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A geometric interpretation for growing networks
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Transcript of A geometric interpretation for growing networks
![Page 1: A geometric interpretation for growing networks](https://reader036.fdocuments.net/reader036/viewer/2022070523/58ed300b1a28abb16c8b45bb/html5/thumbnails/1.jpg)
A geometric interpretation for growing networks
孙佩源2016.12.28
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Real Networks
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• scale free• 节点度分布满足幂律分布
• strong clustering• 与同一节点连接的节点对有较大概率连接
• significant community structure• 通常节点出现集聚现象
Real Networks
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Previous Work
• Erdos-Renyi Model [Erdos and Renyi, 1959]
• 假设任意两点之间的连接为独立同分布的伯努利随机变量• Erdos-Renyi 通常只是作为一个基准模型
• Stochastic Block Model [Nowicki and Snijders, 2001]
• 假设任意两点之间的连接依赖于一个表述节点所属类型的隐变量• 只具有集群现象
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Previous Work
• Preferential Attachment Model [Barabasi and Albert, 1999]
• 网络中新加入节点与已有节点连接的概率与其度成正比• 最成功的地方在于发现 citation 网络中的幂律指数为 3
• Preferential Linking Model [Dorogovtsev, 2000]
• 在 Barabasi 的基础上加入了节点的初始 attractiveness
• 最成功的地方在于获得了初始 attractiveness 与幂律指数的关系表达式
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Geometry Framework• Hyperbolic Geometry Model [Krioukov and Papadopoulos, 2010]
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Geometry Framework
• Hyperbolic Geometry Model [Krioukov and Papadopoulos, 2010]
• 网络节点由极坐标描述 : (angular coordinate, radial coordinate)
• 不同于传统的 Euclidean 空间, Hyperbolic Space 中两点距离为:2' ln
2x r r
节点极半径 = K 两节点角度差
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Geometry Framework
• Hyperbolic Geometry Model [Krioukov and Papadopoulos, 2010]
• 假设我们可以将现实网络中的节点映射到 Hyperbolic Space 中• 两点间产生连接的概率为:
• 则简单推导即可得:( ) ( )p x R x
距离小于 R 的连接概率为 1
12 1,2
( ) ,12,2
p k k r
节点度满足幂律分布
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Geometry Framework
• Hyperbolic Geometry Model [Krioukov and Papadopoulos, 2010]
• 进一步地,如果假设连接概率为:
• 则更进一步的推导可得: 1/
1( )1Tp x
x
与集群系数成反比T
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Geometry Framework
• Hyperbolic Geometry Model [Krioukov and Papadopoulos, 2010]
假设网络节点位于 Hidden Hyperbolic Space 和特定连接概率即可得1. 生成网络中节点度服从幂律分布2. 通过调节连接概率参数可以控制集群系数
Geometry Framework 非常简单并且满足现实网络的特性
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Geometry Framework
• Popularity versus Similarity Model [Papadopoulos, Krioukov etc, 2012]
• 前述模型只考虑了 popularity ,而忽略了 similarity
e.g :新的微博用户除了关注大 V ,还关注与自己兴趣相近的用户• 前述模型只考虑了静态网络,没有考虑网络的动态增长
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Geometry Framework
• Popularity versus Similarity Model [Papadopoulos, Krioukov etc, 2012]
1. 初始网络为空2. 在时刻 : (a) 节点 的极坐标为: ,角坐标为: (b) 早于 的节点更新极坐标为:
3. 新加入节点与已经存在节点连接概率为:
1,2, ,i t 2 lnir i
[0,2 ]U ii
( ) (1 )j j ir i r r
( )2
1( )1 ij i
ijx R
T
p xe
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Geometry Framework
• Node coordinate inference [Papadopoulos, Krioukov etc, 2015]
1. 基于链接的推断算法逐节点使用 MLE 求解 :
原理为:2. 基于公共邻居的推断算法求出公共邻居节点的概率分布,然后使用 MLE 求解似然度函数
1
1
( ) [1 ( )]ij ijiL ij ij
j i
L p x p x
( ) ( , )ij ij i ip x r
https://bitbucket.org/dk-lab/2015_code_hypermap
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Geometry Framework
• Geometric correlations in multiplex network [Kleineberg, 2016]
主要结论:1. 多层网络重叠节点在不同层的极坐标具有很强的相关性
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Geometry Framework
• Geometric correlations in multiplex network [Kleineberg, 2016]
主要结论:2. 多层网络重叠节点在不同层的角坐标具有很强的相关性
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Geometry Framework
• Geometric correlations in multiplex network [Kleineberg, 2016]
主要结论:3. 多层网络节点间链接概率具有很强的关联性
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some preliminary experiments• 数据集
序号 事件 总结点数 总边数 MCC节点 MCC边 节点比例1 郭美美 153744 435515 67680 158551 44.02%
2 李庄 40989 70444 16216 30249 39.56%
3 钱云会 64596 165066 24882 53027 38.52%
4 夏俊峰 56405 79163 21567 29712 38.24%
5 药家鑫 215316 372219 79852 129829 37.09%
6 房价 525083 1292306 192383 432974 36.64%
7 9级地震 173700 286229 49076 75433 28.25%
8 钱明奇 44390 65206 10281 13550 23.16%
9 王功权 173619 165362 27701 32593 15.96%
10 中石化 86504 98116 11442 12832 13.23%
11 李刚 110523 128589 14291 15913 12.93%
12 宜黄 12886 8775 1534 1533 11.90%
13 邓玉娇 18739 16507 2155 2431 11.50%
14 个税起征点 51267 40115 3211 3269 6.26%
15 胶州路大火 107825 113908 5605 5790 5.20%
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some preliminary experiments• 各层重叠节点分布
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some preliminary experiments•验证转发网络是否满足 Hyperbolic 结构( 1)是否满足幂律分布(原图)
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some preliminary experiments•验证转发网络是否满足 Hyperbolic 结构( 1)是否满足幂律分布(最大连通子图)
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some preliminary experiments•验证转发网络是否满足 Hyperbolic 结构( 2)集群系数是否满足(极大连通子图)
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some preliminary experiments•验证转发网络是否满足 Hyperbolic 结构( 2)集群系数是否满足(极大连通子图, 10bin)
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some preliminary experiments•验证转发网络是否满足 Hyperbolic 结构( 3) Hyperbolicity 是否满足(极大连通子图) [Tree-like structure in large social and information networks, 2013, ICDM]
指出:可以使用树分解衡量图的 hyperbolicity ,树分解得到的 tree width越低, hyperbolicity越强
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some preliminary experiments•验证转发网络是否满足 Hyperbolic 结构( 3)双曲性是否满足(极大连通子图)
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some preliminary experiments•验证转发网络是否满足 Hyperbolic 结构( 3)双曲性是否满足(极大连通子图)
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some preliminary experiments•几个事件的 Polar图
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Reference1. Krzysztof Nowicki and Tom A B Snijders. Estimation and prediction for
stochastic blockstructures. Journal of the American Statistical Association, 96(455):1077–1087, 2001.
2. A.L. Barabási and R. Albert, Science 286, 509 (1999).3. S.N.Dorogovtsev, J.F.F.Mendes and A.N. Samukhin, PRL, 2000.4. Krioukov D, Papadopoulos F, Kitsak M, et al. Hyperbolic geometry of
complex networks[J]. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2010, 82(3 Pt 2):98-118.
5. Papadopoulos F, Aldecoa R, Krioukov D. Network Geometry Inference using Common Neighbors[J]. Computer Science, 2015, 92(2).
6. Kleineberg K K, Boguñá M, Serrano M Á, et al. Hidden geometric correlations in real multiplex networks[J]. Nature Physics, 2016.
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谢谢大家!