towards data analytics on attributed graphs ngs qe oral presentation 1 student : qi fan supervisor:...

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TOWARDS DATA ANALYTICS ON ATTRIBUTED GRAPHSNGS QE Oral Presentation

Student : Qi FanSupervisor: Prof. Kian-lee Tan

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Outline

• Attributed Graph Analytic

• Graph Window Query

• Graph Window Query Processing

• Experiments

• Future Works

Graph Analytic Window Query Query Processing Experiments Future Work

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Outline

Attributed Graph Analytic



• Experiments

• Future Works


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Data Analytics

• Data Analytics plays an important part in business [1]• Web analytics for advertising and recommendation• Customer analytics for market optimization• Portfolio analytics for risk control

• Analytics on data yield:• Data products• Data-driven decision support• Insights of data model

[1] Analytics Examples: http://en.wikipedia.org/wiki/Analytics


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Relational Data Analytic

• Table as data representation, SQL as the query language

• Analytic SQL:• Ranking• Windowing• LAG/LEAD• FIRST/LAST• SKYLINE • TOP-K• … …


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Emerging of Large Linked Data

• In real world, linked data are becoming emerging:• Facebook, LinkedIn, Biological network, Phone Call

network, Twitter, etc.

• Modeling linked data in relational way and querying using SQL is inefficient:• Graph queries are often traverse based• SQL based traversal is 100 times slower than adjacent

list based [1]

• Graph model is more fit for linked data!!![1] http://java.dzone.com/articles/mysql-vs-neo4j-large-scale


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Graph Data Model

Vertex Edge

G = (V, E, A)

Attributed Graph Vertices Edges

Graph

Vertex Attr1 Attr2 Attr3

… …

Attribute Table

Attributes

Graph Structure + attribute dimensions


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Graph Data Model

• Graph Data:• Vertex – entities, i.e. User, Webpage, Molecule, etc.• Edge – relationships, i.e. follow, cite, depends-on,

friends-of, etc.• Attribute – profile information for vertex/edge

• Specific model depends on data, thus:• Edge – directed / undirected • Attribute – homogeneous, inhomogeneous


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Graph Data Model Example

People and follow relationships...

People and friends relationships…

Bimolecules and depends-on relationships...

Attributed Graph models a wealth of information


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Graph Data Analytics

• Graph Database environment is growing:• Neo4j, Titan, SPARQL, Pregel etc.

• Graph Data Analytics are becoming popular:• Graph Summarization[1], Graph OLAP [2] etc.

• In our research, we focus on:• Discover needs of native graph analytical queries• Process graph analytical query efficiently

[2] C. Chen, X. Yan, F. Zhu, J. Han, and P. S. Yu, “Graph olap: Towards online analytical processing on graphs,” in Data Mining, 2008. ICDM’08

[1] Tian, Y., Hankins, R. A., & Patel, J. M. (2008, June). Efficient aggregation for graph summarization. In Proceedings of the 2008 ACM SIGMOD


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Outline


Graph Window Query


• Experiments

• Future Works


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SQL Window Query• A SQL window query:

• Partitions a table• Sorts each partition• Implicitly forms window of each tuple

Window of Tuple 7

Window of a tuple contains other tuples related to it


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Graph Window Query

• In graph, a vertex can also have a set of related vertices to be its window.

• The aggregation on window is a personalized analysis over each vertex.


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Graph Window Examples

• These queries focus on the neighborhoods of each user, thus the neighborhoods forms a vertex’s window

Summarizing the age distribution of each user’s friends

Summarizing the activeness of each user’s friends

Analyze the industry distribution of a user potential connections


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Graph Window Examples

• These queries focus on the ancestor-descendent relationship of molecules, thus ancestor-descendent is a vertex’s window

Find how many enzymes are in each molecule’s pathway

Find how many molecules are affected by each enzyme in the pathway


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Graph Window Queries

• We thus identify two types of graph window queries:

• K-hop window (k-window):• A vertex’s k-hop window contains all the vertices that

are its the k-hop neighbors.

• Topological window (t-window):• A vertex’s topological window contains all the vertices

that are its accentors / descendents


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• K-hop Window:• Similar to ego-centric analysis of network analysis

community• For undirected graph:

• all vertices that can connect a vertex

• For directed graph:• In-k-hop, for vertices that reaches a vertex in k-hop• Out-k-hop, for vertices that reached by a vertex in k-hop

• K-hop, union of in-k-hop and out-k-hop

• T-Window:• Requires graph to be DAG


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• Graph Window Query:• INPUT: a specific window (k-hop, topological) and an

aggregation function

• OUTPUT: aggregated value over each vertex’s window


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Outline



Graph Window Query Processing

• Experiments

• Future Works


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Related Work• In [1] a system EAGr has been proposed to process

neighborhood query• Focuses on 1-hop neighbor

• It uses iterative planning methods to share aggregations results between different vertex’s window

• However, it assumes a large intermediate data to reside in memory, which is not reasonable for k-window () and t-window

[1] J. Mondal and A. Deshpande, “Eagr: Supporting continuous ego-centric aggregate queries overlarge dynamic graphs,” SIGMOD, 2015.


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Graph Window Query Processing• Naïve Processing I:

1. Compute vertex’s window sequentially

2. Aggregate each vertex individually

• Advantage:• No large intermediate data generated

• Inefficiencies:• Repeated computation of every vertex’s window:

• k-window is of complexity in arbitrary graph• t-window is of complexity in arbitrary graph

• Slow in individual aggregation:• Each vertex may have window size of • Total aggregation complexity can be


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Graph Window Query Processing

• Naïve Processing II:1. Materialize each vertex’s window

2. On query processing, aggregate each vertex’s window individually

• Advantage:• No computation of windows at run time

• Inefficiencies:• Materialize is not memory efficient

• All the vertex’s window can be as large as

• Query processing is still slow as in Naïve Processing I


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Overview of our approach

• Two index schemes:• Dense Block Index: for general window and k-hop

window• Parent Index: for topological window

• Indexes achieves:• Completely preserve the window information for each

vertex• Space efficiency• Efficient run-time query processing


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Dense Block Index – Matrix View• Window Matrix:

• Records vertex-window mapping• Rows represent vertex• Columns represent window

A B C D E FA 1 1 1 1 1 1B 1 1 0 1 0 1C 1 0 1 1 1 1D 1 1 1 1 0 0E 1 0 1 0 1 0F 1 1 1 0 0 1


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Dense Block Index – Matrix View• Window Matrix Properties:

• Boolean matrix• Completely keeps the vertex-

window information

• Equivalent Matrices:• Window matrix can be applied

with row and column permutations

• Invariant: number of non-zero elements ()

A B C D E FA 1 1 1 1 1 1B 1 1 0 1 0 1C 1 0 1 1 1 1D 1 1 1 1 0 0E 1 0 1 0 1 0F 1 1 1 0 0 1

A C B E D FB 1 0 1 0 1 1D 1 1 1 0 1 0F 1 1 1 0 0 1A 1 1 1 1 1 1C 1 1 0 1 1 1E 1 1 0 1 0 0


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Dense Block Index – Matrix View

• Window matrix based aggregation:• Similar to Naïve Processing II

1. Traverse the matrix vertically

2. Aggregate the cells with value one, ignore cells with value zero

• Space and Query Complexity:• in sparse matrix format• in matrix format• Note that can be as large as


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Dense Block Index• Dense Blocks:

• Given a matrix, dense blocks is the submatrix whose values are all non-zeros

• Properties of Dense Blocks ():• Space complexity

• compared to

• Query complexity• compared to

{𝐴 ,𝐵 }× {𝐴 ,𝐵 ,𝐶 }A B C D

A 1 1 1 0

B 1 1 1 0

C 0 0 0 1

D 1 0 0 1

Store row id and column id i.e. (A,B)(A,B,C) rather than 6 elements

Query: Compute A+B first, then the result is shared for window (A,B,C)

Same asymptotical bounds, thus can optimize both simultaneously


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• Dense Block Index:• For every window to be computed, index all the dense

blocks in a window matrix

• A bipartite graph

A B C D E F

A,F,D B A,CC,ED E F

A C B E D F

B 1 0 1 0 1 1

D 1 1 1 0 1 0

F 1 1 1 0 0 1

A 1 1 1 1 1 1

C 1 1 0 1 1 1

E 1 1 0 1 0 0

Dense Block Index


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Dense Block Index

• Properties:• Preserves every non-zero entry of window matrix• During query, no need to access original window

matrix

• Query Processing:1. compute partial aggregates for each dense block

2. compute final aggregates for every window


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Dense Block Index Query ProcessingSummarizing the activeness of each user’s friends:

Compute On Graph GOver 1-hop Window

A 118B 64C 103D 78E 66F 55


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Dense Block Index• Equivalent matrices may have different optimal partitions

• Find best dense block partition out of all equivalent matrices• Fixed size dense block partition is NP-hard [1]• Heuristics need to be applied

A B C D E FA 1 1 1 1 1 1B 1 1 0 1 0 1C 1 0 1 1 1 1D 1 1 1 1 0 0E 1 0 1 0 1 0F 1 1 1 0 0 1

A C B E D FB 1 0 1 0 1 1D 1 1 1 0 1 0F 1 1 1 0 0 1A 1 1 1 1 1 1C 1 1 0 1 1 1E 1 1 0 1 0 0

[1] V. Vassilevska and A. Pinar, “Finding nonoverlapping dense blocks of a sparse matrix,” Lawrence Berkeley National Laboratory, 2004


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MinHash Clustering for DBI

• Heuristic• Classifies similar windows together, then mining the

dense blocks in each cluster• Clustering + Mining

• Clustering:• Jaccard coefficient is used to measure the similarity

between windows• Since each window is a set of vertices

• MinHash is an efficient way to perform Jaccard coefficient based clustering


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MinHash Clustering for DBI

• Mining:1. Build partial window matrix for each cluster

2. Condense the rows with identical values

3. For uncondensed rows, recursively cluster + mining, until stop condition achieves


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MinHash Clustering for DBIA B C D E F

A 0 0 1 1 1 1B 1 1 1 1 1 0C 0 0 1 1 1 1D 1 1 1 1 0 1E 0 0 1 1 0 0F 0 1 1 1 1 1

A BA 0 0B 1 1C 0 0D 1 1E 0 0F 0 1

C D E FA 1 1 1 1B 1 1 1 0C 1 1 1 1D 1 1 0 1E 1 1 0 0F 1 1 1 1

A BB,D 1 1F 0 1

C D E FA,C,F 1 1 1 1

C D E FB 1 1 1 0D 1 1 0 1E 1 1 0 0

MinHash Clustering

{𝐴 ,𝐶 ,𝐹 }× {𝐶 ,𝐷 ,𝐸 ,𝐹 }

OutputsOutputs

Split

Recursive cluster


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MinHash Clustering for DBI• DBI generation can be summarized into following steps:

• Clustering Step:1. Min-Hash each vertex, based on its window

• Mining Step:1. Generate partial matrix for each window

2. Group identical rows

3. Recursive clustering

Bottlenecks

MINHASH COST: WINDOW COST: for k-window, for t-windowToo HIGH in practice


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Estimated MinHash Clustering

• For K-hop, we developed an estimation scheme to speed up the index creation process.

• The observation is that when hop goes larger, the overlapping between each vertex also goes larger• Thus we can use lower hop window information in the

clustering phase


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Comparison• MinHash Clustering

1. Clustering Step:1. Min-Hash each

vertex, based on its window

2. Mining Step:1. Generate partial

matrix for each window

2. Group Identical rows


• Estimated Clustering1. Clustering Step:

1. Min-Hash each vertex, based on its lower-hop window

2. Mining Step:1. Generate partial

matrix for each window

2. Group Identical rows


The estimation reduces the indexing time since:1. Lower-hop window has less elements, so MinHash is faster2. Lower-hop window generation requires less time


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Topological Window Processing

• Dense Block Index can be used on Topological Window as well• However, more efficient index exists given a T-

window query

• Containment Relationship in T-window• If , then • Thus, when compute window of , ’s result can be

directly used.


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Parent Index• Given , in order to use for computing , we need to

materialize the difference between and

• For a given , the vertex with smallest difference must be one of ’s parent

• Thus, for each vertex, we only index its parent which has the smallest different


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Parent Index• A parent index is a lookup table of three fields:• Vertex: the index entry• Parent: the closest parent

id• Diff: the difference

vertices between Vertex and Parent


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Parent Index based Query Processing

• Topologically process each vertex’ window

• Use the formulae:

• Topological order ensures that when processing a vertex, its parents’ results are ready


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Parent Index Creation

• Efficiently creation based on Topological Scan:• During scan, each vertex passes its current ancestor

information to its child• Child on receiving parents’ ancestor information, union

these ancestors• Child on receiving all parents information, record the

portent with smallest difference


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Outline




Experiments

• Future Works


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Experiments

• Machine: 2.27GHz CPU with 32 GB memory

• Data Synthetic:• SNAP [1] generator for directed graphs• DAGGR [2] generator for DAGs

[2] H. Yildirim, V. Chaoji, and M. J. Zaki, “Dagger: A scalable index for reachability queries in large dynamic graphs,” arXiv preprint arXiv:1301.0977, 2013.

[1] Stanford Networ Analysis Platform, http://snap.stanford.edu/snap/index.html


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Comparing Algorithms• K-hop window:

• MA: materialize ahead algorithm (materialize vertex-window mapping, individual aggregate)

• KBBFS: bounded BFS for computing window of each vertex• MC: MinHash Clustering• EMC: Estimated MinHash Clustering

• Topological window:• MA• DBI: dense block index• TS: Topological Scan to compute window of each vertex• PI: parent index


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Effectiveness of Estimation

Hop = 1 Hop = 2

Hop = 3 Hop = 4


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Benefit of Estimation

Degree 160

Hop MC_HASH MC_BFS EMC_HASH EMC_BFS EMC/MC

2 157,885 241,072 1,666 120,931 0.307294

3 2,281,794 4,494,853 1,637 2,257,493 0.33337

4 4,355,439 8,633,192 1,631 4,414,207 0.339977

Hop MC_HASH MC_BFS EMC_HASH EMC_BFS EMC/MC

2 33,611 19,559 484 9,974 0.19669

3 417,102 742,502 470 374,489 0.323351

4 964,521 184,3078 471 927,751 0.330611

Degree 40


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Index size of MC and EMC

Degree = 40


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Scalability of EMC

V = 100k, hop =1

V = 100k, hop = 2


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Effectiveness of PI

V = 10k


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Index size of PI

Vertex = 10k


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Indexing Time of PI

Degree = 20


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Scalability of PI

Degree = 10


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Outline




• Experiments

Future Works


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Conclusion and Future Work

• Conclusion:• We proposed two graph window queries and two

indexes for efficient processing

• In future:• Extend the query processing to handle large graphs (in

parallel platform / disk resident index)• More complex aggregation processing (include graph

OLAP)• Dynamic graphs (able to handle updates)


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Thank you !


towards data analytics on attributed graphs ngs qe oral presentation 1 student : qi fan supervisor:...

Documents

graph data analytics

graph model

graph olap

data yield

data representation

data mining

graph queries

graph summarization1