This document discusses different damping functions that can be used for link-based ranking algorithms like PageRank. It begins by rewriting the PageRank algorithm and defining branching contribution of a path. It then presents an explicit formula for PageRank and generalizes it to a functional ranking using different damping functions. It proposes a linear damping function and shows how to calculate rankings using this function without storing the full path matrices. Finally, it provides pseudocode for the algorithm. The key points are: rewriting PageRank, generalizing to different damping functions, proposing a linear damping, and giving an algorithm to compute rankings with the linear damping efficiently.

1. Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
The Choice of a Damping Function for
Notation
Propagating Page Importance
Rewriting
PageRank in Link-Based Ranking
Functional
Rankings
Algorithms
Comparison
Ricardo Baeza-Yates1 , Paolo Boldi2 and Carlos Castillo3
Conclusions
1. Yahoo Research – Barcelona, Spain
2. Universit` di Milano – Italy
a
3. Universit` di Roma “La Sapienza” – Italy
a
February 6th, 2005

2. Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
1 Notation
C. Castillo
Notation 2 Rewriting PageRank
Rewriting
PageRank
Functional
Rankings
3 Functional Rankings
Algorithms
Comparison 4 Algorithms
Conclusions
5 Comparison
6 Conclusions

3. •›››››››››››››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Rewriting
PageRank Let PN×N be the normalized link matrix of a graph
Functional
Rankings Row-normalized
Algorithms No “sinks”
Comparison
Conclusions

4. ••››››››››››››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation Definition (PageRank)
Rewriting
PageRank
Stationary state of:
Functional
Rankings (1 − α)
αP + 1N×N
Algorithms N
Comparison
Conclusions

5. ••››››››››››››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation Definition (PageRank)
Rewriting
PageRank
Stationary state of:
Functional
Rankings (1 − α)
αP + 1N×N
Algorithms N
Comparison
Conclusions Follow links with probability α
Random jump with probability 1 − α

6. •••›››››››››››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Rewriting Rewriting PageRank [Boldi et al., 2005]
PageRank
Functional ∞
Rankings (1 − α)
r(α) = (αP)t .
Algorithms N
t=0
Comparison
Conclusions

7. ••••››››››››››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Rewriting
Definition (Branching contribution of a path)
PageRank
Given a path p = x1 , x2 , . . . , xt of length t = |p|
Functional
Rankings
1
Algorithms
branching(p) =
Comparison d1 d2 · · · dt−1
Conclusions where di are the out-degrees of the members of the path

8. •••••›››››››››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Explicit formula for PageRank [Newman et al., 2001]
Notation
(1 − α)α|p|
Rewriting ri (α) = branching(p)
PageRank N
p∈Path(−,i)
Functional
Rankings
Algorithms
Comparison
Conclusions

9. •••••›››››››››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Explicit formula for PageRank [Newman et al., 2001]
Notation
(1 − α)α|p|
Rewriting ri (α) = branching(p)
PageRank N
p∈Path(−,i)
Functional
Rankings
Algorithms Path(−, i) are incoming paths in node i
Comparison
Conclusions

10. •••••›››››››››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Explicit formula for PageRank [Newman et al., 2001]
Notation
(1 − α)α|p|
Rewriting ri (α) = branching(p)
PageRank N
p∈Path(−,i)
Functional
Rankings
Algorithms Path(−, i) are incoming paths in node i
Comparison
Conclusions
General functional ranking
damping(|p|)
ri (α) = branching(p)
N
p∈Path(−,i)

11. ••••••››››››››››››
Damping
Functions for
Link Ranking Distribution of shortest paths
R. Baeza-Yates,
P. Boldi and .it (40M pages) .uk (18M pages)
C. Castillo 0.3 0.3
Notation
0.2 0.2
Frequency
Frequency
Rewriting
PageRank
0.1 0.1
Functional
Rankings
0.0 0.0
5 10 15 20 25 30 5 10 15 20 25 30
Algorithms
Distance Distance
Comparison
.eu.int (800K pages) Synthetic graph (100K pages)
Conclusions
0.3 0.3
0.2 0.2
Frequency
Frequency
0.1 0.1
0.0 0.0
5 10 15 20 25 30 5 10 15 20 25 30
Distance Distance

12. •••••••›››››››››››
Damping
Functions for
0.30
Link Ranking
R. Baeza-Yates, damping(t) with α=0.8
P. Boldi and
C. Castillo
damping(t) with α=0.7
Notation 0.20
Rewriting Weight
PageRank
Functional
Rankings 0.10
Algorithms
Comparison
Conclusions 0.00
1 2 3 4 5 6 7 8 9 10
Length of the path (t)
Exponential damping = PageRank
damping(t) = α(1 − α)t

13. ••••••••››››››››››
Damping
Functions for
0.30
Link Ranking
damping(t) with L=15
R. Baeza-Yates, damping(t) with L=10
P. Boldi and
C. Castillo
0.20
Weight
Notation
Rewriting
PageRank
Functional 0.10
Rankings
Algorithms
Comparison
0.00
Conclusions 1 2 3 4 5 6 7 8 9 10
Length of the path (t)
Linear damping
2(L−t)
L(L+1) t<L
damping(t) =
0 t≥L

14. ••••••••››››››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
For calculating LinearRank we use:
Notation
∞
Rewriting 1
PageRank LinearRank = damping(t)Pt
Functional N
t=0
Rankings
L−1
Algorithms 1 2(L − t) t
Comparison
= P
N L(L + 1)
Conclusions
t=0

15. ••••••••››››››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
For calculating LinearRank we use:
Notation
∞
Rewriting 1
PageRank LinearRank = damping(t)Pt
Functional N
t=0
Rankings
L−1
Algorithms 1 2(L − t) t
Comparison
= P
N L(L + 1)
Conclusions
t=0
However, we cannot hold the temporary Pt in memory!

16. •••••••••›››››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
We have to rewrite to be able to calculate:
Notation
2
Rewriting R(0) =
PageRank L+1
Functional (L − k − 1) (k)
Rankings R(k+1) = R P
Algorithms
(L − k)
Comparison
Conclusions

17. •••••••••›››››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
We have to rewrite to be able to calculate:
Notation
2
Rewriting R(0) =
PageRank L+1
Functional (L − k − 1) (k)
Rankings R(k+1) = R P
Algorithms
(L − k)
Comparison
L−1
Conclusions LinearRank = R(k)
k=0

18. •••••••••›››››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
We have to rewrite to be able to calculate:
Notation
2
Rewriting R(0) =
PageRank L+1
Functional (L − k − 1) (k)
Rankings R(k+1) = R P
Algorithms
(L − k)
Comparison
L−1
Conclusions LinearRank = R(k)
k=0
Now we can give the algorithm . . .

19. ••••••••••››››››››
Damping
Functions for
Link Ranking 1: for i : 1 . . . N do {Initialization}
2
R. Baeza-Yates, 2: Score[i] ← R[i] ← L+1
P. Boldi and
C. Castillo 3: end for
Notation
Rewriting
PageRank
Functional
Rankings
Algorithms
Comparison
Conclusions

20. ••••••••••››››››››
Damping
Functions for
Link Ranking 1: for i : 1 . . . N do {Initialization}
2
R. Baeza-Yates, 2: Score[i] ← R[i] ← L+1
P. Boldi and
C. Castillo 3: end for
Notation
4: for k : 1 . . . L − 1 do {Iteration step}
Rewriting
5: Aux ← 0
PageRank
Functional
Rankings
Algorithms
Comparison
Conclusions

21. ••••••••••››››››››
Damping
Functions for
Link Ranking 1: for i : 1 . . . N do {Initialization}
2
R. Baeza-Yates, 2: Score[i] ← R[i] ← L+1
P. Boldi and
C. Castillo 3: end for
Notation
4: for k : 1 . . . L − 1 do {Iteration step}
Rewriting
5: Aux ← 0
PageRank 6: for i : 1 . . . N do {Follow links in the graph}
Functional
Rankings
7: for all j such that there is a link from i to j do
Algorithms
8: Aux[j] ← Aux[j] + R[i]/outdegree(i)
Comparison 9: end for
Conclusions 10: end for

22. ••••••••••››››››››
Damping
Functions for
Link Ranking 1: for i : 1 . . . N do {Initialization}
2
R. Baeza-Yates, 2: Score[i] ← R[i] ← L+1
P. Boldi and
C. Castillo 3: end for
Notation
4: for k : 1 . . . L − 1 do {Iteration step}
Rewriting
5: Aux ← 0
PageRank 6: for i : 1 . . . N do {Follow links in the graph}
Functional
Rankings
7: for all j such that there is a link from i to j do
Algorithms
8: Aux[j] ← Aux[j] + R[i]/outdegree(i)
Comparison 9: end for
Conclusions 10: end for
11: for i : 1 . . . N do {Add to ranking value}
12: R[i] ← Aux[i] × (L−k−1)
(L−k)
13: Score[i] ← Score[i] + R[i]
14: end for
15: end for
16: return Score

23. ••••••••••››››››››
Damping
Functions for
Link Ranking 1: for i : 1 . . . N do {Initialization}
2
R. Baeza-Yates, 2: Score[i] ← R[i] ← L+1
P. Boldi and
C. Castillo 3: end for
Notation
4: for k : 1 . . . L − 1 do {Iteration step}
Rewriting
5: Aux ← 0
PageRank 6: for i : 1 . . . N do {Follow links in the graph}
Functional
Rankings
7: for all j such that there is a link from i to j do
Algorithms
8: Aux[j] ← Aux[j] + R[i]/outdegree(i)
Comparison 9: end for
Conclusions 10: end for
11: for i : 1 . . . N do {Add to ranking value}
12: R[i] ← Aux[i] × (L−k−1)
(L−k)
13: Score[i] ← Score[i] + R[i]
14: end for
15: end for
16: return Score

24. •••••••••••›››››››
Damping
Functions for
Link Ranking Other functions studied in the paper:
R. Baeza-Yates,
P. Boldi and Hyperbolic damping
C. Castillo
Notation
Rewriting
PageRank
Functional
Rankings
Algorithms
Comparison
Conclusions

25. •••••••••••›››››››
Damping
Functions for
Link Ranking Other functions studied in the paper:
R. Baeza-Yates,
P. Boldi and Hyperbolic damping
C. Castillo
Empirical damping
Notation
Rewriting 0.7
PageRank Average text similarity
Functional
Rankings
0.6
Algorithms
0.5
Comparison
Conclusions
0.4
0.3
0.2
1 2 3 4 5
Link distance

26. ••••••••••••››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Rewriting
PageRank
How to approximate one functional ranking with another?
Functional
Rankings
Algorithms
Comparison
Conclusions

27. ••••••••••••››››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Rewriting
PageRank
How to approximate one functional ranking with another?
Functional Analysis (in the paper): match the first few levels of their
Rankings
damping functions
Algorithms
Comparison In practice the orderings can be very similar . . .
Conclusions

28. •••••••••••••›››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Experimental comparison: 18-million nodes in the U.K. Web
Rewriting
PageRank Graph
Functional
Rankings
Algorithms
Comparison
Conclusions

29. •••••••••••••›››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Experimental comparison: 18-million nodes in the U.K. Web
Rewriting
PageRank Graph
Functional
Rankings
Calculated PageRank with α = 0.1, 0.2, . . . , 0.9
Algorithms
Comparison
Conclusions

30. •••••••••••••›››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Experimental comparison: 18-million nodes in the U.K. Web
Rewriting
PageRank Graph
Functional
Rankings
Calculated PageRank with α = 0.1, 0.2, . . . , 0.9
Algorithms Calculated LinearRank with L = 5, 10, . . . , 25
Comparison
Conclusions

31. •••••••••••••›››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Experimental comparison: 18-million nodes in the U.K. Web
Rewriting
PageRank Graph
Functional
Rankings
Calculated PageRank with α = 0.1, 0.2, . . . , 0.9
Algorithms Calculated LinearRank with L = 5, 10, . . . , 25
Comparison
For certain combinations of parameters, the rankings are
Conclusions
almost equal!

32. ••••••••••••••››››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
Experimental Comparison in the U.K. Web Graph
C. Castillo
Notation
Rewriting
1.00
0.95
τ
PageRank
Functional 0.90
Rankings
0.85
τ ≥ 0.95
Algorithms 0.80
Comparison
Conclusions 25
20
15 0.9
L 10 0.7
0.8
5 0.5 0.6 α

33. •••••••••••••••›››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
Prediction of Best Parameter Combinations (Analysis)
C. Castillo
25
Actual optimum
Notation Predicted optimum with length=5
Rewriting
L that maximizes Kendall’s τ 20
PageRank
Functional
Rankings 15
Algorithms
Comparison 10
Conclusions
5
0.5 0.6 0.7 0.8 0.9
Exponent α

34. ••••••••••••••••››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Rewriting
What have we done?
PageRank
Functional
Rankings
Algorithms
Comparison
Conclusions

35. ••••••••••••••••››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Rewriting
What have we done?
PageRank
Separate the damping from the calculation
Functional
Rankings
Algorithms
Comparison
Conclusions

36. ••••••••••••••••››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Rewriting
What have we done?
PageRank
Separate the damping from the calculation
Functional
Rankings Show that different damping functions can provide the
Algorithms
same ranking
Comparison
Conclusions

37. ••••••••••••••••››
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Rewriting
What have we done?
PageRank
Separate the damping from the calculation
Functional
Rankings Show that different damping functions can provide the
Algorithms
same ranking
Comparison
Conclusions
Analysis and experiments in the paper

38. •••••••••••••••••›
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Rewriting What can we do with this?
PageRank
Functional Fast approximation of PageRank using linear damping
Rankings
Algorithms
Comparison
Conclusions

39. •••••••••••••••••›
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Rewriting What can we do with this?
PageRank
Functional Fast approximation of PageRank using linear damping
Rankings
Algorithms
Fast calculation of other link-based rankings (e.g. HITS)
Comparison
Conclusions

40. •••••••••••••••••›
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Rewriting What can we do with this?
PageRank
Functional Fast approximation of PageRank using linear damping
Rankings
Algorithms
Fast calculation of other link-based rankings (e.g. HITS)
Comparison Spam detection (e.g.: cut the first levels of links)
Conclusions

41. ••••••••••••••••••
Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Notation
Rewriting
PageRank
Functional
Rankings
Thank you!
Algorithms
Comparison
Conclusions

42. Damping
Functions for
Link Ranking
R. Baeza-Yates,
P. Boldi and
C. Castillo
Baeza-Yates, R., Boldi, P., and Castillo, C. (2005).
Notation The choice of a damping function for propagating importance in link-based
ranking.
Rewriting
PageRank
Technical report, Dipartimento di Scienze dell’Informazione, Universit degli
Studi di Milano.
Functional
Rankings Boldi, P., Santini, M., and Vigna, S. (2005).
Algorithms Pagerank as a function of the damping factor.
In Proceedings of the 14th international conference on World Wide Web,
Comparison
pages 557–566, Chiba, Japan. ACM Press.
Conclusions
Newman, M. E., Strogatz, S. H., and Watts, D. J. (2001).
Random graphs with arbitrary degree distributions and their applications.
Phys Rev E Stat Nonlin Soft Matter Phys, 64(2 Pt 2).