    • تطابق گراف‌ها
    • انتشار بر روی درخت‌ها
    • مدل‌های بلوکی تصادفی همبسته
    • ساختار پایان‌نامه
  • انتشار بر روی درخت‌ها
    • معرفی مسئله‌ی انتشار بر روی درخت‌ها و نتایج موجود
    • اثبات اول: نشت اطلاعات
    • اثبات دوم: درخت گیسویی!
  • مدل‌های بلوکی تصادفی
    • انواع بازیابی
    • کران عدم دستیابی بازیابی ضعیف
  • تطابق گراف‌ها
    • مسئله‌ی بازیابی جایگشت
    • تشخیص همبستگی بین درخت‌ها
  • بازیابی ضعیف در مدل‌های بلوکی تصادفی همبسته
    • بیان قضیه‌ی بازیابی ضعیف در مدل‌های بلوکی تصادفی همبسته
    • اثبات به کمک انتشار بر روی درخت‌های همبسته
      • جفت‌شدگی بین همسایگی‌های محلی مدل بلوکی تصادفی همبسته و مدل انتشار بر روی درخت‌های همبسته
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      • اثبات با اطلاعات بادآورده! این عبارت تعبیری فارسی برای عبارت Genie based proof است.
  • تطابق گراف‌ها در مدل‌های بلوکی تصادفی همبسته
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