1-4 曲率视角下的图网络建模与分析.pdf

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1、曲率视角下的图数据分析与学习周敏 华为诺亚方舟实验室 主任工程师|自我介绍|本科毕业于中科大，博士毕业于新加坡国立大学 2017年入职华为诺亚方舟实验室 研究方向:图数据、序列数据模式挖掘和学习产产业业会会战战行行业业军军团团01Curvatures02Curvature on Network03Curvature on Surface04Conclusion目录目录 CONTENT|Curvatures01|Networks|Source:Internet(left):https:/ available in the form of networks are ubiquitous(socia

2、l networks,telecom network,drug interaction etc.)Network(Graph）Analysis&Learning|Graph structureDegree distributionGraph diameterShortest path lengthSparsenessGraph GeometryCurvatureRepresentation Learning Graph embeddingGraph Neural NetworkCurvature?|A geometric property:Flatness of an objectGeomet

3、ric intuition:Measure for growth rate of volume of distance ball“geodesic dispersion”Eidi M,Jost J.Ollivier ricci curvature of directed hypergraphsJ.Scientific Reports,2020,10(1):1-14.Curvature in Geometry|How do we know that the earth is not flat?Figure from internetSectional curvature&Ricci Curvat

4、ure|Consider a tangent vector =and another tangent vector at.Transport along to be a tangent vector at y.Ricci Curvature:averaging over all directions Curvature on Surfaces vs.on Network|Networks as Geometric Objects|Understanding the corresponding complex networks via the lens of curvatureIntrinsic

5、Discrete Ricci Curvature-Ollivier-Ricci Curvature-Forman Ricci Curvature-Other NotionsExtrinsicLearning Space-Euclidean Space-Hyperbolic Space-Mixed-Curvature SpaceCurvature on Network02|Discrete Ricci Curvature|Ollivier-Ricci curvatureForman-Ricci curvature Basic idea：Optimal transport Emphasizes c

6、lustering Basic idea：Dispersion Emphasize network dynamicSource：Comparative analysis of two discretizations of Ricci curvature for complex networksFroman-Ricci Curvature|Quantifies the degree of spread of the vertices Consider trianglesas faces，and uvis the number of triangles that contain u,v.All e