Regularities and dynamics in bisimulation reductions of big graphs - - PowerPoint PPT Presentation
Regularities and dynamics in bisimulation reductions of big graphs Yongming Luo , George Fletcher, Jan Hidders, Paul De Bra and Yuqing Wu GRADES 2013 SIGMOD/PODS 2013 The research of YL, GF, JH and PD is supported by the Nether- lands
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Yongming Luo , George Fletcher, Jan Hidders, Paul De Bra and Yuqing Wu GRADES 2013 • SIGMOD/PODS 2013
The research of YL, GF, JH and PD is supported by the Nether- lands Organisation for Scientific Research. The research of YW is supported by Research Foundation Flan- ders during her sabbatical visit to Hasselt University, Bel- gium.
2/11 department of mathematics and computer science
An example of bisimulation reduction
3/11 department of mathematics and computer science
◮ Bisimulation partitioning is an important concept in many
fields (computer science, modal logic, etc.), in DB research as well (structural index, graph reduction)
◮ It can be seen as a way of clustering nodes
1 2 3 4 5 6 7 8 P1 P3 P4 P2
Figure: Bisimulation partition example, partition block graph (reduction graph) {P2 ↔ P1 → P3 → P4}
3/11 department of mathematics and computer science
◮ Bisimulation partitioning is an important concept in many
fields (computer science, modal logic, etc.), in DB research as well (structural index, graph reduction)
◮ It can be seen as a way of clustering nodes
1 2 3 4 5 6 7 8 P1 P3 P4 P2
Figure: Bisimulation partition example, partition block graph (reduction graph) {P2 ↔ P1 → P3 → P4}
◮ Reduce graph size while preserving structural properties
(e.g., reachability)
◮ Result can be seen as a graph ◮ Many algorithms, no work on analyzing the results
4/11 department of mathematics and computer science
4/11 department of mathematics and computer science
◮ Do graphs under bisimulation reduction also have
4/11 department of mathematics and computer science
◮ Do graphs under bisimulation reduction also have
◮ How would that knowledge help us?
5/11 department of mathematics and computer science
◮ Big graphs, from 1 Million to 1.4 Billion edges
◮ One dynamic social graph, from 17 Million to 33
◮ State-of-the-art I/O efficient algorithm for
◮ We use cumulative distribution function (CDF) to
6/11 department of mathematics and computer science
7/11 department of mathematics and computer science
Partition block size distribution
100 102 104 106 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8
x (# of nodes per PB) cumulative % of PB with ≥ x
real graphs
100 102 104 106 108 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7
x (# of nodes per PB)
synthetic graphs
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8/11 department of mathematics and computer science
Bisimulation graph in/out-degree distribution
100 102 104 106 108 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8
x (in-degree) cumulative % of Nk with ≥ x
real graphs
100 102 104 106 108 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7
x (in-degree)
synthetic graphs
Jamendo LinkedMDB DBLP DBPedia WikiLinks Twitter Flickr-Grow BSBM SP2B Power Random
9/11 department of mathematics and computer science
◮ Does the bisimulation result grow when the
9/11 department of mathematics and computer science
◮ Does the bisimulation result grow when the
9/11 department of mathematics and computer science
◮ Does the bisimulation result grow when the
◮ How fast does it grow?
9/11 department of mathematics and computer science
◮ Does the bisimulation result grow when the
◮ How fast does it grow?
10/11 department of mathematics and computer science
◮ Power-law distributions in bisimulation results ⇒
10/11 department of mathematics and computer science
◮ Power-law distributions in bisimulation results ⇒
◮ Behaviors of graph generators ⇒ some more work
10/11 department of mathematics and computer science
◮ Power-law distributions in bisimulation results ⇒
◮ Behaviors of graph generators ⇒ some more work
◮ Bisimulation result/graph grows ⇒ lower k or
11/11 department of mathematics and computer science
11/11 department of mathematics and computer science
Let k be a non-negative integer and G = N, E, λN, λE be a
iff the following holds:
(v, v′) ∈ E, such that u′ ≈k−1 v′ and λE(u, u′) = λE(v, v′), and
(u, u′) ∈ E, such that v′ ≈k−1 u′ and λE(v, v′) = λE(u, u′).
11/11 department of mathematics and computer science
Let k be a non-negative integer and G = N, E, λN, λE be a
iff the following holds:
(v, v′) ∈ E, such that u′ ≈k−1 v′ and λE(u, u′) = λE(v, v′), and
(u, u′) ∈ E, such that v′ ≈k−1 u′ and λE(v, v′) = λE(u, u′).
1
M
2
M
4
P
3
P
5
P
6
P
w l w l l l l
In this example graph, nodes 1 and 2 are 0- and 1- bisimilar but not 2-bisimilar.
11/11 department of mathematics and computer science
Power-law exists in in/out-degree distribution for most of the examined graphs.
100 102 104 106 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8
x (in-degree) cumulative % of nodes with ≥ x
real graphs
100 102 104 106 108 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8
x (in-degree)
synthetic graphs
Jamendo LinkedMDB DBLP DBPedia WikiLinks Twitter Flickr-Grow BSBM SP2B Power Random
11/11 department of mathematics and computer science
100 102 104 106 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8
x (signature length) cumulative % of nodes with ≥ x
real graphs
100 101 102 103 104 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7
x (signature length)
synthetic graphs
Jamendo LinkedMDB DBLP DBPedia WikiLinks Twitter Flickr-Grow BSBM SP2B Power Random
11/11 department of mathematics and computer science
100 102 104 106 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8
x (out-degree) cumulative % of Nk with ≥ x
real graphs
100 102 104 106 108 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7
x (out-degree)
synthetic graphs
Jamendo LinkedMDB DBLP DBPedia WikiLinks Twitter Flickr-Grow BSBM SP2B Power Random