Laura Zager and George Verghese EECS, MIT March 2005pvassil/downloads/GraphDistance/LauraZa... ·...
Transcript of Laura Zager and George Verghese EECS, MIT March 2005pvassil/downloads/GraphDistance/LauraZa... ·...
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Graph similarity
Laura Zager and George Verghese
EECS, MIT
March 2005
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Words you won’t hear today
impedance matching thyristoroxide layerVARs
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Some quick definitionsG V E( , ) a graph G
V the set of vertices or nodes
E V V⊂ × the set of edges – can be directed or undirected.
ex 1
2 3
4 5
1 2 3 4 5
0 1 1 0 00 0 1 0 10 0 0 1 10 1 0 0 00 0 0 0 0
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
a directed graph and itsnode-node adjacency matrix
1
2
3
4
5
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Graph theory: some perspective
The Königsberg bridge problem(18th c.)
The Four Color Theorem(1976)
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Graph theory: some perspective
The Königsberg bridge problem(18th c.)
The Four Color Theorem(1976)
Erdös and Rényi random graph models (1959)
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Graph theory: some perspective
The Königsberg bridge problem(18th c.)
The Four Color Theorem(1976)
present and future:graphs that arise in the natural world
Erdös and Rényi random graph models (1959)
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ApplicationsComparing biological networks
Deriving phylogenetic trees from metabolic pathway data [Heymans, Singh, 2003].
Social network mappingSmall world phenomena [Milgram, 1967; Watts, 1999].
Web searchingImproving searching results using WWW structure [Kleinberg, 1999].
Chemical structure matching
Finding similar structures in a chemical database [Hattori et al., 2003].
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ApplicationsComparing biological networks
Deriving phylogenetic trees from metabolic pathway data [Heymans, Singh, 2003].
Social network mappingSmall world phenomena [Milgram, 1967; Watts, 1999].
Web searchingImproving searching results using WWW structure [Kleinberg, 1999].
Chemical structure matching
Finding similar structures in a chemical database [Hattori et al., 2003].
one common thread: similarity
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Notions of similarity
Isomorphism – identifying a bijection between the nodes of two graphs which preserves (directed) adjacency.
Corneil & Gotlieb, Journal of the ACM, 1970.Pelillo, Neural Computation, 1999.Ullman, Journal of the Assoc. of Computing Machinery, 1976.
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Notions of similarity
Isomorphism – identifying a bijection between the nodes of two graphs which preserves (directed) adjacency.
Corneil & Gotlieb, Journal of the ACM, 1970.Pelillo, Neural Computation, 1999.Ullman, Journal of the Assoc. of Computing Machinery, 1976.
c aa
b
cd
b d
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Notions of similarity
isomorphism
Edit distance – given a cost function on edit operations (e.g. addition/deletion of nodes and edges), determine the minimum cost transformation from one graph to another.
Bunke, IEEE Trans. Pattern Analysis and Machine Int., 1999.Messmer & Bunke, IEEE Trans. Pattern Analysis and Machine Int., 1998.
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Notions of similarity
Maximum common subgraph – identifying the `largest’ isomorphic subgraphs of two graphs.Minimum common supergraph – identifying the `smallest’ graph that contains both graphs.
Fernandez & Valiente, Pattern Recognition Letters, 2001.Bunke, Jiang & Candel, Computing, 2000.
isomorphism
edit distance
max.
com.
sub
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Notions of similarityedit distance
Statistical methods – assessing aggregate measures of graph structure (e.g. degree distribution, diameter, betweenness measures).
Albert, Barabasi, Reviews of Modern Physics, 2002Dill, Kumar, et al., ACM Transactions on Internet Technology, 2002.Watts, Small Worlds, 1999.
isomorphism max (min) commonsub(super)graph
degree
frequ
ency
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Notions of similarityedit distance ?
isomorphism max (min) commonsub(super)graph
statisticalcomparison
?
Iterative methods:Two graph elements (e.g., edges or nodes) are similar if
their neighborhoods are similar.
Kleinberg, Journal of the ACM, 1999.Blondel, Van Dooren, et al., SIAM Review, 2004.Jeh & Widom, 8th Intl. Conf. on Knowledge Discovery and Data Mining, 2002.Melnik, Garcia-Molina, 18th Intl. Conf. on Data Engineering, 2002.Heymans & Singh, Bioinformatics, 2003.
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Kleinberg, 1999*
Motivated by demands of web searchingStep 1: Use text-based search methods to identify a candidate graph containing relevant websites and their neighbors.
text-based search results
plus the local neighborhood
Kleinberg, J.M. Authoritative sources in a hyperlinked environment. Journal of the ACM. 1999
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Kleinberg, 1999
Relevant search results might be:Hubs – pages which point to many good authoritiesAuthorities – pages which are pointed to by many good hubs
} good hubs
} good authorities
Step 2: Compute hub and authority scores for every node in the candidate graph.
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Kleinberg, 1999Denote:
x1p(k) = hub score of node p at iteration kx2p(k) = authority score of node p at iteration k
Update rule:x k x kp q
q q p E2 11( ) ( )
:( , )+ =
∈∑
x k x kp qq p q E
1 21( ) ( ):( , )
+ =∈
∑
i.e. the sum of hub scores of nodes that point to node p
i.e. the sum of authority scores of nodes that are pointed to by node p
Normalize the scores so that ∑pxip = 1 and repeat.
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Kleinberg, 1999Denote:
x1p(k) = hub score of node p at iteration kx2p(k) = authority score of node p at iteration k
Update rule:x k x kp q
q q p E2 11( ) ( )
:( , )+ =
∈∑
x k x kp qq p q E
1 21( ) ( ):( , )
+ =∈
∑
i.e. the sum of hub scores of nodes that point to node p
i.e. the sum of authority scores of nodes that are pointed to by node p
Normalize the scores so that ∑pxip = 1 and repeat.
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Kleinberg, 1999Denote:
x1p(k) = hub score of node p at iteration kx2p(k) = authority score of node p at iteration k
Update rule:Stack the scores x1p(k) into a vector [x1]k, then stack [x1]k and [x2]k.
Let B be the node-node adjacency matrix of the candidate graph. Then:
xx
BB
xx
k k
1
2 1
1
2
00
⎡
⎣⎢
⎤
⎦⎥ =
′⎡
⎣⎢
⎤
⎦⎥⎡
⎣⎢
⎤
⎦⎥
+
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Kleinberg, 1999Ex.
nodesx1
hub scores
x2
authority scores
1 0.374 0
2 0.242 0
3 0.467 0
4 0 0.365
5 0 0.467
6 0 0.365
7 0 0.308
1
2
3
4
5
6
7
for a good read, see “The Ongoing Search for Efficient Web Search Algorithms,” SIAM News, November 2004.
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Blondel, Van Dooren, et al., 2004*
Views Kleinberg’s iteration as a comparison between the web graph and a hub-authority graph:
hub node
2
authority node
1
Observe that the matrix form of Kleinberg’s update can be written as follows:
( )xx
BB
xx A B A B
xx
k k k
1
2 1
1
2
1
2
00
⎡
⎣⎢
⎤
⎦⎥ =
′⎡
⎣⎢
⎤
⎦⎥⎡
⎣⎢
⎤
⎦⎥ = ⊗ + ′ ⊗ ′
⎡
⎣⎢
⎤
⎦⎥
+
A =⎡
⎣⎢
⎤
⎦⎥
0 10 0
*Blondel, V., Gajardo, A., Heymans, M., Senellart, P., Van Dooren, P. A measure of similarity between graph vertices: applications to synonym extraction and web searching. SIAM Review, v. 46(4), 647-666. 2004.
Is this generalizable to any two graphs GA and GB?
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Blondel, Van Dooren, et al., 2004
A first step toward generalizing Kleinberg’s approach: consider comparing the graph GB to the following graph using a similar update:
A =
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
0 1 00 0 10 0 0hub node
2
central node
1 3
authority node
p
x k x kp qq p q E
1 21( ) ( ):( , )
+ =∈
∑
x k x k x kp qq q p E
qq p q E
2 1 31( ) ( ) ( ):( , ) :( , )
+ = +∈ ∈
∑ ∑
x k x kp qq q p E
3 21( ) ( ):( , )
+ =∈
∑
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Blondel, Van Dooren, et al., 2004
A first step toward generalizing Kleinberg’s approach: consider comparing the graph GB to the following graph using a similar update:
A =
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
0 1 00 0 10 0 0hub node
2
central node
1 3
authority node
p
x k x kp qq p q E
1 21( ) ( ):( , )
+ =∈
∑
x k x k x kp qq q p E
qq p q E
2 1 31( ) ( ) ( ):( , ) :( , )
+ = +∈ ∈
∑ ∑
x k x kp qq q p E
3 21( ) ( ):( , )
+ =∈
∑
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Blondel, Van Dooren, et al., 2004
A first step toward generalizing Kleinberg’s approach: consider comparing the graph GB to the following graph using a similar update:
A =
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
0 1 00 0 10 0 0hub node
2
central node
1 3
authority node
p
x k x kp qq p q E
1 21( ) ( ):( , )
+ =∈
∑
x k x k x kp qq q p E
qq p q E
2 1 31( ) ( ) ( ):( , ) :( , )
+ = +∈ ∈
∑ ∑
x k x kp qq q p E
3 21( ) ( ):( , )
+ =∈
∑
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Blondel, Van Dooren, et al., 2004
A first step toward generalizing Kleinberg’s approach: consider comparing the graph GB to the following graph using a similar update:
A =
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
0 1 00 0 10 0 0hub node
2
central node
1 3
authority node
x k x kp qq p q E
1 21( ) ( ):( , )
+ =∈
∑
x k x k x kp qq q p E
qq p q E
2 1 31( ) ( ) ( ):( , ) :( , )
+ = +∈ ∈
∑ ∑p
x k x kp qq q p E
3 21( ) ( ):( , )
+ =∈
∑
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Blondel, Van Dooren, et al., 2004
A first step toward generalizing Kleinberg’s approach: consider comparing the graph GB to the following graph using a similar update:
hub node
2
central node
1 3
authority nodeA =
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
0 1 00 0 10 0 0
( )xxx
BB B
B
xxx
A B A Bxxx
k k k
1
2
3 1
1
2
3
1
2
3
0 00
0 0
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
= ′′
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
= ⊗ + ′ ⊗ ′
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
+
(use this construction for automatic synonym extraction)
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Blondel, Van Dooren, et al., 2004In general, the nodes of two graphs GA and GB can be compared via the following update:
( )x A B A B xk k+ = ⊗ + ′ ⊗ ′1
Ex.similarity scores
1 2 3nodes 1 2 3
1 0.443 0.104
0.396
0.396
0.049
0.104
0
2 0.280 0.086
3 0.086 0.280
4 0.222 0.222
5 0 0.443
GA
1
2 3
4 5
GB
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Coupled edge and node scoringIdea: use this iterative approach to assign edge similarity scores as well as node similarity scores.Couple the definitions in the following manner:
xij = similarity between node i in GB and node j in GA
= sum of pairwise similarities between adjacent edges
yij = similarity between edge i in GB and edge j in GA.= sum of similarities of source and terminal nodes
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Coupled edge and node scoringIdea: use this iterative approach to assign edge similarity scores as well as node similarity scores.Couple the definitions in the following manner:
xij = similarity between node i in GB and node j in GA
= sum of pairwise similarities between adjacent edges
yij = similarity between edge i in GB and edge j in GA.= sum of similarities of source and terminal nodes
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Coupled edge and node scoringIdea: use this iterative approach to assign edge similarity scores as well as node similarity scores.Couple the definitions in the following manner:
xij = similarity between node i in GB and node j in GA
= sum of pairwise similarities between adjacent edges
yij = similarity between edge i in GB and edge j in GA.= sum of similarities of source and terminal nodes
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Coupled edge and node scoringIdea: use this iterative approach to assign edge similarity scores as well as node similarity scores.Couple the definitions in the following manner:
xij = similarity between node i in GB and node j in GA
= sum of pairwise similarities between adjacent edges
yij = similarity between edge i in GB and edge j in GA.= sum of similarities of source and terminal nodes
[ ]y A B A B xk S S T T k+ = ′ ⊗ ′ + ′ ⊗ ′1
[ ]x A B A B yk S S T T k+ = ⊗ + ⊗1
[ ]As j i
elseS ij=
=⎧⎨⎩
10
( ) [ ]At j i
elseT ij=
=⎧⎨⎩
10
( )
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Example 1
2 3
4 5
21 3GA GB
Blondel, Van Dooren, et al.similarity scores
Coupled modelsimilarity scores
nodes 1 2 31 0.443 0.104
0.3960.3960.0490.104
02 0.280 0.0863 0.086 0.2804 0.222 0.2225 0 0.443
nodes 1 2 31 0.324 0.054
0.5870.5870.0100.054
02 0.177 0.0183 0.018 0.1774 0.127 0.1275 0 0.324
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Application: Graph Matching
Assign a correspondence between nodes and/or edges of each graph to maximize some performance criteria.
The Approach: apply Hungarian algorithm to node similarity matrix to maximize the sum of matched scores.
1 3 7
3 2 4
4 8 3
1 3 7
3 2 4
4 8 3
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Application: Graph Matching
Task: subgraph matchingGenerate a random graph, G.Select a subgraph, S.Compute the node similarity matrices between G and S.Apply the Hungarian algorithm to `best’ match the nodes of S to those in G by finding a matching the maximizes the sum of matched scores.Record successes for nodes that are matched with their original identifier.
G
S
c
d
a
b c’
d’
a’
b’
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Application: Graph Matching
Task: subgraph matchingGenerate a random graph, GSelect a subgraph, SCompute the node similarity matrices between G and SApply the Hungarian algorithm to `best’ match the nodes of S to those in G by finding a matching the maximizes the sum of matched weights.Record successes for nodes that are matched with their original identifier
Yields a lower bound on thesuccess of the matching process
G
a* S
c
db
a
c’
d’
a’
b’
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Application: Graph Matching
Using local edge similarity to improve scores:
a a’ a a’
perform
localedge
matching
xaa’ xaa’* = xaa’ + maa’
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Application: Graph Matching
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Application: Graph Matching
Exploring the impact of node labeling:
cannot be
matched
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Application: Graph Matching
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Current/future work
How does graph structure (e.g., cycles, paths, completeness) impact similarity scores?
What can be inferred about a pair of graphs from a similarity measurement?
What kinds of tasks is this measure appropriate for?
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Acknowledgments
George Verghese, MITSandip Roy, WSUPaul Van Dooren, Université catholique de Louvain
Work supported by a NSF Graduate Research Fellowship.