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Chen Chen, Xifeng Yan, Feida Zhu, Jiawei Han, Philip S. Yu University of Illinois at Urbana-Champaign IBM T. J. Watson Research Center University of Illinois at Chicago

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Outline Motivation Framework Efficient Computation Experiments Conclusion

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Online Analytical Processing Jim Gray, 1997 OLAP as a powerful analytical tool

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The Usefulness of OLAP Multi-dimensional Different perspectives Multi-level Different granularities Can we offer roll-up/drill-down and slice/dice on graph data? Traditional OLAP cannot handle this, because they ignore links among data objects

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The Prevalence of Graphs Chemical compounds, computer vision objects, circuits, XML Especially various information networks Biological networks Bibliographic networks Social networks World Wide Web (WWW)

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Applications WWW >= 3 billion nodes, >= 50 billion arcs Facebook >= 100 million active users Combining topological structures and node/edge attributes Great challenge to view and analyze them We propose Graph OLAP to tackle this issue

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Scenario #1 A bibliographic network The collaboration patterns among researchers for SIGMOD 2004

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Scenario #2

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Outline Motivation Framework Data Model Two types of Graph OLAP Dimension, Measure and OLAP operations Efficient Computation Experiments Conclusion

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Data Model We have a collection of network snapshots G = {G 1, G 2,..., G N } Each snapshot G i = (I 1,i, I 2,i,..., I k,i ; G i ) I 1,i, I 2,i,..., I k,i are k informational attributes describing the snapshot as a whole G i = (V i, E i ) is an attributed graph, with attributes attached with its nodes V i and edges E i Since G 1, G 2,..., G N only represent different observations of a network, V 1, V 2,..., V N actually correspond to the same set of objects

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Two Types of OLAP Informational OLAP (abbr. I-OLAP) Topological OLAP (abbr. T-OLAP)

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Informational OLAP Dimensions come from informational attributes attached at the whole snapshot level, so-called Info-Dims e.g., scenario #1

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I-OLAP Characteristics Overlay multiple pieces of information Do not change the objects whose interactions are being looked at In the underlying snapshots, each node is a researcher In the summarized view, each node is still a researcher

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Topological OLAP Dimensions come from the node/edge attributes inside individual networks, so-called Topo-Dims e.g., scenario #2

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T-OLAP Characteristics Zoom in/Zoom out Network topology changed: “generalized” nodes and “generalized” edges In the underlying network, each node is a researcher In the summarized view, each node becomes an institute that comprises multiple researchers

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Measures in Graph OLAP Measure is an aggregated graph I-aggregated graph T-aggregated graph Other measures like node count, average degree, etc. can be treated as derived Graph plays a dual role Data source Aggregate measure

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Generality of the Framework Measures could be complex e.g., maximum flow, shortest path, centrality Combine I-OLAP and T-OLAP into a hybrid case

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Graph OLAP Operations Graph I-OLAPGraph T-OLAP Roll-up Overlay multiple snapshots to form a higher-level summary via I-aggregated graph Shrink the topology and obtain a T- aggregated graph that represents a compressed view, whose topological elements (i.e., nodes and/or edges) have been merged and replaced by corresponding higher-level ones Drill-down Return to the set of lower- level snapshots from the higher-level overlaid (aggregated) graph A reverse operation of roll-up Slice/dice Select a subset of qualifying snapshots based on Info-Dims Select a subgraph of the network based on Topo-Dims

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Outline Motivation Framework Efficient Computation Measure classification Optimizations Constraint pushing Experiments Conclusion

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Two Categories of Strategies Top-down Generalized cells later How to combine and leverage intermediate results? Bottom-up Generalized cells first How to early-stop?

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Measure Classification How to combine and leverage intermediate results? Distributive The computation of high-level cells can be directly built on low-level cells Algebraic Not distributive, but can be easily derived from several distributive measures Holistic Neither distributive nor algebraic

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Examples Distributive: collaboration frequency Use distributiveness to drive computation up the cuboid lattice Algebraic: maximum flow Will prove later Semi-distributive Holistic: centrality Need to go down to the raw data and start from scratch

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Optimizations Special measures may have special properties that can help optimize the calculations We discuss two of them here, with regard to I-OLAP Localization Attenuation

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Localization During computation, only a neighborhood of the networks needs to be consulted e.g., the collaboration frequency of “R. Agrawal” and “R.Srikant” for [sigmod, all-years] only depends on their collaboration frequencies in each SIGMOD conferences Perfect (i.e., 0-neighborhood) localization k-neighborhood is less ideal, but still useful e.g., # of common friends shared by “R. Agrawal” and “R.Srikant”

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Attenuation Consider the transporting capability (i.e., maximum flow) from source S to destination T Multiple transportation networks, each one is operated by a separate company With regard to I-OLAP, each network is a “snapshot”, and overlaying more than one snapshots means to share link capacities among companies

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Attenuation Data graph C Node: cities Edge: capacity of a link Measure graph F Node: cities Edge: when maximum flow is transmitted, the quantity that passes through a link

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Attenuation Maximum flow is algebraic F can be derived from C Just run the maximum flow algorithm The capacity graph C is obviously distributive Lemma Let F be a flow in C and let C F be its residual graph, where residual means that C F = C - F, then F ′ is a maximum flow in C F if and only if F + F ′ is a maximum flow in C

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Attenuation Consider two snapshots that are overlaid Maximum flow F 1, F 2 already calculated from C 1, C 2 Without attenuation Compute the overall maximum flow F from C 1 + C 2 With attenuation Take F 1 + F 2 as basis Compute the residual maximum flow F ′ from (C 1 - F 1 ) + (C 2 - F 2 ), and augment it onto F 1 + F 2 Thus, our input attenuates from C 1 + C 2 to (C 1 + C 2 ) - (F 1 + F 2 ), which substantially decreases the efforts

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Constraint Pushing Iceberg graph cube Partial materialization Satisfying some interestingness requirement Push the constraints Anti-monotone e.g., maximum flow |f| ≥ δ |f| Monotone e.g., diameter d ≥ δ d

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Outline Motivation Framework Efficient Computation Experiments Conclusion

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OLAP a Bibliographic Network We get the coauthorship data from DBLP Measure Information Centrality Two Info-Dims Area Database (DB): PODS/SIGMOD/VLDB/ICDE/EDBT Data Mining (DM): ICDM/SDM/KDD/PKDD Information Retrieval (IR): SIGIR/WWW/CIKM Time

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OLAP a Bibliographic Network

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Efficiency A test that computes maximum flow as the measure Synthetically generate flow networks Details in the paper, with each “snapshot” representing an individual player in the transportation industry Like the Multi-Way method, calculate low-level cells before merging them into high-level ones One takes advantage of the attenuation heuristic The other does not

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Efficiency

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Outline Motivation Framework Efficient Computation Experiments Conclusion

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We propose a Graph OLAP framework to perform multi-dimensional, multi-level analysis on network data Measure is an aggregated graph Informational/Topological dimensions lead to I-OLAP, T-OLAP

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Conclusion Mainly focusing on I-OLAP, we discuss how a graph cube can be efficiently computed and materialized distributive, algebraic, holistic Optimizations: localization, attenuation Constraint pushing

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Future Works Technical issues for T-OLAP Selective drilling and discovery-driven InfoNet-OLAP

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