[论文解读] The importance of directed triangles with reciprocity: patterns and algorithms.

C. Seshadhri, Ali Pınar+2 more|arXiv (Cornell University)|Feb 1, 2013

Complex Network Analysis Techniques参考文献 25被引用 4

一句话总结

本文引入了有向闭包值作为经典传递性的类比，用于分析现实世界网络中的有向三角形，揭示了互惠边和有向楔形结构在三角形形成中起关键作用。本文提出了一种新颖的楔形采样算法，实现了估算有向三角形的数个数量级的速度提升，从而实现了对复杂有向网络结构的大规模分析。

ABSTRACT

The computation and study of triangles in graphs is a standard tool in the analysis of real-world networks. Yet most of this work focuses on undirected graphs. Real-world networks are often directed and have a significant fraction of reciprocal edges. While there is much focus on directed triadic patterns in the social sciences community, most data mining and graph analysis studies ignore direction. But how to we make sense of this complex directed structure? We propose a collection of directed closure values that are analogues of the classic transitivity measure (the fraction of wedges that participate in triangles). We perform an extensive set of triadic measurements on a variety of massive real-world networks. Our study of these values reveal a wealth of information of the nature of direction. For instance, we immediately see the importance of reciprocal edges in forming triangles and can measure the power of transitivity. Surprisingly, the chance that a wedge is closed depends heavily on its directed structure. We also observe striking similarities between the triadic closure patterns of different web and social networks. Together with these observations, we also present the first sampling based algorithm for fast estimation of directed triangles. Previous estimation methods were targeted towards undirected triangles and could not be extended to directed graphs. Our method, based on wedge sampling, gives orders of magnitude speedup over state of the art enumeration. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. ∗This work was funded by the GRAPHS Program at DARPA and the Laboratory Directed Research and Development program at Sandia National Laboratories.

研究动机与目标

• 理解方向性和互惠性在现实世界有向网络中三角形形成过程中的作用。
• 为有向图设计一种有原则的传递性度量扩展，捕捉有向楔形结构的影响。
• 设计一种可扩展的基于采样的算法，实现对有向三角形的快速估算，克服先前枚举方法的局限性。
• 通过在多样化的真实世界数据集上进行广泛测量，揭示有向网络中的结构模式。
• 证明三元闭包行为在很大程度上取决于有向楔形的配置，而不仅仅是边的数量。

提出的方法

• 提出有向闭包值作为无向传递性的类比，通过测量形成三角形的有向楔形比例来衡量。
• 引入一种专为有向图设计的楔形采样技术，其中每个采样到的楔形都会被检查是否形成三角形。
• 使用重要性采样高效估算有向三角形的总数，相比完整枚举显著降低计算成本。
• 基于楔形类型（如互惠、前馈等）采用分层采样策略，以提高估算精度。
• 在大型网络上将采样方法与精确枚举进行验证，结果表明其在实现显著加速的同时保持了高精度。
• 分析一系列多样化的真实世界网络，以测量并比较不同领域中定向三元闭包模式。

实验结果

研究问题

• RQ1 有向楔形的结构（如互惠、前馈）如何影响三角形闭包的可能性？
• RQ2 在现实世界有向网络中，互惠边在三角形形成中所起的作用有多大？
• RQ3 不同类型的网络和社交网络中，有向三元闭包模式有何异同？
• RQ4 基于采样的方法能否在大规模网络中实现对有向三角形的准确且可扩展的估算？
• RQ5 在不同类型的现实世界网络中，有向传递性在数量上存在哪些差异？

主要发现

• 互惠边在三角形形成中起主导作用，互惠楔形的闭包率显著高于非互惠楔形。
• 楔形闭包的概率在很大程度上取决于楔形的具体有向结构，而不仅仅是边的存在。
• 在多样化网络（如网络和社交网络）中观察到显著相似的三元闭包模式，表明存在共同的结构原则。
• 所提出的楔形采样算法相比最先进的枚举方法实现了数个数量级的速度提升，同时保持了高精度。
• 有向传递性在不同网络类型中存在显著差异，社交网络和网络表现出基于楔形配置的截然不同的闭包动态。
• 本研究揭示了传统无向传递性度量无法捕捉有向网络结构的复杂性，因此需要专门针对有向图的度量。