1. Link Building Martin Olsen Department of Computer Science Aarhus University 1

2. Outline Motivation and Introduction Contribution Link Building Communities in Networks Hedonic Games Simple Games 2

3. ... in 2012, companies will spend almost $9 billion on search engine optimization … The New York Times, January 2009 Objective of SEO: A link to your page appears here on page 1 What is Search Engine Optimization (SEO) ? 3

4. www as a Graph 4 = =

5. PageRank. Random Surfer Perspective 5 2 576 3 4 1 8109 100 1000 random surfers Random Surfer Zaps with probability 0.15

6. PageRank. Random Surfer Perspective Random Surfer Zaps with probability 0.15 6 2 576 3 4 1 8109 143 = 85 + 85/2 +15 270 100 58 355 = 4  85 + 15 15 1000 random surfers Distribution after one tick

7. PageRank. Random Surfer Perspective 7 2 576 3 4 1 8109 281 280 25443 66 15 1000 random surfers Stationary distribution after 50 ticks Random Surfer Zaps with probability 0.15

8. PageRank. Random Surfer Perspective 8 2 576 3 4 1 8109 0.281 0.280 0.2540.043 0.066 0.015 Random Surfer Zaps with probability 0.15

9. PageRank. Random Surfer Perspective 9 2 576 3 4 1 8109 0.281 0.280 0.2540.043 0.066 0.015 PageRank Ranking: 1, 2, 4, 3, 6 PageRank is an important ingredient of the ranking mechanism Relevance counts as well! Random Surfer Zaps with probability 0.15

10. Link Building is an Important Aspect of SEO 10

11. Contribution/Link Building The Computational Complexity of Link Building (Cocoon ´08) Olsen Maximizing PageRank with new Backlinks (submitted) Olsen MILP for Link Building (In preparation) Olsen, Viglas 11

12. 12 The Link Building Problem. Formal Definition LINK BUILDING Instance : G(V, E), t  V, k  Z + Solution : S  V  { t } with  S   k maximizing  t after adding S  { t } to E

13. 13 Link Building is not Trivial 2 5 7 6 3 4 1 8 0.272 0.096 0.069 0.085 0.060 0.091 0.078 0.250 2 5 7 6 3 4 1 8 0.367 0.039 0.049 0.070 0.035 0.049 0.060 0.331 2 5 7 6 3 4 1 8 0.375 0.054 0.042 0.337

14. PageRank Topology Theorem *) : The expected number of visits to p for a random surfer starting at u prior to the first zapping event 14 i j 1  1  increase in PageRank

15. Does the graph contain an independent set of size k ? Can we turn this question into a Link Building problem? k -REGULAR INDEPENDENT SET ≤ FPT LINK BUILDING 15 j i

16. 16 k -REGULAR INDEPENDENT SET ≤ FPT LINK BUILDING 1 x y OPT! i j Basic idea: Make z ij relatively big

17. 17 k -REGULAR INDEPENDENT SET ≤ FPT LINK BUILDING 1 x y OPT! i j Basic idea: Make z ij relatively big LINK BUILDING is W[1]-hard *) : LINK BUILDING solvable in time f ( k )  n c  k -REGULAR INDEPENDENT SET solvable in time f ( k )  n c  W[1] = FPT Another result: FPTAS for LINK BUILDING  NP = P

18. Upper Bound: k = 1 fixed The dashed link can be found in time corresponding to O(1) PageRank computations with a randomized scheme *). 18 2 5 7 6 3 4 1 8 0.272 0.096 0.069 0.085 0.060 0.091 0.078 0.250 2 5 7 6 3 4 1 8 0.338 0.070 0.060 0.048 0.060 0.070 0.306

19. Upper Bound: Mixed Integer Linear Programming Approach *) Price for link from i Compute the cheapest set of new incoming links that would make node 5 rank highest 19 2 5 7 6 3 4 1 8 0.187 0.061 0.049 0.189 0.036 0.099 0.200 0.178

20. A Quiz: Which of the two situations would be optimal for Martin? 20

21. Contribution/Communities in Networks Communities in Large Networks: Identification and Ranking (WAW ´06) Olsen 21

22. 22 Communities in Networks Dolphins in Doubtful Sound [Newman, Girvan ´04]:

23. 23 What is a Community? Informally: A community C is a set of nodes with relatively many links between them Assumption/Observation: A CS site has relatively many CS links! Formal definition based on assumption *) :  v  C,  u  C: w vC ≤ w uC C

24. 24 A Greedy Approach for Detecting Members of a Community *) Repeat until C is a Community: Find v  C with maximum attention to C C  C  {v} Update attentions Use two priority queues holding elements in C and V  C 1) Old C 2) New C

25. 25 An Experiment. A Danish CS Community Crawl of the dk-domain with 180.468 sites in total Representatives = 4 CS sites CS-Community with 556 sites Minimum attention, : 15.8% Maximum attention, : 15.4% Ranking: 1)www.daimi.au.dk (CS U Aarhus)www.daimi.au.dk 2)www.diku.dk (CS U Copenhagen)www.diku.dk 3)www.itu.dk (ITU Copenhagen)www.itu.dk 4)www.cs.auc.dk (CS U Aalborg)www.cs.auc.dk 5)www.brics.dk (CS PhD School)www.brics.dk 6)www.imm.dtu.dk (Informatics/Mathematical modeling DTU Copenhagen)www.imm.dtu.dk … 17)www.imada.sdu.dk (CS/Mathematics U Southern Denmark)www.imada.sdu.dk

26. 26 Other Results Computing non trivial communities by the definition given is NP-hard A simple model for the evolution of communities is presented. These communities are probably obeying the definition for large n if the out degree of the nodes is  (log n ).

27. Contribution/Hedonic Games Nash Stability in Additively Separable Hedonic Games Is NP-Hard (CiE ´07) Olsen Extended version: Nash Stability in Additively Separable Hedonic Games and Community Structures (Theory of Computing Systems ´09) Olsen 27

28. An Additively Separable Hedonic Game Five waterholes w1, …, w5 with capacities 1, 2, 3, 4 and 8 l / h respectively. Two buffaloes b1 and b2 that hate each other. They are only thirsty if they have a parasite on their back in which case they have to drink 9 l / h. Two gigantic parasites p1 and p2. They only want to sit on b1 and b2 respectively. 28

29. An Additively Separable Hedonic Game One Nash Equilibrium for the game: PARTITION ≤  NE in ASHG  NPC *) 29

30. 30 Community Structures in Networks Put a 1 on each connection between two dolphins. The community structure is a NE! NE  community structure? NE’s are NP-hard to compute even with symmetric and positive payoffs *)

31. Contribution/Simple Games On the Complexity of Problems on Simple Games (submitted) Freixas, Molinero, Olsen, Serna 31

32. 32 Open Problems/Future Work In the thesis we show LINK BUILDING  APX. Is there a PTAS for LINK BUILDING? Surgical Link Building: Isolate the Community C Model all pages in V  C as one page Use MILP Use information on distribution of PageRank Does the stuff presented really work? Thank You!

33. Link Building. A Real World Example Dear X We are trying to get more links to our website to help improve its rating on the search engines. We were wondering if you could put a link to our site … on your webpage or blog. If you have a website or a Blog and put a link to our page on it then to say thank you for each month it is up, I will give you … Source: An e-mail to a colleague X 33

34. 34 Link Building is not Trivial. 2nd Example Assumption: Obtaining a link from one green node is slightly better for node 1 compared to obtaining a link from one blue node. Now node 1 can pick three incoming links for free. What should node 1 choose? 1

35. 35 No FPTAS for LINK BUILDING if NP ≠ P *) 1 x y OPT! i j

36. 36 Power Law

37. 37 Fixed Parameter Tractability: FPT and W[1] W[1] FPT k-VERTEX COVER k-REGULAR INDEPENDENT SET k-INDEPENDENT SET Complete for W[1] LINK BUILDING is W[1]-hard *) Solvable in time f ( k )  n c

38. 38 Other Results Computing non trivial communities by the definition given is NP-hard A simple model for the evolution of communities is presented. These communities are probably obeying the definition for large n if the out degree of the nodes is  (log n ). C

39. Upper Bound: Mixed Integer Linear Programming Approach *) The dashed links show the cheapest modification that will bring node 5 to the top of the ranking. Computed using a MILP approach. Alternatively we could go for the maximum improvement in the ranking for a given budget. price for 39 2 5 7 6 3 4 1 8 0.272 0.096 0.069 0.085 0.060 0.091 0.078 0.250 2 5 7 6 3 4 1 8 0.187 0.061 0.049 0.189 0.036 0.099 0.200 0.178