Challenges in Querying Graph Streams for Friendly Compression

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Query-Friendly Compression of Graph
Streams
 
Graph Streams
1
/23
•
Graph Stream:
 Continuous stream of graph edges 
 
Telephone
network, communication network, social media data, IP traffic
e
2
e
1
e
5
e
9
e
11
e
1
e
2
e
3
e
4
e
5
e
6
e
8
e
9
e
10
e
11
e
12
e
13
e
15
e
16
Edge Stream
Graph Structure
With Edge Frequency
….
A.
 Khan
, C. Aggarwal
 
Graph Streams
1
/23
•
Graph Stream:
 Continuous stream of graph edges 
 
Telephone
network, communication network, social media data, IP traffic
•
Massive volume and high speed
•
Construct summary to support future queries
e
2
e
1
e
5
e
9
e
11
e
1
e
2
e
3
e
4
e
5
e
6
e
8
e
9
e
10
e
11
e
12
e
13
e
15
e
16
Edge Stream
Graph Structure
With Edge Frequency
….
Challenges in Data Streams
Querying
2
/23
•
Trade-off among Space, Accuracy, and Efficiency:
       -- Increasing space increases accuracy, but reduces throughput
•
Other requirements:
       -- Build summary in one pass over the stream
       --  Incremental updates in summary
A.
 Khan
, C. Aggarwal
Additional Challenges in Graph Streams
Querying: Query Expressibility
3
/23
       
•
Compute reachability formed by heavy-hitter edges
e
2
e
1
e
5
e
9
e
11
e
1
e
2
e
3
e
4
e
5
e
6
e
8
e
9
e
10
e
11
e
12
e
13
e
15
e
16
Edge Stream
Graph Data:
Red Edges are heavy-hitter edges
….
A.
 Khan
, C. Aggarwal
Additional Challenges in Graph Streams
Querying: Query Expressibility
3
/23
       
•
Compute reachability formed by heavy-hitter edges
e
2
e
1
e
5
e
9
e
11
e
1
e
2
e
3
e
4
e
5
e
6
e
8
e
9
e
10
e
11
e
12
e
13
e
15
e
16
Edge Stream
Graph Data:
Red Edges are heavy-hitter edges
….
V
1
V
2
A.
 Khan
, C. Aggarwal
Additional Challenges in Graph Streams
Querying: Query Expressibility
3
/23
       
•
Compute reachability formed by heavy-hitter edges
e
2
e
1
e
5
e
9
e
11
e
1
e
2
e
3
e
4
e
5
e
6
e
8
e
9
e
10
e
11
e
12
e
13
e
15
e
16
Edge Stream
Graph Data:
Red Edges are heavy-hitter edges
….
V
1
V
2
Related Work
4
/23
•
Graph Summarization:
              
- Query Preserving Graph Compression (SIGMOD 2012)
               
- Graph Summarization with Bounded Error (SIGMOD 2008)
               
- Representing Web Graphs (ICDE 2003)
               - The Transitive Reduction of a Directed Graph (SIGCOMP 1972)
•
Data Stream Summarization:
               - Sketches (SIGMOD 2002, VLDB 2002, SIGMOD 2004)
               - Histograms (SIGMOD 1996, VLDB 1998)
               - Wavelets (SIAM Rev. 1996)
               - Space Saving (ICDT 2005)
•
Graph Streams Querying: 
               - gSketches (VLDB 2012)
               - Analyzing Graph Structure via Linear Measurements (SODA 2012)
               
- Graph Sketches: Sparsification, Spanners, and Subgraphs (PODS 2012)
               -
 
TCM Sketch (SIGMOD 2016)
Related Work
5
/23
•
Graph Summarization:
              
- Query Preserving Graph Compression (SIGMOD 2012)
               
- Graph Summarization with Bounded Error (SIGMOD 2008)
               
- Representing Web Graphs (ICDE 2003)
               - The Transitive Reduction of a Directed Graph (SIGCOMP 1972)
•
Data Stream Summarization:
               
- Sketches (SIGMOD 2002, VLDB 2002, SIGMOD 2004)
               - Histograms (SIGMOD 1996)
               - Wavelets (SIAM Rev. 1996)
               - Space Saving (ICDT 2005)
•
Graph Streams Querying: 
               
- gSketches (VLDB 2012)
               - Analyzing Graph Structure via Linear Measurements (SODA 2012)
               
- Graph Sketches: Sparsification, Spanners, and Subgraphs (PODS 2012)
               - 
TCM Sketch (SIGMOD 2016)
Related Work
6
/23
•
TCM Sketch (SIGMOD 2016):
              
- Does not provide theoretical error bounds
               
               - Difficult to answer reachability over heavy-hitter edges
A.
 Khan
, C. Aggarwal
Count-Min Sketch
h
w
( e, f ) 
+ f
+ f
+ f
H
1
(e)
H
w
(e)
•
“h”  much smaller than total no of edges
•
Estimate frequency of an edge, find heavy-hitter edges
•
Cannot answer structural queries: are these two nodes
connected by only high-frequency edges?
7
/23
Our Solution: GMatrix Synopsis
h
h
 k-th Hash Function
hashes into
( H
k
(i), H
k
(j))
w
(H
1
(i), H
1
(j))
H
1
(
.
)
H
3
(
.
)
H
4
(
.
)
H
2
(
.
)
•
incoming edge:  e = (i,j)
 
•
“h”  much smaller than 
         total no of nodes
 
8/23
A.
 Khan
, C. Aggarwal
GMatrix Compression
       
•
Contract nodes into a total of h super-nodes
•
Different hash functions create different contractions 
⇒
 Holds key to
effective query processing
•
A graph with 10
8 
nodes, 10
10
 edges 
⇒
 Storage  40 GB
 
•
GMatrix with h = 10
3
 and w = 10 
⇒
 Storage  40 MB
9
/23
A.
 Khan
, C. Aggarwal
Choice of Hash Functions
10
/23
       
•
Pair-wise independent, e.g., modular hash function
•
P is a prime number larger than any node id:  (1, 2, … , n)
•
a, b chosen uniformly from (1, P-1)
A.
 Khan
, C. Aggarwal
Reverse Hash Mapping
11
/23
•
Modular hash function:    reverse hash mapping size ⌊P/h⌋
•
Reverse hash mapping 
⇒
 small size and computed efficiently
•
7
x
  mod 9 = 1
•
x
= 4
•
7*4 = 3*9 + 1
•
Can be computed in time O(⌊P/h⌋  log P) using extended Euclidean
algorithm
A.
 Khan
, C. Aggarwal
Other Synopsis Options with Same
Functionality as GMatrix
12
/23
•
Reverse hash mapping computes   w . n
2
/h
2
  intersections
h
2
w
( ij, f ) 
+ f
+ f
+ f
H
1
(ij)
H
w
(ij)
•
In GMatrix, reverse hash mapping computes  2. w . n/h  intersections
A.
 Khan
, C. Aggarwal
Queries supported by GMatrix
(not a comprehensive list)
13
/23
       
•
Edge Frequency Query
•
Heavy-hitter Edge Query
•
Node Frequency Query
•
Sub-graph Edge Frequency Query
•
Heavy-hitter Node Query
•
Reachability Query over High-frequency Edges
A.
 Khan
, C. Aggarwal
Queries supported by GMatrix
(not a comprehensive list)
13
/23
       
•
Edge Frequency Query
•
Heavy-hitter Edge Query
•
Node Frequency Query
•
Sub-graph Edge Frequency Query
•
Heavy-hitter Node Query
•
Reachability Query over High-frequency Edges
Edge-Frequency Estimation Query
14
/23
       
•
For edge (i, j), compute the frequencies of w different cells: (H
k
(i), H
k
(j), k)
•
The minimum of these values is returned as the estimated frequency
•
Estimation is good for high-frequency edges
•
If true frequency is significant fraction of total stream size, then relative
error is small
A.
 Khan
, C. Aggarwal
Heavy-Hitter Edge Query
15
/23
       
•
Find all edges with frequency greater than F
•
No false negative, but false positive
•
Find all hash-edges with frequency at least F
•
Reverse hash mapping to find real edges
•
Intersection of edge sets
A.
 Khan
, C. Aggarwal
Heavy-Hitter Edge Query: Optimization
16
/23
       
•
If a node does not appear as the source node of some potential
frequent edge in at least one of the w hash functions, that node and
its outgoing edges can be safely eliminated.
First Optimization
Second
 Optimization
A.
 Khan
, C. Aggarwal
Heavy-Hitter Edge Query: Time Complexity
17
/23
A.
 Khan
, C. Aggarwal
Reachability Query
18
/23
       
•
Find if two query nodes are connected by a path with edges
having frequency at least F
•
Determine all edges for which frequency is at least F using heavy-
hitter edge query
•
Answer reachability query with these edges
A.
 Khan
, C. Aggarwal
Experimental Results
19
/23
Friendster Stream (Zipf Frequency Distribution with Varying Skew)
•
Experiments were performed on a single core of 10GB, 2.4GHz Xeon server
 Skew 1.0
 Skew 1.2
 Skew 1.4
A.
 Khan
, C. Aggarwal
Edge Frequency Estimation Query
Query over top-500 
frequent edges
20
/23
A.
 Khan
, C. Aggarwal
Heavy Hitter Edge Query
21
/23
Frequency Threshold 
= 0.01% of Total Stream Size
Query Answering Time
Reachability Query
22
/23
•
Each reachability query can be processed in 0.1 sec
Frequency Threshold 
= 0.01% of Total Stream Size
A.
 Khan
, C. Aggarwal
Conclusions
23
/23
       
•
GMatrix synopsis for summarizing rapid graph streams
•
Can be leveraged for a variety of frequency and structural queries
A.
 Khan
, C. Aggarwal
•
Future Work: 
Improving accuracy by hashing high- and low-frequency
edges separately
?

								

								

								

										

												
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Graph streams pose challenges in querying due to trade-offs among space, accuracy, and efficiency. The need to balance space and accuracy while maintaining throughput presents obstacles in constructing summaries and incorporating incremental updates. Additional challenges include query expressibility and preserving connectivity information of heavy-hitter edges.


								
    
																		
  •   Graph Streams
    • 
																	
  •    Querying
    • 
																	
  •    Challenges
    • 
																	
  •    Friendly Compression
    • 
																	
  •    Connectivity
    • 
																

								

								

									

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  1. Query Query- -Friendly Compression  Friendly Compression of  Streams of Graph  Graph  Streams Arijit Khan Charu C. Aggarwal Nanyang Technical University Singapore IBM T. J. Watson Research Lab NY, USA
    2. 
            
            
            
  2. Graph Streams Graph Stream: Continuous stream of graph edges     Telephone  network, communication network, social media data, IP traffic e1 e2 e5 e4 e3    . e2 e1 e5 e9 e11 e6 e8 e11 e9e10 e15 e13 Edge Stream e12 e16 Graph Structure  With Edge Frequency 1/23 A. Khan, C. Aggarwal
    3. 
            
            
            
  3. Graph Streams Graph Stream: Continuous stream of graph edges     Telephone  network, communication network, social media data, IP traffic e1 e2 e5 e4 e3    . e2 e1 e5 e9 e11 e6 e8 e11 e9e10 e15 e13 Edge Stream e12 e16 Graph Structure  With Edge Frequency Massive volume and high speed Construct summary to support future queries 1/23
    4. 
            
            
            
  4. Challenges in Data Streams  Querying Trade-off among Space, Accuracy, and Efficiency:         -- Increasing space increases accuracy, but reduces throughput  Other requirements:         -- Build summary in one pass over the stream        --  Incremental updates in summary 2/23 A. Khan, C. Aggarwal
    5. 
            
            
            
  5. Additional Challenges in Graph Streams  Querying: Query Expressibility Compute reachability formed by heavy-hitter edges e1 e2 e5 e4 e3    . e2 e1 e5 e9 e11 e6 e8 e11 e9e10 e15 e13 Edge Stream e12 e16 Graph Data:  Red Edges are heavy-hitter edges 3/23 A. Khan, C. Aggarwal
    6. 
            
            
            
  6. Additional Challenges in Graph Streams  Querying: Query Expressibility Compute reachability formed by heavy-hitter edges V1 e1 e2 e5 e4 e3    . e2 e1 e5 e9 e11 e6 e8 e11 e9e10 V2 e15 e13 Edge Stream e12 e16 Graph Data:  Red Edges are heavy-hitter edges 3/23 A. Khan, C. Aggarwal
    7. 
            
            
            
  7. Additional Challenges in Graph Streams  Querying: Query Expressibility Compute reachability formed by heavy-hitter edges V1 e1 e2 e5 e4 e3    . e2 e1 e5 e9 e11 e6 e8 e11 e9e10 V2 e15 e13 Edge Stream e12 e16 Graph Data:  Red Edges are heavy-hitter edges Need to preserve connectivity  information of the edges in the  graph data 3/23
    8. 
            
            
            
  8. Related Work Graph Summarization: - Query Preserving Graph Compression (SIGMOD 2012)                - Graph Summarization with Bounded Error (SIGMOD 2008)                - Representing Web Graphs (ICDE 2003)                  - The Transitive Reduction of a Directed Graph (SIGCOMP 1972) Data Stream Summarization:                - Sketches (SIGMOD 2002, VLDB 2002, SIGMOD 2004)                - Histograms (SIGMOD 1996, VLDB 1998)                - Wavelets (SIAM Rev. 1996)                - Space Saving (ICDT 2005)              Graph Streams Querying:                 - gSketches (VLDB 2012)                - Analyzing Graph Structure via Linear Measurements (SODA 2012)                - Graph Sketches: Sparsification, Spanners, and Subgraphs (PODS 2012) - TCM Sketch (SIGMOD 2016) 4/23
    9. 
            
            
            
  9. Related Work Graph Summarization: - Query Preserving Graph Compression (SIGMOD 2012)                - Graph Summarization with Bounded Error (SIGMOD 2008)                - Representing Web Graphs (ICDE 2003)                  - The Transitive Reduction of a Directed Graph (SIGCOMP 1972) Data Stream Summarization:                - Sketches (SIGMOD 2002, VLDB 2002, SIGMOD 2004)                - Histograms (SIGMOD 1996)                - Wavelets (SIAM Rev. 1996)                - Space Saving (ICDT 2005)              Graph Streams Querying:                 - gSketches (VLDB 2012)                - Analyzing Graph Structure via Linear Measurements (SODA 2012)                - Graph Sketches: Sparsification, Spanners, and Subgraphs (PODS 2012) - TCM Sketch (SIGMOD 2016) 5/23
    10. 
            
            
            
  10. Related Work TCM Sketch (SIGMOD 2016): - Does not provide theoretical error bounds                - Difficult to answer reachability over heavy-hitter edges 6/23 A. Khan, C. Aggarwal
    11. 
            
            
            
  11. Count-Min Sketch h + f H1(e) ( e, f )  + f w Hw(e) + f    h     much smaller than total no of edges Estimate frequency of an edge, find heavy-hitter edges Cannot answer structural queries: are these two nodes  connected by only high-frequency edges? 7/23
    12. 
            
            
            
  12. Our Solution: GMatrix Synopsis H4(.) incoming edge:  e = (i,j) H3(.) w    h     much smaller than           total no of nodes H2(.) H1(.) h k-th Hash Function  hashes into  ( Hk(i), Hk(j)) h (H1(i), H1(j)) 8/23 A. Khan, C. Aggarwal
    13. 
            
            
            
  13. GMatrix Compression Contract nodes into a total of h super-nodes Different hash functions create different contractions     Holds key to  effective query processing A graph with 108 nodes, 1010 edges     Storage  40 GB  GMatrix with h = 103 and w = 10     Storage  40 MB  9/23 A. Khan, C. Aggarwal
    14. 
            
            
            
  14. Choice of Hash Functions Pair-wise independent, e.g., modular hash function ( ( ) ) = + ( ) mod mod H i ai b P h P is a prime number larger than any node id:  (1, 2,     , n)  a, b chosen uniformly from (1, P-1) 10/23 A. Khan, C. Aggarwal
    15. 
            
            
            
  15. Reverse Hash Mapping             = = 1 ( ) : ( ) H p i H i p 7x  mod 9 = 1 x= 4  7*4 = 3*9 + 1 Reverse hash mapping     small size and computed efficiently  Modular hash function:    reverse hash mapping size    P/h    Can be computed in time O(   P/h     log P) using extended Euclidean algorithm 11/23 A. Khan, C. Aggarwal
    16. 
            
            
            
  16. Other Synopsis Options with Same  Functionality as GMatrix h2 + f H1(ij) ( ij, f )  + f w Hw(ij) + f Reverse hash mapping computes   w . n2/h2  intersections In GMatrix, reverse hash mapping computes  2. w . n/h  intersections 12/23 A. Khan, C. Aggarwal
    17. 
            
            
            
  17. Queries supported by GMatrix  (not a comprehensive list) Edge Frequency Query Heavy-hitter Edge Query Node Frequency Query Sub-graph Edge Frequency Query  Heavy-hitter Node Query Reachability Query over High-frequency Edges 13/23 A. Khan, C. Aggarwal
    18. 
            
            
            
  18. Queries supported by GMatrix  (not a comprehensive list) Edge Frequency Query Heavy-hitter Edge Query Node Frequency Query Sub-graph Edge Frequency Query  Heavy-hitter Node Query Reachability Query over High-frequency Edges 13/23
    19. 
            
            
            
  19. Edge-Frequency Estimation Query For edge (i, j), compute the frequencies of w different cells: (Hk(i), Hk(j), k) The minimum of these values is returned as the estimated frequency  Estimation is good for high-frequency edges If true frequency is significant fraction of total stream size, then relative  error is small         +     . L , ( i ) , ( i ) , ( i ) Q j Q j Q j 14/23 A. Khan, C. Aggarwal
    20. 
            
            
            
  20. Heavy-Hitter Edge Query Find all edges with frequency greater than F No false negative, but false positive Find all hash-edges with frequency at least F ( p H V k ), ( )     ( ), H q k F k Reverse hash mapping to find real edges     : ) , ( j i E k =     ) q                 1 1 ( ), ( i H p j H k k Intersection of edge sets w     k = E k 1 15/23 A. Khan, C. Aggarwal
    21. 
            
            
            
  21. Heavy-Hitter Edge Query: Optimization First Optimization If a node does not appear as the source node of some potential  frequent edge in at least one of the w hash functions, that node and  its outgoing edges can be safely eliminated. Second Optimization     , (     =         ( , ) : ) j , S Q Q i i Q j Q 1 2 1 2 = ( , ) ( , ) ( , ) S Q Q S P P S Q P Q P             1 2 1 2 1 1 2 2 16/23 A. Khan, C. Aggarwal
    22. 
            
            
            
  22. Heavy-Hitter Edge Query: Time Complexity                 w P     = k            +            2 2 log O h w P k c               h 1 17/23 A. Khan, C. Aggarwal
    23. 
            
            
            
  23. Reachability Query Find if two query nodes are connected by a path with edges  having frequency at least F Determine all edges for which frequency is at least F using heavy- hitter edge query Answer reachability query with these edges 18/23 A. Khan, C. Aggarwal
    24. 
            
            
            
  24. Experimental Results #Nodes #Edges Agg. Edge  Freq. Max. Edge  Freq. 4.43    108 Flat Stream  Size Compressed  Stream Size 16.47 GB 2.37 GB 250 MB Skew 1.0 66M 3612M 1010 80GB Skew 1.2 1.81    109 3.22    109 Skew 1.4 Friendster Stream (Zipf Frequency Distribution with Varying Skew) GMatrix Size GMatrix Update Time 40MB (h=1000, w=10) 10-6 sec Experiments were performed on a single core of 10GB, 2.4GHz Xeon server 19/23 A. Khan, C. Aggarwal
    25. 
            
            
            
  25. Edge Frequency Estimation Query 8.0E-04 5 Stream Update Time  Observed Error 4 (micro seconds) 6.0E-04 3 GMatrix 4.0E-04 2 2.0E-04 1 Count-Min Sketch 0 1.0E-07 3 # Hash Functions (w) 5 10 1 Skew (Zipf) 1.2 1.4         Query i     | Est  Freq True Freq | i i = Observed Error Query over top-500  frequent edges         Query i True Freq i 20/23 A. Khan, C. Aggarwal
    26. 
            
            
            
  26. Heavy Hitter Edge Query 10 Frequency Threshold  = 0.01% of Total Stream Size False  Positive  Rate 5 GMatrix Count Min Sketch 0 1 1.2 1.4 Skew (ZipF)     # Edges incorrectl reported y      heavy hitter = False Positive Rate # True heavy hitter  edges Frequency Threshold (%  of Total Stream Size) 1 0.1 0.01 GMatrix Count-Min Sketch 28 sec 149 sec 771 sec 1 sec 2 sec 7 sec Query Answering Time 21/23
    27. 
            
            
            
  27. Reachability Query Skew (ZipF) 1.0 1.2 1.4 Reachability Error 0.012 0.008 0.004 Frequency Threshold  = 0.01% of Total Stream Size # incorrectl pairs Node reported y  reachable = Reachabili Error ty  # Node pairs Each reachability query can be processed in 0.1 sec 22/23 A. Khan, C. Aggarwal
    28. 
            
            
            
  28. Conclusions GMatrix synopsis for summarizing rapid graph streams Can be leveraged for a variety of frequency and structural queries Future Work: Improving accuracy by hashing high- and low-frequency  edges separately? 23/23 A. Khan, C. Aggarwal
    29. 
            
            
            
							

							

					

					
					
					

						
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              Solving Train Track Problems Using Interval Graphs and Graph Coloring
            

            

              
Solving Train Track Problems Using Interval Graphs and Graph Coloring

              
rossief

            

          

        
      


     							

						

					

					

				


				

			

		
		

			

				
				

					
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