Graph similarity. Laura Zager and George Verghese EECS, MIT. March 2005

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Graph similarity Laura Zager and George Verghese EECS, MIT March 2005

Words you won t hear today impedance matching thyristor oxide layer VARs

Some quick definitions GV (, 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 a directed graph and its node-node adjacency matrix 1 2 3 4 5 1 2 3 4 5 0 1 1 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0

Graph theory: some perspective The Königsberg bridge problem (18 th c.) The Four Color Theorem (1976)

Graph theory: some perspective The Königsberg bridge problem (18 th c.) The Four Color Theorem (1976) Erdös and Rényi random graph models (1959)

Graph theory: some perspective The Königsberg bridge problem (18 th c.) The Four Color Theorem (1976) Erdös and Rényi random graph models (1959) present and future: graphs that arise in the natural world

Applications Comparing biological networks Deriving phylogenetic trees from metabolic pathway data [Heymans, Singh, 2003]. Social network mapping Small world phenomena [Milgram, 1967; Watts, 1999]. Web searching Improving searching results using WWW structure [Kleinberg, 1999]. Chemical structure matching Finding similar structures in a chemical database [Hattori et al., 2003].

Applications Comparing biological networks Deriving phylogenetic trees from metabolic pathway data [Heymans, Singh, 2003]. Social network mapping Small world phenomena [Milgram, 1967; Watts, 1999]. Web searching Improving 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

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.

Notions of similarity Isomorphism identifying a bijection between the nodes of two graphs which preserves (directed) adjacency. a c a d c b d b Corneil & Gotlieb, Journal of the ACM, 1970. Pelillo, Neural Computation, 1999. Ullman, Journal of the Assoc. of Computing Machinery, 1976.

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.

Notions of similarity edit distance isomorphism Maximum common subgraph identifying the `largest isomorphic subgraphs of two graphs. Minimum common supergraph identifying the `smallest graph that contains both graphs. max. com. sub Fernandez & Valiente, Pattern Recognition Letters, 2001. Bunke, Jiang & Candel, Computing, 2000.

Notions of similarity edit distance isomorphism max (min) common sub(super)graph Statistical methods assessing aggregate measures of graph structure (e.g. degree distribution, diameter, betweenness measures). frequency degree Albert, Barabasi, Reviews of Modern Physics, 2002 Dill, Kumar, et al., ACM Transactions on Internet Technology, 2002. Watts, Small Worlds, 1999.

Notions of similarity edit distance? isomorphism max (min) common sub(super)graph? statistical comparison 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, 8 th Intl. Conf. on Knowledge Discovery and Data Mining, 2002. Melnik, Garcia-Molina, 18 th Intl. Conf. on Data Engineering, 2002. Heymans & Singh, Bioinformatics, 2003.

Kleinberg, 1999 * Motivated by demands of web searching Step 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 leinberg, J.M. Authoritative sources in a hyperlinked environment. Journal of the ACM. 1999

Kleinberg, 1999 Relevant search results might be: Hubs pages which point to many good authorities Authorities 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.

Kleinberg, 1999 Denote: x 1p (k) = hub score of node p at iteration k x 2p (k) = authority score of node p at iteration k Update rule: x ( k + 1) = x ( k) 2p 1q q:( q, p) E i.e. the sum of hub scores of nodes that point to node p x ( k + 1) = x ( k) 1p 2q q:( p, q) E i.e. the sum of authority scores of nodes that are pointed to by node p Normalize the scores so that p x ip = 1 and repeat.

Kleinberg, 1999 Denote: x 1p (k) = hub score of node p at iteration k x 2p (k) = authority score of node p at iteration k Update rule: x ( k + 1) = x ( k) 2p 1q q:( q, p) E i.e. the sum of hub scores of nodes that point to node p x ( k + 1) = x ( k) 1p 2q q:( p, q) E i.e. the sum of authority scores of nodes that are pointed to by node p Normalize the scores so that p x ip = 1 and repeat.

Kleinberg, 1999 Denote: x 1p (k) = hub score of node p at iteration k x 2p (k) = authority score of node p at iteration k Update rule: Stack the scores x 1p (k) into a vector [x 1 ] k, then stack [x 1 ] k and [x 2 ] k. Let B be the node-node adjacency matrix of the candidate graph. Then: x x 1 = 2 k+ 1 0 B B x 0 x 1 2 k

Kleinberg, 1999 Ex. 1 2 5 3 6 7 nodes x 1 hub scores x 2 authority scores 1 0.374 0 2 0.242 0 3 0.467 0 4 0 0.365 5 0 0.467 4 6 0 0.365 7 0 0.308 for a good read, see The Ongoing Search for Efficient Web Search Algorithms, SIAM News, November 2004.

Blondel, Van Dooren, et al., 2004 * Views Kleinberg s iteration as a comparison between the web graph and a hub-authority graph: 1 hub node 2 authority node A = 0 1 0 0 Observe that the matrix form of Kleinberg s update can be written as follows: x x 1 = 0 B B x 0 x 1 ( A B A B ) x = + x 2 k+ 1 2 k 2 k Is this generalizable to any two graphs G A and G B? 1 * 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.

Blondel, Van Dooren, et al., 2004 A first step toward generalizing Kleinberg s approach: consider comparing the graph G B to the following graph using a similar update: 1 2 3 hub node central node authority node A = 0 1 0 0 0 1 0 0 0 x ( k + 1) = x ( k) 1p 2q q:( p, q) E p x ( k + 1) = x ( k) + x ( k) 2p 1q 3q q:( q, p) E q:( p, q) E x ( k + 1) = x ( k) 3p 2q q:( q, p) E

Blondel, Van Dooren, et al., 2004 A first step toward generalizing Kleinberg s approach: consider comparing the graph G B to the following graph using a similar update: 1 2 3 hub node central node authority node A = 0 1 0 0 0 1 0 0 0 x ( k + 1) = x ( k) 1p 2q q:( p, q) E p x ( k + 1) = x ( k) + x ( k) 2p 1q 3q q:( q, p) E q:( p, q) E x ( k + 1) = x ( k) 3p 2q q:( q, p) E

Blondel, Van Dooren, et al., 2004 A first step toward generalizing Kleinberg s approach: consider comparing the graph G B to the following graph using a similar update: 1 2 3 hub node central node authority node A = 0 1 0 0 0 1 0 0 0 x ( k + 1) = x ( k) 1p 2q q:( p, q) E p x ( k + 1) = x ( k) + x ( k) 2p 1q 3q q:( q, p) E q:( p, q) E x ( k + 1) = x ( k) 3p 2q q:( q, p) E

Blondel, Van Dooren, et al., 2004 A first step toward generalizing Kleinberg s approach: consider comparing the graph G B to the following graph using a similar update: 1 2 3 hub node central node authority node A = 0 1 0 0 0 1 0 0 0 x ( k + 1) = x ( k) 1p 2q q:( p, q) E p x ( k + 1) = x ( k) + x ( k) 2p 1q 3q q:( q, p) E q:( p, q) E x ( k + 1) = x ( k) 3p 2q q:( q, p) E

Blondel, Van Dooren, et al., 2004 A first step toward generalizing Kleinberg s approach: consider comparing the graph G B to the following graph using a similar update: 1 2 3 hub node central node authority node A = 0 1 0 0 0 1 0 0 0 x x x 1 2 0 B 0 x = B 0 B x 0 B 0 x 1 2 ( ) = A B + A B x x x 3 k+ 1 3 k 3 k 1 2 (use this construction for automatic synonym extraction)

Blondel, Van Dooren, et al., 2004 In general, the nodes of two graphs G A and G B can be compared via the following update: ( ) xk+ 1 = A B + A B x k Ex. G A G B 1 2 3 1 2 3 similarity scores nodes 1 2 3 1 0.443 0.104 0 2 0.280 0.396 0.086 3 0.086 0.396 0.280 4 0.222 0.049 0.222 5 0 0.104 0.443 4 5

Coupled edge and node scoring Idea: use this iterative approach to assign edge similarity scores as well as node similarity scores. Couple the definitions in the following manner: x ij = similarity between node i in G B and node j in G A = sum of pairwise similarities between adjacent edges y ij = similarity between edge i in G B and edge j in G A. = sum of similarities of source and terminal nodes

Coupled edge and node scoring Idea: use this iterative approach to assign edge similarity scores as well as node similarity scores. Couple the definitions in the following manner: x ij = similarity between node i in G B and node j in G A = sum of pairwise similarities between adjacent edges y ij = similarity between edge i in G B and edge j in G A. = sum of similarities of source and terminal nodes

Coupled edge and node scoring Idea: use this iterative approach to assign edge similarity scores as well as node similarity scores. Couple the definitions in the following manner: x ij = similarity between node i in G B and node j in G A = sum of pairwise similarities between adjacent edges y ij = similarity between edge i in G B and edge j in G A. = sum of similarities of source and terminal nodes

Coupled edge and node scoring Idea: use this iterative approach to assign edge similarity scores as well as node similarity scores. Couple the definitions in the following manner: x ij = similarity between node i in G B and node j in G A = sum of pairwise similarities between adjacent edges y ij = similarity between edge i in G B and edge j in G A. = sum of similarities of source and terminal nodes [ ] x + 1 = A B + A B y k S S T T k [ ] + y A B A B x k+ 1 = S S T T k [ A ] S ij = 1 0 s( j) = else i [ A ] T ij = 1 0 t( j) = else i

Example 1 G A 1 2 3 G B 2 3 4 5 Blondel, Van Dooren, et al. similarity scores nodes 1 2 3 1 0.443 0.104 2 0.280 0.396 0.086 3 0.086 0.396 0.280 4 0.222 0.049 0.222 5 0 0.104 0.443 0 Coupled model similarity scores nodes 1 2 3 1 0.324 0.054 2 0.177 0.587 0.018 3 0.018 0.587 0.177 4 0.127 0.010 0.127 5 0 0.054 0.324 0

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 1 3 7 3 2 4 3 2 4 4 8 3 4 8 3

Application: Graph Matching Task: subgraph matching Generate 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 d c a b d S c a b

Application: Graph Matching Task: subgraph matching Generate 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 weights. Record successes for nodes that are matched with their original identifier G a* a S d c Yields a lower bound on the success of the matching process b d c a b

Application: Graph Matching Using local edge similarity to improve scores: perform a a local edge matching a a x aa x aa * = x aa + m aa

Application: Graph Matching

Application: Graph Matching Exploring the impact of node labeling: cannot be matched

Application: Graph Matching

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?

Acknowledgments George Verghese, MIT Sandip Roy, WSU Paul Van Dooren, Université catholique de Louvain Work supported by a NSF Graduate Research Fellowship.

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