Upgrade Report - Department of Informatics - King's College London

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ContentsI Introduction 21 Online Social Networks (OSNs) 22 Large Online Social Networks and Generative Models 33 Graph sampling and crawling 3II Related Work 44 Network properties and metrics 44.1 Diameter and degree distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2 Degree correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.3 Expander graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.4 Connected Components And Community Structure . . . . . . . . . . . . . . . . . . . . . . . . 54.5 Communities: Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.6 Communities: Centrality and modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.7 Communities: Overlapping communities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.8 Property testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Graph Generation Models 75.1 Erdós-Rényi Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.2 Preferential attachment model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.3 Edge Copying model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.4 Triangle Closing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.5 Forest Fire Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.6 Stochastic Kronecker Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.7 Affiliation Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Graph sampling and crawling 146.1 Simple Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Weighted Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.3 Metropolis Hastings Random Walks (MHRWs) . . . . . . . . . . . . . . . . . . . . . . . . . . 17III Our Work 197 Graph generation 197.1 Random Walk Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2 Preferential attachment and message propagation . . . . . . . . . . . . . . . . . . . . . . . . . 207.3 Grow, Back-Connect, Densify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.4 Implicit Graph Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22I
7.5 Partitioned Preferential Attachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Existing Data Set Analysis 258.1 Ground truth. Complete twitter snapshot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.2 Other Real Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2.1 Degree Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.2.2 Degree-Based Cut Conductunce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.2.3 Degree Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Graph Sampling 299.1 Sampling Real World Networks: Case study <strong>of</strong> twitter . . . . . . . . . . . . . . . . . . . . . . 299.1.1 Our initial crawling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.1.2 Second crawling, unbiased sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.2 Uniform Sampling (UNI): A study <strong>of</strong> efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 329.3 Sampling Manufactured Graphs: Weighted Random Walk . . . . . . . . . . . . . . . . . . . . 339.3.1 Theoretical foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.3.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37IV Future Work 4110 Graph Analysis 4110.1 Real world network characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.2 Property Testing and Estimation Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211 Graph sampling 4211.1 Algorithm Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.1.1 Algorithm complexity reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.1.2 Run-time reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.1.3 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.2 Improvement Of Sampling Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412 Graph Generation models 44V References 4613 Acronyms 49VI Appendix 50A Sampling Manufactured Graphs: BFS Tree Filtering 50A.1 Crawler Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50A.2 Measured properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50A.2.1 Crawlers With Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51II
Page 1: King’s College LondonDepartment o
Page 5 and 6: List of Figures1 Erdós-Rényi Grap
Page 7 and 8: 2Part IIntroduction1 Online Social
Page 9 and 10: 4Part IIRelated Work4 Network prope
Page 11 and 12: 4 NETWORK PROPERTIES AND METRICS 6W
Page 13 and 14: 5 GRAPH GENERATION MODELS 8Figure 1
Page 15 and 16: 5 GRAPH GENERATION MODELS 10Figure
Page 17 and 18: 5 GRAPH GENERATION MODELS 12Figure
Page 19 and 20: 6 GRAPH SAMPLING AND CRAWLING 14on
Page 21 and 22: 6 GRAPH SAMPLING AND CRAWLING 16In
Page 23 and 24: 6 GRAPH SAMPLING AND CRAWLING 18{a(
Page 25 and 26: 7 GRAPH GENERATION 20(a) Constant m
Page 27 and 28: 7 GRAPH GENERATION 22Figure 9: Grow
Page 29 and 30: 7 GRAPH GENERATION 24Figure 10: Imp
Page 31 and 32: 8 EXISTING DATA SET ANALYSIS 26From
Page 33 and 34: 8 EXISTING DATA SET ANALYSIS 28From
Page 35 and 36: 9 GRAPH SAMPLING 30calls. We presen
Page 37 and 38: 9 GRAPH SAMPLING 32Figure 19: Real
Page 39 and 40: 9 GRAPH SAMPLING 34where b > 0 cons
Page 41 and 42: 9 GRAPH SAMPLING 36Theorem 3. For c
Page 43 and 44: 9 GRAPH SAMPLING 38Figure 22: Plots
Page 45 and 46: 9 GRAPH SAMPLING 40wide range of re
Page 47 and 48: 11 GRAPH SAMPLING 42In addition to
Page 49 and 50: 12 GRAPH GENERATION MODELS 4411.1.3
Page 51 and 52: 46Part VReferencesReferences[1] Dim
Page 53 and 54:
REFERENCES 48[36] Jure Leskovec, La
Page 55 and 56:
50Part VIAppendixASampling Manufact
Page 57 and 58:
A SAMPLING MANUFACTURED GRAPHS: BFS
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