To Randomize or Not To

Transcription

1 To Randomize or Not To Randomize: Space Optimal Summaries for Hyperlink Analysis Tamás Sarlós, Eötvös University and Computer and Automation Institute, Hungarian Academy of Sciences Joint work with András A. Benczúr, Károly Csalogány, Dániel Fogaras, and Balázs Rácz

2 Contents Fully Personalized PageRank 1. Efficient algorithms for Fully Personalized PageRank Definition, motivation and preliminaries Rounding Sketching Lower bounds Experiments 2. Link-based similarity search with SimRank Definition Reduction of SimRank to Personalized PageRank

3 Personalized PageRank Definition and Motivation Definition: random surfer with teleportation distribution r and tel. probab. c 0.15 PPR r (u) = c r(u)+(1 c) PPR r (v) Motivation: Search engines Improved ranking Fighting link spam v:(vu) E Slow to compute naively with the power method

4 Personalized PageRank Linearity Linearity: PPR α1 r 1 +α 2 r 2 (u) = α 1 PPR r1 (u)+α 2 PPR r2 (u) Single page teleportation suffices: PPR r (u) = v r(v) PPR v (u)

5 Personalized PageRank Preliminaries Two-phase algorithm 1. precomputes a PPR database 2. answers PageRank queries using the database Exact PPR on a graph of n millions...billions of vertices: Storage requirement Person. Topic sensitive O(t n) words t [Haveliwala 02] topics Hub decomp. O(h n) words h [Jeh Widom 03] pages Lower bound of Ω(n 2 ) bits, infeasible all pages [Fogaras Rácz 04]

6 Sampling Fully Personalized PageRank Express PPR u (v) as probability of random walk starting at u ending in v Sample ending points of random walks as above First algorithm with no restriction on u Additive error ±ɛ; out of bounds prob. δ Uses O(n ɛ 2 log 1/δ log n) bits of space

7 Power Iteration and Dynamic Programming Example u v 1 v 2 v 3 v 4 v 6 v 5 w Power iteration amplifies the error downwards Dynamic programming [Jeh Widom WWW 2003] averages the error upward PPR (k+1) u = cχ u + (1 c) v:(uv) E PPR (k) v /d + (u) Problem: small world, number of non-zeroes grow quickly in u s neighborhood

8 Rounded Dynamic Programming Repeat k max = 2 log 1 c ɛ times for all u ) PPR u = Round k (cχ u +(1 c) PPR v /d + (u) v:(uv) E Space: n sparse PPR u vectors in O(n 1/ɛ log n) bits optimal for top queries Can gradually decrease rounding error ɛ k from ɛ 1 = 1 to ɛ kmax = ɛ Deterministic output; inductive proof shows PPR u (v) 2ɛ/c PPR u (v) PPR u (v) Preprocessing: linear O((n + m)/(cɛ)) time

9 Dynamic Programming with Sketches Drunken Surfer Mix up memories by random hash h(v) of pages v SPPR u (i) = PPR u (v) for i = 1,..., 2e/ɛ v:h(v)=i Use surfers for j = 1,..., log1/δ and use minimum vote: Count-Min Sketch [Cormode Muthukrishnan 05] PPR u (v) = min j=1,...,log 1/δ SPPR(j) u (h j(v))

10 Dynamic Programming with Sketches Cont d Dynamic programming over sketches by their linearity A variant also gives linear time preprocessing O(n 1/ɛ log 1/δ) bits of space optimal for value queries PPR u (v) 2ɛ/c ɛ PPR u (v) PPR u (v) + 2ɛ/c

11 Lower Bounds Fully Personalized PageRank Reduction to one-way communication complexity of bit-vector probing Alice bit string y {0, 1} s 1. creates G(y) Bob index i, output: y i 2. transmits the PPR database of G(y) 3. queries the database for PPR u(i) (v(i))

12 Experiments Fully Personalized PageRank Stanford WebBase: 80M nodes, 800M edges Measured accuracy over 1000 random nodes Effect of rounding with k max = 35 iterations. Maximum Error e-04 DP with rounding Worst case bound 1e e-04 1e-05 Rounding error ɛ

13 Quality of Approximate t Precision = Recall: Kendall s Tau: approximate top-t true top-t t #inversions in approximate top-t 1 2 ( t2 )

14 Precision Fully Personalized PageRank Precision Rounding ɛ = 10 5 Rounding ɛ = Sketch Monte Carlo BFS Size of top list t 1000

15 Kendall s Tau Fully Personalized PageRank Kendall s τ Rounding ɛ = 10 5 Rounding ɛ = Sketch Monte Carlo BFS Size of top list t 1000

16 SimRank Preliminaries and Sampling Two pages are similar if pointed to by similar pages [Jeh Widom 02] { Sim (k) (1 c) Sim (k 1) (u 1,u 2 ) (v 1, v 2 ) = d (v 1 ) d (v 2 if v ) 1 v 2 1 if v 1 = v 2. (1 c) k -weighted path pair summation (incl. sampling [Fogaras Rácz 05]) over v 1 = w 0, w 1,..., w k 1, w k v 2 = w 0, w 1,..., w k 1, w k = u = u

17 SimRank Reduction to Personalized PageRank Version 0 reduction: count path pairs from v 1 and v 2 that may meet several times Sim (0) v 1,v 2 = (1 c) k k>0 u RP [k] v 1 (u)rp [k] v 2 (u) Recursively define self-similarity SimRank of at least t + 1 inner meeting points as SSim (t+1) (v)

18 SimRank Reduction to Personalized PageRank Obtain SimRank by inclusion-exclusion of self-similarities Sim(v 1, v 2 ) = k>0 (1 c) k u RP [k] v 1 (u)rp [k] v 2 (u) SSim(u) SSim(u) = 1 SSim (0) (u) + SSim (1) (u) SSim (2) (u) +... Converges for 1 c < 1/2, technicalities to carry through approximation

19 Conclusion Fully Personalized PageRank Efficient algorithms + lower bounds = space-optimal summaries for Fully Personalized PageRank and for SimRank with decay factor < 1/2 At the heart of it: low space approximation of large vectors in the... norm Works well in practice

20 Thank you! Fully Personalized PageRank

21 Algorithms Compared Algorithm Dynamic Programming with ɛ = and ɛ = 10 5 rounding to varying ɛ k Dynamic Programming with ɛ = , δ = sketches Monte Carlo sampling with N = samples Breadth First Search heuristic Running time 1.5 and 2.25 days 6 days 6 days 3.5 days

22 SimRank Example Fully Personalized PageRank v1 u3 u2 u1 v2 1 3 k k>0 u RP [k] v 1 (u)rp [k] v 2 (u) = ( ) +... = SSim (0) (u i ) = = 1 2 SSim (1) (u i ) = 1 4 SSim(u i ) = = 2 3