Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Effective Index Construction Algorithm for Optimal $(k,η)$-cores Computation

Shengli Sun, Peng Xu, Guanming Jiang, Philip S. Yu, Yi Li

TL;DR

This paper tackles the problem of efficiently and correctly computing online $(k,\eta)$-cores on uncertain graphs, where current UCF-Index constructions suffer from correctness issues due to floating-point divisions in $k$-probability updates. It introduces OptiUCF, a correct index construction framework that combines on-demand $k$-probability recalculation with tight probabilistic bounds (Beta-function and Top-K bounds), a lazy refreshing strategy, and a progressive refinement scheme across $\,\eta$-threshold layers to minimize recomputation and search space. The Baseline algorithm establishes a correct DP-based refresh, while OptiUCF significantly reduces overhead and yields substantial speedups (1–2 orders of magnitude over the baseline, 5–10x over a non-optimized variant) and robust accuracy. Extensive experiments on eight real-world graphs demonstrate improvements in efficiency, accuracy, and scalability, confirming the practical impact for uncertain-graph analysis tasks such as community detection, fraud analysis, and influence maximization.

Abstract

Computing $(k,η)$-cores from uncertain graphs is a fundamental problem in uncertain graph analysis. UCF-Index is the state-of-the-art resolution to support $(k,η)$-core queries, allowing the $(k,η)$-core for any combination of $k$ and $η$ to be computed in an optimal time. However, this index constructed by current algorithm is usually incorrect. During decomposition, the key is to obtain the $k$-probabilities of its neighbors when the vertex with minimum $k$-probability is deleted. Current method uses recursive floating-point division to update it, which can lead to serious errors. We propose a correct and efficient index construction algorithm to address this issue. Firstly, we propose tight bounds on the $k$-probabilities of the vertices that need to be updated, and the accurate $k$-probabilities are recalculated in an on-demand manner. Secondly, vertices partitioning and progressive refinement strategy is devised to search the vertex with the minimum $k$-probability, thereby reducing initialization overhead for each $k$ and avoiding unnecessary recalculations. Finally, extensive experiments demonstrate the efficiency and scalability of our approach.

Effective Index Construction Algorithm for Optimal $(k,η)$-cores Computation

TL;DR

This paper tackles the problem of efficiently and correctly computing online -cores on uncertain graphs, where current UCF-Index constructions suffer from correctness issues due to floating-point divisions in -probability updates. It introduces OptiUCF, a correct index construction framework that combines on-demand -probability recalculation with tight probabilistic bounds (Beta-function and Top-K bounds), a lazy refreshing strategy, and a progressive refinement scheme across -threshold layers to minimize recomputation and search space. The Baseline algorithm establishes a correct DP-based refresh, while OptiUCF significantly reduces overhead and yields substantial speedups (1–2 orders of magnitude over the baseline, 5–10x over a non-optimized variant) and robust accuracy. Extensive experiments on eight real-world graphs demonstrate improvements in efficiency, accuracy, and scalability, confirming the practical impact for uncertain-graph analysis tasks such as community detection, fraud analysis, and influence maximization.

Abstract

Computing -cores from uncertain graphs is a fundamental problem in uncertain graph analysis. UCF-Index is the state-of-the-art resolution to support -core queries, allowing the -core for any combination of and to be computed in an optimal time. However, this index constructed by current algorithm is usually incorrect. During decomposition, the key is to obtain the -probabilities of its neighbors when the vertex with minimum -probability is deleted. Current method uses recursive floating-point division to update it, which can lead to serious errors. We propose a correct and efficient index construction algorithm to address this issue. Firstly, we propose tight bounds on the -probabilities of the vertices that need to be updated, and the accurate -probabilities are recalculated in an on-demand manner. Secondly, vertices partitioning and progressive refinement strategy is devised to search the vertex with the minimum -probability, thereby reducing initialization overhead for each and avoiding unnecessary recalculations. Finally, extensive experiments demonstrate the efficiency and scalability of our approach.
Paper Structure (21 sections, 4 theorems, 8 equations, 8 figures, 1 table, 3 algorithms)

This paper contains 21 sections, 4 theorems, 8 equations, 8 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Core Decomposition on Uncertain Graphs
  4. UCF-Index
  5. Baseline Algorithm to Build a Correct UCF-Index
  6. Optimization Road Map of Baseline.
  7. Effective UCF-Index Construction
  8. Lazy Refreshing Strategy
  9. The Beta-Function Based Bound.
  10. The Top-K Based Lower Bound.
  11. Lazy Refreshing Strategy.
  12. Progressive Refinement Strategy
  13. Sketch of OptiUCF
  14. Experimental Evaluation
  15. Experimental Setup.
  16. ...and 6 more sections

Key Result

Lemma 1

Given an uncertain graph $\mathcal{G}$, an integer $k$, and a vertex $u \in \mathcal{G}$, we have $k$-$prob(u,\mathcal{G})$$\ge \prod\nolimits_{e\in N_{u}^{k}(\mathcal{G})}{{{p}_{e}}}$.

Figures (8)

  • Figure 1: The $(k,\eta)$-core of $\mathcal{G}$ for $k=2$ and $\eta=0.107$
  • Figure 2: The $(k,\eta)$-core and $\eta$-$\text{tree}_2$ of $\mathcal{G}$
  • Figure 3: Lazy refreshing strategy
  • Figure 4: The vertices partitioning framework
  • Figure 5: Time cost for UCF-Index construction
  • ...and 3 more figures

Theorems & Definitions (15)

  • Definition 1: $(k,\eta)$-core
  • Example 1
  • Definition 2: $\eta$-threshold
  • Definition 3: $k$-probability
  • Definition 4: Associative Vertex
  • Lemma 1
  • proof
  • Definition 5: Indefinite/Definite vertex
  • Example 2
  • Definition 6: Partition of Vertices
  • ...and 5 more