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1 2.5K-Graphs: from Sampling to Generation Minas Gjoka, Maciej Kurant ‡, Athina Markopoulou UC Irvine, ETZH ‡

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Motivation Generate synthetic topologies that resemble real ones – development and testing of algorithms, protocols – anonymization Obtaining full topology is not always possible – data owner not willing to release it – exhaustive measurement impractical due to size 2

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Our methodology 3 Node Samples Synthetic Graph Real Graph ( unknown ) Crawl using UIS, WIS, RW Estimated Graph Properties Estimate Generate

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Which graph properties? dk-series framework [Mahadevan et al, Sigcomm ’06] – 0K specifies the average node degree – 1K specifies the node degree distribution – 2K specifies the joint node degree distribution (JDD) – 3K specifies the distribution of subgraphs of 3 nodes –.. – nK specifies the entire graph Accuracy vs practicality 4 2.5K

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2.5K-Graphs Joint Node Degree Distribution (2K): Degree-dependent Average Clustering Coefficient Distribution: 5 all triangles using node u all possible triangles 1a 2a 4a 3b 3a 1b 4b k all nodes of degree k c 3a = 2 3

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Related Work Network Modeling – Exponential Random Graph Models [Handcock et al, ‘10] – Kronecker graphs [Kim & Leskovec, ‘11] – dk-Series [Mahadevan et al, Sigcomm ’06] Graph Generation – Simple 1K graphs [Molloy & Reed ’95] – 1K + [Bansal et. al. ’09] – 1K + Triangle Sequence [Newman ’09] – 1K + [Serrano & Boguna, ’05] – 2K [Krioukov et. al. ‘06, Stanton & Pinar ’11] – 3K [Krioukov et. al. ’06] 6 2.5K

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7 Node Samples Synthetic Graph Real Graph (unknown) Crawl using UIS, WIS, RW Estimated c(k) & JDD Estimate Generate

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Estimation Node Samples 8 UIS – Uniform Independence Sample WIS – Weighted Independence Sample sampling probability w(v) proportional to node degree RW – Random Walk Sample

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Estimation c(k) 9 all triangles using node a all possible triangles UIS WIS - all nodes of degree k - neighbors of node a - # shared partners for nodes (a,b) - all sampled nodes of degree k - number of samples for node b - sampling weight of node a when ((RW)

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Estimation JDD 10 UIS WIS - all nodes of degree k - all edges - all sampled nodes of degree k - sampling weight of node a when (RW)

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Estimation Random Walks RW Traversed Edges Induced Edges Induced Edges with safety margin M=1 Hybrid estimator combines Traversed Edges + Induced Edges with margin M with safety margin M=0 = WIS

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12 Node Samples Real Graph (unknown) Crawl using UIS, WIS, RW Estimated c(k) & JDD Generate Estimate Synthetic Graph

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Node Samples 2.5K Graph Real Graph (unknown) Crawl using UIS, WIS, RW Estimate Target c(k) Target JDD simple graph exact JDD many triangles 2K Construction Smart Double Edge Swaps Generate

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2K Construction (1) Step K k k# K

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2K Construction (2) Step k# vk v 1a 1b 2a 3a 3b 4a 4b 1a1a 2a2a 4a4a 3b3b 3a3a 1b1b 4b4b 2K 1K

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2K Construction (1) Step a 2a 4a 3b 3a 1b 4b v k v r v 1a 1b 2a 3a 3b 4a 4b a 2a b 4b 90 1b 4a 70 3a k

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2K Construction (2) Step a 2a 4a 3b 3a 1b 4b v k v r 1a 1b 2a 3a 3b 4a 4b (3b,4b) (1a,4a) (1a,2a) a 2a b 4b 90 1b 4a 70 3a k (1b,2a) (1b,4b) (2a,4b) (2a,3b) (1b,3a) (2a,4a) (1a,3b) (1b,3b) (4a,4b) (3b,4a)(1a,4b) (2a,3a) (3a,3b) (3a,4b) (3a,4a) (1a,1b)(1b,4a) (1a,3a)

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2K Construction Step a 2a 4a 3b 3a 1b 4b v k v r 1a 1b 2a 3a 3b 4a 4b k b 4a 3a 3b k=3 k=4 Degree pair (4,3) Construction stuck. Iterate over all unsaturated degree pairs Iterate over all unsaturated node pairs for the degree pair (4,3) Move 1: Add edge (4b, 3a) Move 2: Add edge (4b,3*) or (3a,4*) Move 3: Add edge (4*, 3*)

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Node Samples 2.5K Graph Real Graph (unknown) Crawl using UIS, WIS, RW Estimate Target c(k) Target JDD simple graph exact JDD many triangles 2K Construction Smart Double Edge Swaps Generate

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“Smart” double edge swaps 2K-preserving c(k)-targeting 20 degree k 1 Target Destroy triangles Create triangles k1 k2 k1 k3 Double edge swap [ Mahadevan et. al., Sigcomm ‘06 ] How we select edges for double edge swaps – Triangle creation » select edges with low number of shared partners – Triangle destruction » select random edges Current at end of 2K JDD(k1,k2) JDD(k1,k3) unchanged

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21 Node Samples Real Graph (unknown) Crawl using UIS, WIS, RW Generate Estimate Evaluation 2.5K Graph Estimated c(k) & JDD

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Evaluation Total Number of Triangles Average Clustering Coefficient Number of Nodes Number of Edges Average Degree

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Evaluation Efficiency of Estimators (RW) 23 Error Metric : Hybrid combines advantages of IE & TE Facebook New Orleans

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Evaluation Speed of 2.5K Graph Generation 24 our 2.5K prior 2.5K Higher gain for graphs with more triangles

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Evaluation From Sampling to Generation 25 Facebook New Orleans

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Conclusion Complete and practical methodology to generate synthetic graphs from node samples – novel estimators that measure c(k) and JDD – novel and fast 2.5K generator Python implementation – 26 Node Samples Synthetic Graph Real Graph (unknown) Crawl using UIS, WIS, RW c(k) & JDD Estimate Generate

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