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CMU SCS I2.2 Large Scale Information Network Processing INARC 1 Overview Goal: scalable algorithms to find patterns and anomalies on graphs 1. Mining Large Graphs: Algorithms, Inference, and Discoveries 2. Spectral Analysis of Billion-Scale Graphs: Discov eries and Implementation 3. Patterns on the Connected Components of Terabyte-Scale Graphs PI: Christos Faloutsos (CMU) Students: Leman Akoglu, Polo Chau, U Kang

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CMU SCS I2.2 Large Scale Information Network Processing INARC 2 Mining Large Graphs: Algorithms, Inference, and Discoveries U Kang Duen Horng Chau Christos Faloutsos School of Computer Science Carnegie Mellon University

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CMU SCS I2.2 Large Scale Information Network Processing INARC 3 Outline Problem Definition Proposed Method Experiment Conclusion

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CMU SCS I2.2 Large Scale Information Network Processing INARC 4 Motivation Inference on graph: “guilt by association” Adult sites tend to be connected to adult sites, while edu. sites are connected to educational ones Given labels(adult or edu) on a subset of the nodes, infer the labels of other unlabeled nodes on graph Tool: Belief Propagation(BP) red nodes connected to red nodes blue nodes connected to blue nodes

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CMU SCS I2.2 Large Scale Information Network Processing INARC Prior prob Messages from neighbors Node belief Propagation matrix ~Messages from neighbors Messsage from node i to node j Message computation Belief computation Prior prob Belief Propagation 5

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CMU SCS I2.2 Large Scale Information Network Processing INARC A Challenge in BP Scalability! Existing works assume that all the nodes (and/or edges) of the input graph fit in memory Problem: what if the graph is too large to fit in memory? Challenge: Scaling up the inference algorithm for very large graphs whose nodes do not fit in memory 6

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CMU SCS I2.2 Large Scale Information Network Processing INARC Problem Definition How can we scale up the BP algorithm to very large graphs? Goal Scalability: to billions of nodes and edges Efficiency: fast algorithm 7

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CMU SCS I2.2 Large Scale Information Network Processing INARC 8 Outline Problem Definition Proposed Method Experiment Conclusion

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CMU SCS I2.2 Large Scale Information Network Processing INARC Main Idea Our approach Use Hadoop to scale-up BP Challenge How can we formulate BP using a simple, efficient operation supported by Hadoop? 9

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CMU SCS I2.2 Large Scale Information Network Processing INARC Main Idea Key observation BP message update equation = local message exchange 10 m 13 m 31 m 01 m 10 m 12 m 21 m 24 m 42 A message is updated from its neighboring messages. For example, m 12 is updated from m 01 and m 31

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CMU SCS I2.2 Large Scale Information Network Processing INARC BP message update can be expressed by a generalized matrix-vector multiplication on a line graph L(G) induced from the original graph G Nodes in L(G) are edges in G Two nodes in L(G) are connected if they are adjacent in G Main Idea 11

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CMU SCS I2.2 Large Scale Information Network Processing INARC BP message update can be expressed by a generalized matrix-vector multiplication on a line graph L(G) induced from the original graph G Proposed: HA-LFP algorithm 12 New message vector Old message vector Line graph of G Generalized m-v multiplication Multiply repeatedly until convergence

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CMU SCS I2.2 Large Scale Information Network Processing INARC Complexity One Iteration of HA-LFP on L(G) One Matrix Vector Multiplication on G = Time : O((V+E) / M) Space: O(V + E) V : # of nodes E : # of nodes M : # of machines 13

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CMU SCS I2.2 Large Scale Information Network Processing INARC 14 Outline Problem Definition Proposed Method Experiment Conclusion

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CMU SCS I2.2 Large Scale Information Network Processing INARC 15 Questions Q1: How fast is HA-LFP? Q2: How does HA-LFP scale-up? Q3: How can we find `good’ and `bad’ sites in a web graph?

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CMU SCS I2.2 Large Scale Information Network Processing INARC Running Time Q1: How fast is HA-LFP? [10 iteration] 16

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CMU SCS I2.2 Large Scale Information Network Processing INARC Scale Up Q2: How does HA-LFP scale-up? Linear on the number of machines, edges 17

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CMU SCS I2.2 Large Scale Information Network Processing INARC Advantage of HA-LFP Scalability The only solution when the node information cannot fit in memory. Near-linear scale up Running Time Faster than the single-machine, for large graphs Fault Tolerance 18

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CMU SCS I2.2 Large Scale Information Network Processing INARC Analysis of Web Graph Q3: How can we find `good’ and `bad’ sites in a web graph? Pages whose goodness scores < 0.9 are likely to be adult pages 19

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CMU SCS I2.2 Large Scale Information Network Processing INARC 20 Outline Problem Definition Proposed Method Experiment Conclusion

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CMU SCS I2.2 Large Scale Information Network Processing INARC 21 Conclusion HA-LFP Belief Propgation for billion-scale graphs on Hadoop Near-linear scalability on # of machines, edges Many applications Finding `good’ and `bad’ web sites Fraud detection …

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CMU SCS I2.2 Large Scale Information Network Processing INARC 22 Spectral Analysis of Billion-Scale Graphs: Discoveries and Implementation U Kang Brendan Meeder Christos Faloutsos School of Computer Science Carnegie Mellon University

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CMU SCS I2.2 Large Scale Information Network Processing INARC 23 Outline Problem Definition Proposed Method Experiment Conclusion

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CMU SCS I2.2 Large Scale Information Network Processing INARC 24 Problem Definition Eigensolver Computes top-k eigenvalues and eigenvectors Application: SVD, triangle counting, spectral clustering, … Existing eigensolver Can handle up to millions of nodes How can we scale up eigensolvers to billion- scale graphs?

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CMU SCS I2.2 Large Scale Information Network Processing INARC 25 Outline Problem Definition Proposed Method Experiment Conclusion

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CMU SCS I2.2 Large Scale Information Network Processing INARC Main Idea HEigen algorithm (Hadoop Eigen-solver) Selective parallelize ‘Lanczos’ algorithm Expensive operation: on Hadoop for scalability Inexpensive operation: on a single-machine for accuracy Block encoding Block encoding, and then do matrix-vector multiplication Exploiting skewness in matrix-matrix mult. In matrix-matrix multiplication when a matrix is very large and the other is very small 26

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CMU SCS I2.2 Large Scale Information Network Processing INARC Application of HEigen Triangle Counting Real social networks have a lot of triangles Friends of friends are friends But: triangles are expensive to compute (3-way join; several approx. algos) Q: Can we do that quickly? A: Yes! #triangles = 1/6 Sum ( λ i 3 ) (and, because of skewness in eigenvalues, we only need the top few eigenvalues!) [Tsourakakis ICDM 2008]

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CMU SCS I2.2 Large Scale Information Network Processing INARC 28 Outline Problem Definition Proposed Method Experiment Conclusion

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CMU SCS I2.2 Large Scale Information Network Processing INARC 29 Questions Q1: How does HEigen scale-up? Q2: Which Matrix-Matrix multiplication algorithm runs the fastest? Q3: How can we find anomalous sites in a web graph?

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CMU SCS I2.2 Large Scale Information Network Processing INARC Running Time Q1: How does HEigen scale-up? Heigen-BLOCK is faster than PLAIN ver. Linear on the number of machines, edges

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CMU SCS I2.2 Large Scale Information Network Processing INARC Scale Up Cache-based MM runs the fastest! Q2: Which Matrix-Matrix multiplication algorithm runs the fastest?

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CMU SCS I2.2 Large Scale Information Network Processing INARC 32 Results Triangle counting on Twitter social network [Twitter 2009; ~ 3 billion edges] U.S. politicians: moderate number of triangles vs. degree Adult sites: very large number of triangles vs. degree Q3: How can we find anomalous sites in a web graph?

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CMU SCS I2.2 Large Scale Information Network Processing INARC 33 Outline Problem Definition Proposed Method Experiment Conclusion

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CMU SCS I2.2 Large Scale Information Network Processing INARC 34 Conclusion HEigen Eigensolver for billion-scale graphs on Hadoop Near-linear scalability on # of machines, edges Cache-based Matrix-Matrix multiplication: fastest! Anomalies in triangle counts Many applications Triangle counting SVD ……

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CMU SCS I2.2 Large Scale Information Network Processing INARC 35 Patterns on the Connected Components of Terabyte-Scale Graphs U Kang* Mary McGlohon* † Leman Akoglu* Christos Faloutsos* (*) SCS, Carnegie Mellon University (†) Google

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CMU SCS I2.2 Large Scale Information Network Processing INARC 36 Outline Problem Definition Static Patterns Evolution Patterns Model Conclusion

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CMU SCS I2.2 Large Scale Information Network Processing INARC A large graph is composed of many connected components 37 Problem Definition Q2: evolution patterns? Q3: model? Size Q1: static patterns? Count YahooWeb graph |V| = 1.4 billion |E| = 6.7 billion 120 GBytes

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CMU SCS I2.2 Large Scale Information Network Processing INARC 38 Outline Problem Definition Static Patterns Evolution Patterns Model Conclusion

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CMU SCS I2.2 Large Scale Information Network Processing INARC 39 Q1: Static Patterns What are the regularities in the connected components of a static graph? How do they look like? Do the GCC and the other connected components look similar? Chain? Clique? Idea: use ‘density’ and ‘radius’ to find patterns

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CMU SCS I2.2 Large Scale Information Network Processing INARC Density of Connected Component What is a good metric for the density of a connected component? A candidate: |E| / |V| (“average degree”) Problem: it increases over time 40 Number of Nodes Number of Edges

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CMU SCS I2.2 Large Scale Information Network Processing INARC Density of Connected Component We want a metric that can measure the ‘intrinsic’ density of a component Proposed: Graph Fractal Dimension(GFD) log |E| / log |V| 41 [Leskovec+ KDD05] Number of Nodes Number of Edges Number of Edges

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CMU SCS I2.2 Large Scale Information Network Processing INARC Density of Connected Component Graph Fractal Dimension(GFD) log |E| / log |V| 42 Chain: GFD ~1 Star: GFD ~1 Bipartite Core: 1 < GFD < 2 Clique: GFD ~2

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CMU SCS I2.2 Large Scale Information Network Processing INARC Density of Connected Component 43 What are the GFDs of connected components in a large, real graph?

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CMU SCS I2.2 Large Scale Information Network Processing INARC Density of Connected Component GFDs of CCs in YahooWeb graph GFDs of CCs are slightly denser than the tree 44 Slope= 1.08 GFDs of CCs are constant on average Number of Nodes Number of Edges Number of Edges

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CMU SCS I2.2 Large Scale Information Network Processing INARC Radius of Connected Component 45 Q1.1: What does the GCC look like? Q1.2: What do the rest CC’s look like? ( What are the GFDs?)

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CMU SCS I2.2 Large Scale Information Network Processing INARC Radius of Connected Component What are the patterns of radii in connected components? A1.2: Chain-like disconnected components 46 Slope= 1.38 Core Chain Average Radius A1.1: GCC looks like a ‘kite’ Max. Radius Avg. Max.

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CMU SCS I2.2 Large Scale Information Network Processing INARC 47 Outline Problem Definition Static Patterns Evolution Patterns Model Conclusion

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CMU SCS I2.2 Large Scale Information Network Processing INARC 48 Q2: Evolution Patterns How do the connected components evolve? Do largest connected components grow with the same rate? How often does a newcomer join the disconnected components? newcomer ? ?

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CMU SCS I2.2 Large Scale Information Network Processing INARC Gelling Point Gelling Point [McGlohon+ KDD08] Diameter starts to shrink 49

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CMU SCS I2.2 Large Scale Information Network Processing INARC Growth of Connected Component GFDs of Top 3 CC’s over time 50 Before “gelling point”: GFDs of Top 3 CC’s stay constant, “tree” like. After “deviation point”: GFD of GCC takes off, becomes denser.

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CMU SCS I2.2 Large Scale Information Network Processing INARC ‘Rebel’ Probability What are the chances that a newcomer doesn’t belong to GCC? (“rebel” prob.) 51 newcomer ? GCC DCs

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CMU SCS I2.2 Large Scale Information Network Processing INARC ‘Rebel’ Probability What are the chances that a newcomer doesn’t belong to GCC? (“rebel” prob.) 52 newcomer d: degree of a newcomer s: size (|V|) of DC But, how exactly?

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CMU SCS I2.2 Large Scale Information Network Processing INARC ‘Rebel’ Prob. power of |V| in dc ‘Rebel’ Probability 53 ‘Rebel’ Prob. exponential to the degree d: degree of a newcomer s: size (|V|) of DC

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CMU SCS I2.2 Large Scale Information Network Processing INARC 54 Outline Problem Definition Static Patterns Evolution Patterns Model Conclusion

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CMU SCS I2.2 Large Scale Information Network Processing INARC 55 Q3: Model How can we explain the static and the evolution patterns by a generative model? Modeling Goals (G1) Constant GFDs (G2) ERP (Exponential Rebel Probability) (G3) Disconnected Components

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CMU SCS I2.2 Large Scale Information Network Processing INARC CommunityConnection Model CommunityConnection model Defines a behavior of a new node joining the network 1. Chooses a host to link to. 2. Visits the neighbors Repeat the two processes! 56

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CMU SCS I2.2 Large Scale Information Network Processing INARC CommunityConnection Model How does the CommunityConnection model match reality? 57

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CMU SCS I2.2 Large Scale Information Network Processing INARC CommunityConnection Model Results (G1) Constant GFDs 58 Number of Nodes Number of Edges Number of Nodes Number of Edges

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CMU SCS I2.2 Large Scale Information Network Processing INARC CommunityConnection Model Results (G2) ERP (Exponential Rebel Probability) (G3) Disconnected Components 59 Degreelog(|V| in DC) log( Rebel Prob.) log( Rebel Prob.)

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CMU SCS I2.2 Large Scale Information Network Processing INARC 60 Outline Problem Definition Static Patterns Evolution Patterns Model Conclusion

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CMU SCS I2.2 Large Scale Information Network Processing INARC 61 Conclusion Patterns in the Connected Components Goal 1 : Static Patterns Chain-like disconnected components ‘Kite’-like GCC Goal 2 : Evolution Patterns Constant, low GFD(“density”) until the gelling point ERP (Exponential Rebel Probability) Goal 3 : Model CommunityConnection Model (matches reality)

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CMU SCS I2.2 Large Scale Information Network Processing INARC Hadoop/PEGASUS Degree Distr. Pagerank Diameter Conn. Comp Eigensolver Belief Propagation Clustering, … Future Plan 62

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CMU SCS I2.2 Large Scale Information Network Processing INARC 63 Thank you! http://www.cs.cmu.edu/~pegasus

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