Efficient Top-k Search for Random Walk with Restart

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1 DEIM Forum 2011 D3-1 Random walk with restart Top-k, Random walk with restart (RWR) RWR K Random walk with restart Top-k Efficient Top-k Search for Random Walk with Restart Yasuhiro FUJIWARA,, Makoto NAKATSUJI, Makoto ONIZUKA, and Masaru KITSUREGAWA NTT Cyber Space Laboratories, 1-1 Hikarinooka, Yokosuka, Kanagawa, Japan NTT Cyber Solution Laboratories, 1-1 Hikarinooka, Yokosuka, Kanagawa, Japan Institute of Industrial Science, The University of Tokyo, Komaba 4-6-1, Meguro, Tokyo Japan {fujiwara.yasuhiro,nakatsuji.makoto,onizuka.makoto}@lab.ntt.co.jp, kitsure@tkl.iis.u-tokyo.ac.jp 1. [1], [2], [3] Random walk with restart (RWR) RWR [4] RWR q q RWR u q q u RWR [5], [6], [7], [] [], [9] q [] q K q K

2 2. 3. RWR RWR Pan [5] RWR 10% [10] Konstas RWR [6] Konstas Sun RWR RWR [] RWR RWR 0 Tong RWR B LIN NB LIN [9] RWR NB LIN RWR Sun RWR Tong O(n 2 ) O(n 2 ) 3. Random walk with restart RWR 1 RWR RWR q [4] c p n 1 p u u 1 q K n m c p u n 1 q q 1 0 n 1 A q n 1 q q 1 0 A A u,v u v p = (1 c)ap + cq (1) RWR p p u u q t RWR O(mt) RWR RWR [6] (1) O(n 2 ) LU

3 [11] K RWR O(1) (1) p = c{i (1 c)a} 1 q = cw 1 q (2) I W = I (1 c)a W 1 W [12] W LU W = LU p = cu 1 L 1 q (3) L 1 U 1 L 1 U 1 L U L 1 U 1 [12] >< L 1 ij = >: U 1 ij = >< >: 0 (i < j) 1/L ij (i = j) 1/L ii P i 1 k=j L ikl 1 kj (i > j) 0 (i > j) 1/U ij (i = j) 1/U ii P j k=i+1 U iku 1 kj (i < j) L U W [12] >< 0 (i < j) L ij = 1 (i = j) >: 1/U jj W ij P j 1 k=1 L iku kj (i > j) >< 0 (i > j) U ij = W ij (i < = j i = 1) >: W ij P i 1 k=1 L iku kj (i < = j i = 1) (4), (5), (6), (7) L 1, U 1, L, U L 1 ij L L 1 L ij W, L, U (4) (5) (6) (7) (1) L 1 ij U 1 ij L U 0 0 (2) L U W 0 0 (3) W A 0 0 A 0 L 1 ij U 1 ij A 0 Newman clustering [13] κ Newman clustering κ+1 1 κ κ+1 A 1 κ κ+1

4 (1) (2) (3) 1. 1 A 0 [11] i i V V s u l u l u V (l u) V (l u) = {v : (v V s) (l v = l u)} A A max A max = max{a ij : i, j V } u A max(u) A max(u) = max{a iu : i V } A max A max(u) u p u 1 q u p u < p u = c : X v V (l u 1) p va max(v) + X v V (l u) + 1 X p va max(v) v V s p v! A max ) () c = (1 c)/(1 A uu + ca uu) u p u = 1 1 O(n) V (l u 1) V (l u) V s O(n) O(1) u p u > = p u 2 l u < = l v u v p u > = p v O(n) u u u u p u,1 p u,2 p u,3 () p u = c ( p u,1 + p u,2 + p u,3) u 2 u ( pu p u,1 =,1 if l(u) = l(u ) p u,2 + p u A max(u ) otherwise ( pu p u,2 =,2 + p u A max(u ) if l(u) = l(u ) (9) 0 otherwise p u,3 = ( p u,3/a max p u ) A max u p u,1 = p qa max(q) p u,2 = 0 p u,3 = (1 p q)a max(u) 3 2 u

5 Algorithm 1 Input: q, K, L 1, L U 1, U Output: V a, 1: θ = 0; 2: V s = ; 3: V a = ; 4: K V a ; 5: q ; 6: while V s = V do 7: u := argmin(l v v V \V s); : u p u ; 9: if p u < θ then 10: return V a; 11: else 12: L 1 U 1 p u ; 13: if p u > θ then 14: v := argmin(p w w V a); 15: v V a ; 16: u V a ; 17: θ := min(p w w V a); 1: end if 19: end if 20: u V s ; 21: end while 22: return V a; RWR O(1) K θ V a K 0 θ 1 θ 2 θ V a θ 5. Tong NB LIN [9] NB LIN 3. Sun [] Tong B LIN Tong c 0.95 [9], [14] Wall clock time [s] Proposed(5) Proposed(25) Proposed(50) NB_LIN(100) NB_LIN(1,000) 2 Dictionary Internet Citation Dictionary 1 : FOLDOC 2 u v u v 13, , 23 Internet 3 : Oregon Route Views Project 4 BGP 22, 963 4, 436 Citation 5 : Condensed Matter E-Print 6 31, , 029 CPU Intel Xeon Quad-Core 3.33GHz 32GB Linux GCC 5. 1 NB LIN 2 K Propased(K) NB LIN 100 NB LIN(100) 1, 000 NB LIN(1,000) NB LIN K NB LIN mejn/netdata/as-22july06.zip mejn/netdata/cond-mat-2003.zip 6

6 Precision 3 Number of non-zero elements Target rank of SVD NB_LIN Proposed Degree Cluster Hybrid Random Dictionary Internet Citation Wall clock time [s] 4 Wall clock time [s] Target rank of SVD NB_LIN Proposed Proposed Without pruning Dictionary Internet Citation 3 4 NB LIN NB LIN Dictionary 3 1 NB LIN 4 NB LIN NB LIN Degree Clustering Hybrid Random O(m) Without pruning 1, NB LIN OS Dictionary NB LIN 1, Microsoft Windows Microsoft Windows W2K Windows/36 Windows 3.0 Windows 3.11 Microsoft OS Microsoft OS Mac OS Apple Macintosh user interface Apple PC GUI Macintosh file system Mac OS Linux Linux Linux Documentation Project NB LIN 6. RWR [1] Y. Koren, S. C. North and C. Volinsky: Measuring and extracting proximity in networks, KDD, pp (2006). [2] H. Tong, C. Faloutsos and Y. Koren: Fast direction-aware proximity for graph mining, KDD, pp (2007). [3] D. Lizorkin, P. Velikhov, M. N. Grinev and D. Turdakov: Accuracy estimate and optimization techniques for simrank computation, PVLDB, 1, 1, pp (200). [4] H. Tong and C. Faloutsos: Center-piece subgraphs: problem definition and fast solutions, KDD, pp (2006). [5] J.-Y. Pan, H.-J. Yang, C. Faloutsos and P. Duygulu: Automatic multimedia cross-modal correlation discovery, KDD, pp (2004). [6] I. Konstas, V. Stathopoulos and J. M. Jose: On social networks and collaborative recommendation, SIGIR, pp (2009).

7 2 NB LIN Microsoft Windows, Mac OS, Linux. Microsoft Windows Mac OS Linux Proposed Microsoft Windows W2K Windows/36 Windows 3.0 Windows 3.11 NB LIN Microsoft Windows Microsoft Networking Proposed Mac OS Macintosh user interface Microsoft Network W2K Thumb Macintosh file system multitasking Proposed Linux Linux Documentation Project NB LIN Linux Linux Documentation Project NB LIN Mac OS Rhapsody SORCERER Macintosh Operating System Macintosh Operating System PowerOpen Association Unix lint Linux Network Administrators Guide SL5 debianize SLANG [7] D. Liben-Nowell and J. M. Kleinberg: The link prediction problem for social networks, CIKM, pp (2003). [] J. Sun, H. Qu, D. Chakrabarti and C. Faloutsos: Neighborhood formation and anomaly detection in bipartite graphs, ICDM, pp (2005). [9] H. Tong, C. Faloutsos and J.-Y. Pan: Fast random walk with restart and its applications, ICDM, pp (2006). [10] J. L. Herlocker, J. A. Konstan, A. Borchers and J. Riedl: An algorithmic framework for performing collaborative filtering, SIGIR, pp (1999). [11] T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein: Introduction to Algorithms, The MIT Press (2009). [12] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery: Numerical Recipes 3rd Edition, Cambridge University Press (2007). [13] A. Clauset, M. E. J. Newman and C. Moore: Finding community structure in very large networks, Physical Review E, pp. 1 6 (2004). [14] J. He, M. Li, H. Zhang, H. Tong and C. Zhang: Manifoldranking based image retrieval, ACM Multimedia, pp (2004).

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