Trends In Graph Data Management And Mining

ETDT Symposium © Srinath Srinivasa, IIIT Bangalore 1

Trends in Graph Data Management and Mining

Srinath SrinivasaIIIT [email protected]


No data is an island…


Outline

• Graph Data and its characteristics• Structural Queries• Storage Models for Graphs • Data Models for Graph Databases• Structural Indexes• Mining Frequent Subgraphs

– gSpan– FBT


Graph Data

A graph G = (V,E) is a collection of nodes (vertices) and edges.

A graph represents a “relationship structure” among different data elements.

A graph database is a collection of different graphs representing different relationship structures.


Graph database versus Relational database

A relational database maintains different instances of the same relationship structure (represented by its ER schema)

A graph database maintains different relationship structures


Graph Database Applications

• Software Engineering– UML diagrams, flowcharts, state machines,

…

• Knowledge Management– Ontologies, Semantic nets, …

• Bioinformatics– Molecular structures, bio-pathways, …

• CAD– Electrical circuits, IC designs, …

• Cartography, XML Bases, HTML Webs, …


Queries over Graph Databases

• Attribute Queries– Queries over attributes and values in

nodes and edges. Equivalent to a relational query within a given schema

• Structural Queries– Queries over the relationship structure

itself. Examples: Structural similarity, substructure, template matching, etc.


Structural Queries on Graph Data

• Undirected Graphs– Structural similarity, substructure

• Directed Graphs– Structural similarity, substructure, reachability

• Weighted Graphs– Shortest paths, “best” matching substructure

• Labeled Graphs– Labeled structural similarity, unlabeled

structural similarity


Structural Queries• Substructure query

– Given a graph database G = {G1, G2, … Gn} and a query graph Q, return all graphs Gi where Q is a

subgraph of Gi.

• Structural similarity – Given a graph database G = {G1, G2, … Gn} and a

query graph Q and a threshold t, return all graphs Gi where the edit distance between Q and Gi is at most t.

– The edit distance between two graphs is the number of edge modifications (additions, deletions) required to rewrite one graph into the other


Structural Queries

• (Sub)graph isomorphism is believed to be neither in P nor in NP-complete

• In graph databases structure matching has to be performed against a set of graphs!

• Proper storage, pre-processing and index structures crucial if structural searches are to be practical


Storing Graph DataAttributed Relational Graphs (ARGs)

A

B

C D

pq

r

s t

A B q

B C s

B D t

A C p

A D r


Storing Graph Data

• ARGs– ARGs store a graph as a set of rows,

each depicting an edge – Amenable to storage in an RDBMS and

easy attribute searches using SQL– Costly structural searches, requiring

complex nesting of SELECT statements– Each graph needs a separate table


Storing Graph Data

A

B

C D

pq

r

s t

Maximum walks:

A r D t B s C p A q B


Storing Graph Data

• Maximum walks– Stores all walks of maximum possible length in

the graph – Traversable graphs stored as a single sequence– Easy to answer attribute queries and

reachability queries– Non-traversable graphs need multiple

sequences – Variable record length for sequences – Significant pre-processing time for reducing

graph to the best set of sequences


Storing Graph DataLinear DFS Tree: (Example: Glide http://www.cs.nyu.edu/cs/faculty/shasha/papers/graphgrep/)

A

B

C D

pq

r

s t

A%1 /p/ C /s/ B%1q /t/ D%1r


Storing Graph Data

• Linear DFS Tree– A sequence form of depth-first traversal of

the graph – Suitable for any kind of undirected graphs

(but not necessarily for directed graphs) – Suitable for attribute queries – Some techniques proposed for substructure

queries over linear DFS trees– Large pre-processing time


Storing Graph DataXML with IDREFS:

A

B

C D

<node id=“A”, adj=“C D”><node id=“B”>

<node id=“C”></node><node id=“D”></node>

</node></node>


Storing Graph Data

• XML with IDREFS– Reduces graph database to an XML base– Use XPath / XQuery engines for structural

queries – Widely supported by a variety of XML

parsers – Costly structure/sub-structure matching – Needs distinction between IDREF edges and

hierarchy edges


Graph Database Models

• “Schema-less” collection of graphs– Example: GraphGrep, Daylight ACD,

gIndex

• Database as a graph– Example: SUBDUE

• Database with schema and views – Example: GRACE


Structural Indexes

• Used for fast structure-based retrieval of graphs

• Primarily meant for labeled undirected graphs

• Usually support substructure and structural similarity searches

• May either return exact matches (NP-complete) or inexact matches based on heuristics (P)


Structural IndexesGraphGrep (Guigno and Shasha 2002)

Two index files: “Fingerprint” file holding label-paths “Path” file holding id-paths

… paths from length 1 up to a maximum lp



A

B

A D

1

2

3 4

G1

Path G1 G2

AA 2 0

AB 2 1

AD 1 1

BD 1 0

AAB 2 1

ABA 2 0

Database Fingerprint file



A

B

A D

1

2

3 4

G1

Path G1

AA {1-3, 3-1}

AB {1-2, 3-2}

AD {1-4}

BD {2-4}

AAB {1-3-2, 3-1-2}

ABA {1-2-3, 3-2-1}

Database Paths file


Structural Indexes

• GraphGrep– Stores all paths in member graphs up to a

maximum length – Signature file narrows search space – Exact substructure matching possible when

node id in query matches node id in member graphs

– Exponential preparation time – Running time increases exponentially as

max path length increases


Structural IndexesHierarchical Conceptual Clusters (SUBDUE) (Jonyer, Cook, Holder 2001)

Database

Graph 1 Graph 2

Concept 1

Concept 2Rest ofGraph 1

Rest ofGraph 2

Concept 1.1


Structural Indexes

• Hierarchical Conceptual Clusters– Clusters the database into commonly occurring

substructures– Database is organized as a hierarchical index – Clustering based on substructures that

perform “best compression” by reducing graph description length

– Number of clusters may increase exponentially– Compression / search time significant


Structural IndexesHierarchical Vector Spaces (Grace 1) (Srinivasa, Acharya, Khare, Agrawal, 2002)

A

B

A D

A:A 1A:B 2A:D 1B:D 1

Level 1 vector


Structural IndexesHierarchical Vector Spaces (Grace 1)

A

B

A D

Level 2 graphs and vectors

AA BD

AB AD

AA:BD 1AB:AD 1


Structural Indexes

• Hierarchical vector spaces– Hashes a graph onto vectors in a hierarchy

of vector spaces – Higher level graphs are formed by replacing

edges (vectors) of lower level by nodes – Compression of a graph may lead to several

higher level graphs – Fast structural similarity searches; but

based on inexact matching – View explosion anomaly during refinement


Structural Indexes gIndex (Yan, Yu, Han, 2004)

1. Mine database for frequent substructures (using gSpan)

2. Maintain index structure containing (size, substructure) pairs

3. Increase minsup as the size of the indexed substructure increases


Structural Indexes gIndex (Yan, Yu, Han, 2004)

Given a query graph q:

1. Mine database along with q, and determine all frequent substructures F in q

2. Reduce search space to all graphs containing all frequent substructures of F

3. Perform graph matching against all graphs in the reduced search space


Graph Mining

• Given a database of graphs find all frequently occurring substructures in the database


Notes on Frequent Item-set Mining

• The Apriori algorithm is useful for mining frequent item-sets from transaction logs

• Apriori is based on the fact that in order to construct a frequent L item-set it is sufficient to know only the set of all frequent L-1 item-sets

• Apriori property holds for frequent subgraphs • However, apriori algorithm on a graph

database requires several sub-graph isomorphism checks!


Apriori Based Graph Mining

• Strategy for Apriori-based graph mining– Use a re-write strategy to represent all

graphs in the database as a unique sequence

– Substructure search reduces to a sub-sequence search

– Use AprioriAll (Apriori for sequences) to mine the database

– Best known rewrite mechanism to date is proposed in gSpan.


gSpan

A

B

A D

p q

rp

0

1

2 3

1. First build a DFS tree (shown in thick lines)

2. Mark each node by its visiting time in the DFS run (shown by numeral)

3. Write the graph as a sequence based on node visiting time. Append all back links from a node after the first forward link into the node.


gSpan

A

B

A D

p q

rp

0

1

2 3

Sequence:

(0,1,A,q,B)(1,2,B,r,A)(2,0,A,p,A)(1,3,B,p,A)

Since a graph has many DFS trees, consider only the DFS tree which yields sequence with the least lexicographic value.


Filtration Based Technique (FBT)

• Proposed by Srinivasa and BalaSundaraRaman (Submitted after first revision to IEEE TKDE)

• Opposite of Apriori construction on graphs but equivalent to Apriori on walks

• Starts with an assertion that all graphs in the database are isomorphic

• Filters away all edges that contradict such an assertion

• Algorithm converges to the maximal common (frequent) subgraph.


Filtration Based Technique (FBT)

• Filtration is based on enumerating label-walks in the graphs. Label walks accentuate differences between graphs as the length of the walks increase…


FBT

A

B C

A

B

A C

B

Length-1Walks

AB, AB, BC, AC AB, AB, BC, AC


FBT

A

B C

A

B

A C

B

Length-2Walks

ABA, ABC, BCA, BAC, ABA, ACB

ABC, ACB, BCA, BAC, BAB, BAC


FBT

1. i = 1 2. Enumerate walks of length i from member

graphs and organize them into different buckets based on label sequence

3. Discard buckets that don’t have minsup 4. i++ 5. Remove as intermediate results all graphs

that don’t have walks of length i6. Go to step 2 until no more walks exist


FBT

• Very fast convergence, but can find only maximal common substructures

• If two or more common substructures overlap, FBT cannot separate the substructures

• Applied successfully to carcinogen dataset from US NTP, protein structures from PDB and Web traversal logs from Yahoo.


GRACE2 and Safari

• Second version of GRACE • Supports a query algebra for graph

queries, views, and dynamic schemas

• Query language called Safari


GRACE2 Data Model

• Member graphs • Node, edge and graph attributes • The “default” graph • Schema graphs and meta-graphs


Safari Constructs

• selectin <cond> <graphref> – Use graphref as a schema and return a view

of the schema based on cond

• selecton <cond> <graphref> – Search for cond within graph referred by

graphref and return a subgraph • selectgraph <cond> <graphref>

– Retrieve graph matching cond from the schema or meta-graph referred by graphref. If more than one graph matches cond, another view is returned.


References

1. I. Jonyer, D.J. Cook, L.B. Holder. Graph-Based Hierarchical Conceptual Clustering. Journal of Machine Learning Research, Vol 2, 2001.

2. Rosalba Guigno, Dennis Shasha. GraphGrep: A Fast and Universal Method for Substructure Searches. Proc of ICCV 2002.

3. Srinath Srinivasa, Sumit Acharya, Rajat Khare, Himanshu Agrawal. Vectorization of Structure for Indexing Graph Databases. Proc of IASTED Int’l Conf on Information Systems and Databases, ISDB 2002, Tokyo, Japan.

4. Srinath Srinivasa, Sujit Kumar. A Platform Based on the Multi-Dimensional Data Model for Analysis of Bio-Molecular Structures. Proc of VLDB 2003, Berlin, Germany.


References

5. Xifeng Yan, Jiawei Han. gSpan: Graph-Based Substructure Pattern Mining.

6. Xifeng Yan, Philip S. Yu, Jiawei Han. Graph Indexing: A Frequent Substructure Based Approach. Proc of SIGMOD 2004.

7. Srinath Srinivasa, Martin Meier, Mandar R. Mutalikdesai, Gopinath P.S., Gowrishankar K.A. LWI and Safari: A New Index Structure and Query Model for Graph Databases.


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