Random Surfjng on Multipartite Graphs Athanasios N. Nikolakopoulos, - PowerPoint PPT Presentation

Random Surfjng on Multipartite Graphs Athanasios N. Nikolakopoulos, Antonia Korba and John D. Garofalakis Department of Computer Engineering and Informatics, University of Patras December 07, 2016 IEEE International Conference on Big Data, IEEE BigData 2016

Outline 1. Introduction & Motivation 2. Block Teleportation Model: Defjnition, Algorithm and Theoretical Analysis 3. Experimental Evaluation 4. Conclusions & Future Work 1

Introduction & Motivation

Revisiting the Random Surfer Model I PageRank Model • Has received much attention (Generalized Damping Functions (Functional Rankings) [1], Multidamping [5], ...) • Little have been done towards its generalization [8]. 2 G = α H + (1 − α ) E Primitivity Adjustment of the Row-Normalized Adjacency Matrix H : • Damping Factor α • Teleportation matrix E

Revisiting the Random Surfer Model II Graphs behavior per partite set. Problems With Traditional Teleportation teleportation model • We modify the traditional • Difgerent Teleportation Overview of Our Approach • Blind to the spectral characteristics of the underlying graph • Restrictive or even counter-intuitive (eg. structured graphs) • Treats nodes in a “leveling way” 3 u 1 v 1 • We focus on Multipartite w 1 u 2 v 2 w 2 u 3 v 3

Block Teleportation Model: Defjnition, Algorithm and Theoretical Analysis

Model Defjnition Sparse and Low-Rank Factors currently in. the same partite set he is jumps to a node belonging to edges of the graph The Random Surfer: Interpretation Random Surfjng 4 Each partite set is a Teleportation Block! S = η H + µ M H ≜ diag( A G 1 ) − 1 A G , { 1 • With probability η follows the if v j ∈ M i , |M i | , [ M ] ij ≜ 0 , otherwise , • With probability µ ≜ 1 − η he where M i the origin partite set of v i . R∆R ⊺ M = � �� �

Decomposability and Time-Scale Dissociation Theorem (Decomposability) Study each Connected Component in Isolation, and then combine the results. Computational Implications... in brush strokes! • Long-term Equilibrium. • Long-term Dynamics. • Short-term Equilibrium. 5 into difgerent connected components. with respect to the partition of the nodes of the initial graph, u 1 u 1 When the value of the teleportation parameter µ is small u 2 v 1 u 2 v 1 enough, the Markov chain corresponding to matrix S is NCD u 3 v 2 u 3 v 2 u 4 v 3 u 4 v 3 u 5 u 5 S = ˜ S ≜ diag { S ( G 1 ) , . . . , S ( G L ) } ˜ S + ε C , n ( I ) L L ∑ ∑ ∑ S t λ t λ t = Z 11 + 1 I Z 1 I + mI Z mI , ���� I =2 I =1 m =2 • Short-term Dynamics. Term A � �� � � �� � Term B Term C n ( I ) L L ∑ ∑ ∑ S t (˜ λ ) t ˜ ˜ mI ˜ = Z 1 I + Z mI . I =1 I =1 m =2 � �� � � �� � Term ˜ Term ˜ A C

BT-Rank Algorithm and Computational Analysis Block-Teleportation Rank When the BT-Rank Markov chain is lumpable wrt to Theorem (Perron Vector ) The BT-Rank Markov chain that corresponds to a Theorem (Lumpability ) Theorem (Eigenvalue Property ) deeper! per iteration General Cost: step 2. 6 2: Compute 1: Let the initial approximation be log ϵ Θ(nnz( H )) log | λ 2 ( S ) | � �� � Input: H , M ∈ R n × n , ϵ , scalars η, µ > 0 such that η + µ = 1 . If χ ( G ) = 2 we can dig a little Output: π ⊺ π ⊺ (0) . Set k = 0 . Assuming G is a connected graph for which χ ( G ) = 2 π ⊺ π ⊺ ← ( k ) H holds, the spectrum of the stochastic matrix S is such ( k +1) that − η + µ ∈ λ ( S ) . φ ⊺ π ⊺ ← ( k ) M π ⊺ η π ⊺ ( k +1) + µ φ ⊺ ← ( k +1) 3: Normalize π ⊺ ( k +1) and compute 2-chromatic graph, is lumpable wrt A = {A 1 , A 2 } . r = ∥ π ⊺ ( k +1) − π ⊺ ( k ) ∥ 1 . 4: If r < ϵ , quit with π ⊺ ( k +1) , otherwise k ← k + 1 and go to partition A , for the left Perron eigenvector of matrix S it holds π ⊺ 1 1 A 1 = π ⊺ 2 1 A 2

Experimental Evaluation

Computational Experiments Time(sec) TREC # of iterations Yahoo!Music Wikipedia(en) Time(sec) Jester MovieLens20M Digg votes # of iterations TREC Time(sec) Yahoo!Music Wikipedia(en) BT-Rank BT-Rank(NoLump) PageRank Digg votes 7 MovieLens20M # of iterations Jester 3 10 200 200 2 5 1 100 100 0 0 0 . 8 0 . 85 0 . 9 0 . 95 0 . 8 0 . 85 0 . 9 0 . 95 0 . 8 0 . 85 0 . 9 0 . 95 0 . 8 0 . 85 0 . 9 0 . 95 2 60 200 1 . 5 200 40 150 1 20 0 . 5 100 100 0 0 50 0 . 8 0 . 85 0 . 9 0 . 95 0 . 8 0 . 85 0 . 9 0 . 95 0 . 8 0 . 85 0 . 9 0 . 95 0 . 8 0 . 85 0 . 9 0 . 95 3 80 200 200 60 2 40 100 100 1 20 0 0 0 . 8 0 . 85 0 . 9 0 . 95 0 . 8 0 . 85 0 . 9 0 . 95 0 . 8 0 . 85 0 . 9 0 . 95 0 . 8 0 . 85 0 . 9 0 . 95

Qualitative Experiments: Ranking-Based Recommendation I Genres movies. Our Approach 8 Movies Users • We model the recommender as a tripartite graph ( ) 1 e ⊺ i , 1 ω ⊺ i , 1 ϖ ⊺ • Personalization through matrix M ≜ diag i • ω i : the normalized vector of the users’ ratings over the set of • ϖ i : the normalized vector of his mean ratings per genre.

Qualitative Experiments: Ranking-Based Recommendation II Recall@N CT MFA FP Katz BT-Rank MRR CT MFA FP Katz BT-Rank NDCG@N Methodology MovieLens1M 9 Cumulative Gain • Mean Reciprocal Rank • Normalized Discounted (NDCG@N) • Randomly select another 1000 unrated items of the Metrics • Randomly sample 1.4% of the ratings ⇒ probe set P • Recall@N • Use each item v j , rated with 5 stars by user u i in P ⇒ test set T same user for each item in T • Form ranked lists by ordering all the 1001 items 0 . 4 0 . 2 0 . 2 0 0 . 15 5 10 15 20 0 . 3 0 . 1 0 . 2 0 . 1 0 5 10 15 20 L † L †

Conclusions & Future Work

Conclusion & Future Work Synopsis We proposed a simple alternative teleportation component for Random Surfjng on Multipartite Graphs. • Can be handled effjciently • Entails nice theoretical properties • Allows for richer modeling Future Directions models that match better the underlying graphs • For the Web-Graph: NCDawareRank (WSDM’13) 10 • Propose a Systematic Framework for the defjnition of teleportation

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Questions?

Appendix: Eigenvalue Theorem #nodes of 1st partite set (1) transformation Back #nodes of 2nd partite set that reveals the desired eigenvalue. Proof Sketch. Theorem (Eigenvalue Property) Assuming G is a connected graph for which χ ( G ) = 2 holds, the spectrum of the stochastic matrix S is such that − η + µ ∈ λ ( S ) . � �� � � �� � • We defjne a vector v ≜ [ 1 1 1 · · · 1 − 1 − 1 · · · − 1 ] • We show that ( − 1 , v ) is an eigenpair of matrix H , and (1 , v ) is an eigenpair of matrix M . ( ) • Then, using any nonsingular matrix, Q ≜ 1 v X , allows us to perform a similarity Q − 1 ( η H + µ M ) Q = · · · = Q − 1 SQ =   η y 1 ⊺ HX + µ y 1 ⊺ MX 1 0 η y 2 ⊺ HX + µ y 2 ⊺ MX   = 0 − η + µ   η Y ⊺ HX + µ Y ⊺ MX 0 0