[论文解读] Generating Practical Random Hyperbolic Graphs in Near-Linear Time and with Sub-Linear Memory

Penschuck, Manuel|arXiv (Cornell University)|Jan 1, 2016

Graph Theory and Algorithms被引用 3

一句话总结

本文提出了几何异质随机图（GIRGs），作为双曲随机图的推广模型，通过忽略常数因子简化了边概率的表达式。该文提出了一种采样算法，可在期望线性时间内生成GIRGs，相比现有最佳双曲随机图采样算法的效率提升了O(√n)倍，同时证明了GIRGs具有常数阶聚类系数、小分隔符，并且可使用线性空间实现高效压缩。

ABSTRACT

Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. In fact, the behavior of real-world networks and random graph models can be the complete opposite of one another, depending on the considered property. Brach, Cygan, Lacki, and Sankowski [SODA 2016] introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both deterministic properties and exploit them to design faster algorithms for a number of classical graph problems like transitive closure, maximum matching, determinant, PageRank, matrix inverse, counting triangles and maximum clique. We complement the work of Brach et al. by showing that some well-studied random graph models exhibit both the mentioned PLB properties and additionally also a power-law lower bound on the degree distribution (PLB-L). All three properties hold with high probability for Chung-Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability for Chung-Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. In the second part of this work we study three classical NP-hard combinatorial optimization problems on PLB networks. It is known that on general graphs, a greedy algorithm, which chooses nodes in the order of their degree, only achieves an approximation factor of asymptotically at least logarithmic in the maximum degree for Minimum Vertex Cover and Minimum Dominating Set, and an approximation factor of asymptotically at least the maximum degree for Maximum Independent Set. We prove that the PLB-U property suffices such that the greedy approach achieves a constant-factor approximation for all three problems. We also show that all three combinatorial optimization problems are APX-complete, even if all PLB-properties hold. Hence, a PTAS cannot be expected, unless P=NP.

研究动机与目标

• 解决现有双曲随机图采样算法计算效率低下的问题。
• 开发一种理论上更简洁但定性上等价于双曲随机图的模型，同时保留其关键结构特性。
• 建立新模型的基本算法与结构特性，包括采样效率、聚类性、分隔符和可压缩性。
• 为未来复杂网络模型的理论与实验研究提供基础。

提出的方法

• 提出几何异质随机图（GIRGs）作为双曲随机图的推广，其中顶点具有权重和在d维环面上的位置。
• 定义边概率与顶点权重成正比，与距离成反比，省略常数因子以简化分析。
• 设计一种采样算法，通过利用几何与权重相关的稀疏性，实现期望线性时间复杂度。
• 采用阈值模型与泰勒近似分析不同距离与权重条件下边概率的行为。
• 运用集中与概率论论证，证明聚类系数为常数、分隔符较小等结构特性。
• 基于图的几何与幂律结构，采用压缩技术实现线性空间表示。

实验结果

研究问题

• RQ1 是否能够在线性期望时间内对具有底层几何结构的随机图模型进行采样？
• RQ2 所提出的模型是否保留了高聚类与小分隔符等关键网络特性？
• RQ3 能否使用次线性空间高效压缩该模型？
• RQ4 在边概率中忽略常数因子，对模型与双曲随机图相比的定性行为有何影响？

主要发现

• 所提出的采样算法在期望线性时间内运行，相比现有最佳双曲随机图采样算法效率提升O(√n)倍。
• GIRGs的聚类系数为Ω(1)，表明其保持了高聚类性，这是现实世界网络的关键特征。
• GIRGs具有小分隔符，意味着仅需移除子线性数量的边即可使最大连通分量断开。
• GIRGs可使用期望线性数量的比特实现无损压缩，表明其具有极强的可压缩性。
• 双曲随机图是GIRGs的一个特例，验证了该模型在现实世界网络建模中的相关性。
• 理论分析确认，GIRGs保留了双曲随机图的无标度性、小世界性与高聚类性，同时显著简化了技术复杂度。