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arXiv 0812.1811physics.soc-phcs.IRphysics.data-an
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跨时间尺度图社区的稳定性

Stability of graph communities across time scales

J. -C. Delvenne, S. N. Yaliraki, M. Barahona

摘要:

生物、社会和工程网络的复杂性使得人们希望找到可以作为简化描述的社区的自然分区,并提供对整个系统的结构和功能的洞察。虽然社区检测方法很多,但在如何量化和排名分区质量方面缺乏共识。我们在这里表明,分区的质量可以衡量其稳定性,定义在图上发生的马尔可夫过程的聚类自协方差。由于稳定性对图的时间尺度有内在的依赖性,它允许我们在每个时间比较和排名分区,并建立分区最佳的时间跨度。因此,马尔可夫时间有效地作为一个内在的分辨率参数,建立了一个层次结构的日益粗糙的聚类。在我们的框架内,我们可以提供一个统一的观点,几个标准的分区措施:模块化和归一化切割大小可以被解释为一步的时间措施,而费德勒的谱聚类出现在很长的时间。我们应用我们的方法来表征的相关性和持久性分区随着时间的推移建设性和真实的网络,包括层次图和社交网络。我们还获得了原子级蛋白质结构在不同的时间尺度上减少的描述。

英文摘要:

The complexity of biological, social and engineering networks makes it desirable to find natural partitions into communities that can act as simplified descriptions and provide insight into the structure and function of the overall system. Although community detection methods abound, there is a lack of consensus on how to quantify and rank the quality of partitions. We show here that the quality of a partition can be measured in terms of its stability, defined in terms of the clustered autocovariance of a Markov process taking place on the graph. Because the stability has an intrinsic dependence on time scales of the graph, it allows us to compare and rank partitions at each time and also to establish the time spans over which partitions are optimal. Hence the Markov time acts effectively as an intrinsic resolution parameter that establishes a hierarchy of increasingly coarser clusterings. Within our framework we can then provide a unifying view of several standard partitioning measures: modularity and normalized cut size can be interpreted as one-step time measures, whereas Fiedler's spectral clustering emerges at long times. We apply our method to characterize the relevance and persistence of partitions over time for constructive and real networks, including hierarchical graphs and social networks. We also obtain reduced descriptions for atomic level protein structures over different time scales.

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