SimRank: A Measure of Structural-Context Similarity Glen Jeh and Jennifer Widom Stanford University ACM SIGKDD 2002 January 19, 2011 Taikyoung Kim SNU IDB Lab.

Outline  Introduction  Basic Graph Model  SimRank  Random Surfer-Pairs Model  Conclusion  Future Work 2

Introduction  Many applications require a measure of “similarity” between objects –“find-similar-document” query in search engine –Collaborative filtering in a recommender system 3

Introduction  Propose a general approach that exploits the object-to-object relationships in many domains –An algorithm to compute similarity scores between nodes based on the structural context  Intuition behind the algorithm –Similar objects are related to similar objects –The base case is that objects are similar to themselves 4 “Two objects are similar if they are referenced by similar objects”

Basic Graph Model  G = (V, E) [vertex, edge] –Nodes in V: objects in the domain –Directed edges in E: relationships between objects – : from object p to object q  For a node v, denote: –I(v): the set of in-neighbors of v –O(v): the set of out-neighbors of v –I i (v): individual in-neighbor ( 1 ≤ i ≤ |I(v)| ) –O i (v): individual out-neighbor ( 1 ≤ i ≤ |O(v)| ) 5 O (Univ) I (ProfB)

Outline  Introduction  Basic Graph Model  SimRank  Random Surfer-Pairs Model  Conclusion  Future Work 6

SimRank  Motivation –Two objects are similar if they are referenced by similar object –Consider an object maximally similar to itself (similarity score of 1) 7 Similar nodes: {ProfA, ProfB}, {StudentA, StudentB}, {Univ, ProfB}, …

SimRank Basic SimRank Equation  The similarity between objects a and b: s(a, b) ∈ [0, 1] –C is a constant between 0 and 1  Confidence level or decay factor  C gives the rate of decay as similarity flows across edges (since C < 1) –If a or b may not have any in-neighbors, s(a,b) = 0 –SimRank scores are symmetric, i.e., s(a,b) = s(b,a)  Similarity between a and b is the average similarity between in- neighbors of a and in-neighbors of b 8

SimRank Basic SimRank Equation  Similarity can be thought of as “propagating” from pair to pair –Consider the derived graph G 2 =(V 2, E 2 ) where  V 2 =V x V, represents a pair (a,b) of nodes in G  An edge from (a,b) to (c,d) exists in E 2, iff the edges and exist in G 9

SimRank Bipartite SimRank  Bipartite domains consist of two types of objects  Recommender system –People are similar if they purchase similar items –Items are similar if they are purchased by similar people 10

SimRank Bipartite SimRank  Bipartite Equation –Directed edges go from people to items –s(A,B) denote the similarity between persons A and B, (A≠B) –s(c,d) denote the similarity between items c and d, (c≠d) –The similarity between persons A and B is the average similarity between the items they purchased –The similarity between items c and d is the average similarity between the people who purchased them 11

SimRank Computing SimRank - Naïve Method  R k (a,b) gives the score between a and b on iteration k  The values R k (*,*) are non-decreasing as k increase   In experiments, when K = 5, R k is rapidly converged  Complexity –Space: O(n 2 ) to store the result R k, –Time: O(Kn 2 d 2 ), d 2 is the average of |I(a)||I(b)| over all node pairs (a,b) 12

SimRank Computing SimRank - Pruning  Pruning the logical graph G 2 –In naïve method,  All n 2 nodes of G 2 are considered  Similarity score are computed for every node-pair –Nodes far from a node v has less similarity score with v than nodes near v  Pruning –Set the similarity between two nodes far apart to be 0 –Consider node-pairs only for nodes which are near each other in the range of radius r –Complexity  space: O(nd r ), d r is average nodes which are near from a node  time: O(Knd r d 2 ) 13

Outline  Introduction  Basic Graph Model  SimRank  Random Surfer-Pairs Model  Conclusion  Future Work 14

Random Surfer-Pairs Model  For the intuition of similarity scores, provide an intuitive model –Based on “random surfers” –Show the SimRank score s(a,b) measures how soon two random surfers are expected to meet at the same node  Expected Distance –u and v are nodes in strongly connected graph –The ED from u to v is exactly the expected number of steps a random surfer would take before he first reaches v, starting from u –Tour t = –l[t]: length of t –P[t]: probability of traveling t 15

Random Surfer-Pairs Model  Expected Meeting Distance (EMD) –EMD is symmetric –EMD m(a,b) is simply the expected distance in G 2 from (a,b) to any singleton node(x,x) ∈ V 2 16 m(v,w)=1 m(u,v)=∞ m(u,w)=∞ m(*,*)= ∞m(*,*)= 3

Random Surfer-Pairs Model  Expected-f Meeting Distance –Our approach to circumvent the “infinite EMD” problem  Map all distances to a finite interval: instead of computing expected length l(t) of a tour  Equivalence to SimRank –S’(*,*) is exactly models that our original definition of SimRank scores 17

Outline  Introduction  Basic Graph Model  SimRank  Random Surfer-Pairs Model  Conclusion  Future Work 18

Conclusion  Main contribution –A formal definition for SimRank similarity scoring over arbitrary graphs, sev eral useful derivatives of SimRank, and an algorithm to compute SimRank –A graph-theoretic model for SimRank that gives intuitive mathematical insig ht into its use and computation –Experimental results using an in-memory implementation of SimRank over two real data sets shows the effectiveness and feasibility of SimRank 19

Future Work  Address efficiency and scalability issues –Including additional pruning heuristics and disk-based algorithms  Consider ternary (or more) relationships in computing structural- context similarity  Explore the combination of SimRank with other domain-specific similarity measures 20