chapter 7 pagerank · background web graph google’s matrix teleportation 1 − α personalised...
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Chapter 7 PageRank
Angsheng Li
Institute of SoftwareChinese Academy of Sciences
Advanced AlgorithmsU CAS
1st, April, 2017
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Outline
1. Backgrounds
2. Web graph
3. Google’s matrix
4. Teleportation
5. Personalised vector
6. Sensitivity
7. Proofs
8. Local algorithms
9. Exrcises
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
The new phenomena
Brin and Page, 1995 - 1998
1. The current-generation search engine
2. Billions of queries everyday
3. What is the principle behind?
4. How good is the current-generation search engine?
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
The graph
• Massive directed graph
• Nodes: webpages
• Directed edges, hyperlines, including inlinks and outlinks
• The question: Rank the web pages by importance.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
The PageRank thesis
A page is important, if it is pointed to by many important pages.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Brin and Page, 1998
Established the equation of the PageRank thesis.The PageRank of a page Pi , written r(Pi), is the sum of thePageRanks of all the pages pointing to Pi , that is,
r(Pi) =∑
Pj∈Bi
r(Pj)
|Pj |, (1)
• Bi : the set of pages pointing to Pi ,
• |Pj |: the number of outlinks from page Pj .
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Recurrence of the PageRank
rk+1(Pi) =∑
Pj∈Bi
rk (Pj )
|Pj |
r0(Pi) =1n
(2)
The stationary solution of the recursive equation in Equation (2)gives rise to the PageRank of a graph G.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Matrix representation
Hij =
1|Pi |
if there is an edge from node i to node j ,
0 o.w.(3)
|Pi |: The number of outlinks from node i .H = (Hij) is the PageRank matrix of G.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
PageRank solution
Let πT be a 1× n vector.Set
π(k+1)T = π(k)TH,
π(0)T = 1n eT,
(4)
where eT = (1,1, · · · ,1).For the equation (4), we require:
• convergence and the interpretation of the solution
• Uniqueness of the solution
• Invariance of π(0)
• The number of iterations of the convergent solution
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Rank sinks
1 2
ցւ3
All the PageRanks go to node 3.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Matrix S
To solve the sink problem, define a vector a,
ai =
1 if node i has no outgoing links,
0 o.w.(5)
DefinitionDefine
S = H +1n
aeT,
where eT = (1,1, · · · ,1).Intuition : If node i has no outgoing link, then from node i , therandomly walks to any other nodes uniformly.S is the transition probability matrix of a Markov chain.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Google’s matrix G
DefinitionDefine the Google’s matrix by
G = αS + (1− α)J,
where Jij =1n .
• J is called teleportation matrix
• 1− α is called the teleportation parameter.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Expander
Recall: If G is a graph with λ = λ(G) < 1, then for A = AG,
A = (1− λ)J + λC,
for some C with ‖C‖ ≤ 1.We thus know that Google’s matrix is an expander. However,the parameter α is chosen arbitrarily. Of course, α determinesthe spectral gap of the graph.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Properties of G - I
(1) G is stochasticIt is a convex combination of two stochastic matrices S andJ.
(2) G is irreducible.Every page is directly connected to every other page.
(3) G is aperiodic.Gii > 0. Every node has a self-loop.
(4) G is primitive.There exists a k such that Gk > 0Because: G is an expander. There is a unique πT such that
‖pGl − πT‖ ≈ 0
for a small l . – Power method works
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Properties of G - II
(5) G is rank-one updated
G = αS + (1− α)1n
eeT
= α(H +1n
aeT) + (1− α)1n
eeT
= αH + (α1n
a + (1− α)1n
e)eT. (6)
• H is sparse• α 1
n a + (1− α) 1n e is dense, but only one-dimensional vector.
(6) G is artificial due to the choice of α.G may not well reflect the real world H.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Computation of πT
Power method
π(k+1)T = π(k)TG
= απ(k)TS +1− α
nπ(k)TeeT
= απ(k)TH + (απ(k)Ta + (1− α)e)eT/n. (7)
Suppose that 1, λ2, · · · , λn are the eigenvalues of G with1 > |λ2| ≥ · · · ≥ |λn|.Then:
G = G1 + λ2G2 + · · · + λnGn,
– G2i = Gi ,
– For i 6= j , GiGj = 0.Then
Gl = G1 + λl2G2 + · · · λl
nGn
Since λ2 < 1, Gl quickly converges to G1.Furthermore, for any probabilistic vector p,
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
λ(G)
LemmaFor the Google matrix G = αS + (1− α)J,
|λ2(G)| ≤ α.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
λ(G) again
LemmaIf the spectrum of the stochastic matrix S is 1, λ2, · · · , λn,then the spectrum of the Google matrix G = αS + (1− α)evT is
1, αλ2, · · · , αλn,where vT is the personalised vector.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Proofs - I
Since S is stochastic, (1,e) is an eigenpair of S. Let Q = (eX )be a nonsingular matrix that has the eigenvector e as its firstcolumn.Set
Q−1 =
(
yT
Y T
)
(8)
Then:
Q−1Q =
(
yTe yTXY Te Y TX
)
=
(
1 00 I
)
(9)
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Proofs - II
Similarly,
Q−1SQ =
(
yTe Y TSXY Te Y TSX
)
=
(
1 yTSX0 Y TSX
)
(10)
This implies that Y TSX contains the remaining eigenvalues ofS, i.e., λ2, · · · , λn.In addition,
Q−1GQ =
(
1 αyTSX + (1− α)vTX0 αY TSX
)
(11)
The eigenvalues of G are
1, αλ2, · · · , αλn.Since λ2 ≤ 1, αλ2 ≤ α.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
The role α
G = (1− α)J + αS.
If α is small, then 1− α is large, G is basically an artificialrandom graph, failing to reflect the real world matrix S.If α is large, then
• there is no unique stationary distribution
• even if there is a stationary distribution, it is hard tocompute
• the power method fails
Google’s choice : α = 0.85.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Personalised PageRank
For a personalised probability vector vT,
G = αS + (1− α)evT.
The power method works as before.The stationary distribution is a personalised PageRank.Significance : Real applications.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
The stationary distribution
TheoremThe Pagerank πT(α) of Gα is
πT(α) =1
n∑
i=1Di(α)
(D1(α),D2(α), · · · ,Dn(α))
where Di(α) is the i-th principal minor determinant of ordern − 1 in I −Gα.Furthermore, every Di(α) is differentiable for α.
Proof.By definition.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Differential
TheoremIf πT(α) = (π1(α), π2(α), · · · , πn(α)), then
1. For each j,
|dπj(α)
dα| ≤ 1
1− α.
2.
‖dπT(α)
dα‖1 ≤
21− α
.
• If α is small, then the PageRank πT(α) is not sensitive.
• If α is large, then the upper bounds 11−α and 2
1−α are bothapproaching to infinity.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Representation
Theorem
dπT(α)
dα= −vT(I − S)(I − αS)−2.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Sensitive to H
1.dπT(hij)
dhij= απi(e
Tj − vT )(I − αS)−1
2.(I − αS)−1 →∞,
as α goes to 1.
πT is sensitive to perturbations in H is α ≈ 1.Therefore, if α ≈ 1, then πT is sensitive to small changes of thematrix H.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Sensitive to vT
dπT(vT )
dvT = (1− α+ α∑
i∈D
πi)(I − αS)−1,
D is the set of nodes that have no outgoing links.The same as before, as α goes to 1, (I − αS)−1 goes to∞.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Summary of sensitivity
If α ≈ 1, then
1. Computing πT(α) is hard, since the power method fails
2. πT(α) is sensitive to the perturbation of H
3. πT(α) is sensitive to the personalised vector vT
Google’s tradeoff:
α = 0.85
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Proof of upper bounds - ITheoremIf πT(α) = (π1(α), π2(α), · · · , πn(α)), then
1. For each j,
|dπj(α)
dα| ≤ 1
1− α.
2.
‖dπT(α)
dα‖1 ≤
21− α
.
πT (α) is a probability vector, so
n∑
i=1
πi(α) = 1
givingπT(α)e = 1, eT = (1,1, · · · ,1).
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Proof of upper bounds- IIBy definition,πT (α) = πT (α)G(α) = πT (α)(αS + (1− α)evT ).By differential,
dπT (α)
dα= πT (α)(S − evT )(I − αS)−1. (12)
For (1). For every real x , xT⊥e, i.e.,∑
xi = 0, and for all realvector y , column vector,
|xTy | = |n
∑
i=1
xiyi |
≤ ‖xT‖1 ·ymax − ymin
2. (13)
By Equation (12),
dπj(α)
dα= πT(α)(S − evT)(I − αS)−1ej .
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Prof of upper bounds - III
Since πT(α)(S − evT)e = 0, set xT = πT(α)(S − evT) andy = (I − αS)−1ej .By Inequality (13),
|dπj(α)
dα| ≤ ‖πT(α)(S − evT)‖1 ·
ymax − ymin
2.
Since ‖πT(α)(S − evT)‖1 ≤ 2,
|dπj (α)dα | ≤ ymax − ymin.
Since (I − αS)−1 ≥ 0 and (I − αS)e = (1− α)e, and hence(I − αS)−1 = (1− α)−1e.This shows that ymin ≥ 0.For ymax, we haveymax ≤ maxi ,j [(I − αS)−1]ij ≤ 1
1−α .(1) follows.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Proof of upper bounds - IV
For (2).
‖dπT(α)
dα‖1 = ‖πT(α)(S − evT)(I − αS)−1‖1
≤ ‖πT(α)(S − evT)‖1 · ‖(I − αS)−1‖∞≤ 2
11− α
=2
1− α. (14)
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Conductance
Given a graph G = (V ,E) and S ⊂ V , the conductance of S inG is:
Φ(S) =|E(S, S)|
minvol(S), vol(S).
The conductance of G isΦ = minΦ(S) | |S| ≤ n
2.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Push(u)
Andersen, Chung and Lang, FOCS, 2006.Define an operatorPush(u):
1. p(u)← p(u) + αr(u)
2. r(u)← (1− α)r(u)/2
3. For each v with v ∼ u,set
r(v)← r(v) + (1− α)r(u)/(2d(u)).
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Approximate PageRank
Given a node v ,
1. set p = 0, r(v) = 1, and r(u) = 0 for all u 6= v .
2. For every u, if r(u) ≥ ǫd(u), then:– Apply push(u).
3. Otherwise, Then output p and r .
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
ACL local algorithm
1. To find the RageRank from a given input vertex v ,
2. To rank the pages by decreasing of the normalisedPageRank, i.e., pv
d(v) . Suppose that v1, v2, · · · , vl
is listed such that
pv1
d(v1)≥ pv2
d(v2)≥ · · · ≥ pvl
d(vl).
3. (Pruning) To take an initial segment of the list as acommunity associated with the given input v .Let j be such that
Φ(Xj) = minΦ(Xi) | 1 ≤ i ≤ l,where φ(X ) is the conductance of X in G, andXk = v1, · · · , vk.Output Xj .
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Question for local algorithm
For every query Q, we rank the set of answers for the query byPageRank, however, the list is a too long list.The question is to determine a short list of ranks as the outputof the query.Still open.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
The great idea
• The PageRank thesis
• The teleportation parameter 1− α.This is a great idea, which may be used in many otherareas, such as learning, data processing.The essence of the idea here is to make sure that theRanking matrix is a well-defined stochastic procedure sothat PageRank exists and can be computed.We may also regard the introduction of 1− α as amplifyingnoises, playing a role similar to that in the error correctingcodes.
• Google’s success: Making big money by randomness
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
A grand challenge
• What is the principle for determining α? Is there a metric ofnetworks which determines the optimum α?
• What are principles for structuring the unstructured andnoisy data?
• Making money by connection and interaction???
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Reference
1. Amy N. Langville and Carl D. Meyer, Google’s PageRankand Beyond: The Science of Search Engine Ranking,Princeton University Press, 2006.2. Andersen, Chung and Lang, Local graph partitioning usingPageRank vectors, FOCS, 2006.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Natural rank
The natural rank based on the structural information theory isthe answer.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 1
Let X1, · · · ,Xn be independent random variables such that Xi isequal to 1 with probability 1− δ and equal to 0 with probability
δ. Let X =n∑
i=1Xi( mod 2). Prove that
Pr[X = 1] =
12 + (1− 2δ)n/2, n is odd,12 − (1− 2δ)n/2, n is even.
Significance?
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 1 - proof 1Let Yi = (−1)Xi , and Y =
n∏
i=1Yi .
Assume n odd.Let Pr[X = 1] = α.Then
E [Y ] = 1− 2α.
Since Xi and then Yi are independent,
E [Y ] = (−1 + 2δ)n
Therefore
(−1 + 2δ)n = 1− 2α
α =12+
(1− 2δ)n
2.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 1 - proof 2Proof.
(1 − 2δ)n = ((1 − δ) − δ)n
=
n∑
i=0
(
ni
)
(1 − δ)i (−δ)n−i .
Case 1. n odd(1− 2δ)n = Pr[X = 1]− Pr[X = 0],
Pr[X = 1] =12+
12(1 − 2δ)n
Case 2. n even (1− 2δ)n = Pr[X = 0]− Pr[X = 1],
Pr[X = 1] =12− 1
2(1 − 2δ)n
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 2
Prove that if there exists a δ-density distribution H such thatPr
x∈RH[C(x) = f (x)] ≤ 1
2 + ǫ for every circuit C of size at most s
with s ≤√
ǫ2δ2n/100, then there exists a subset I ⊆ 0,1n ofsize at least δ
22n such that
Prx∈RI
[C(x) = f (x)] ≤ 12+ 2ǫ
for every circuit C of size at most s.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 2 - Proof
Some problems?Leave this to Mingji
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 3
1. Let f : F→ F be any function. Suppose integer d ≥ 0 and
number ǫ > 2√
d|F| . Prove that there are at most 2/ǫ degree
d polynomials that agree with f on at least an ǫ fraction ofits coordinates.Significance?
2. Prove that if Q(x , y) is a bivariate polynomial over somefield F and P(x) is a univariate polynomial over F such thatQ(x ,P(x)) is the zero polynomial, thenQ(x , y) = (y − P(x))A(x , y) for some polynomial A(x , y).
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 3 - proof 1Suppose that
P1,P2, · · · ,Pl
are the all degree d polynomials that agree with f in at least ǫfraction of coordinates.For each i , define a vector vi by
vi(j) =
1, if Pi(j) = f (j),
0,otherwise
for every j ∈ F.Then for every i ,
‖vi‖1 ≥ ǫ ·m,
where m = |F|.
ǫm ≤ 〈vi , vi〉 ≤ m
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 3 - proof 2
Set
v =l
∑
i=1
vi
Then
〈v , v〉 =l
∑
i=1
〈vi 〉+∑
i 6=j
〈vi vj〉
≤ l ·m + (l2 − l)d .
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 3 - proof 3
And
〈v , v〉 =∑
k∈F
(v(k))2
=∑
k
(
l∑
i=1
vi(k))2
≥ (∑
k∑
i vi(k))2
m
=(∑
i∑
k vi(k))2
m
≥ (lǫm)2
m.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 3 - proof 4
This gives
l ≤ 1− dm
ǫ2 − dm
for ǫ >√
dm .
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 3 - proof 5For
ǫ > 2
√
dm
and
l ≤ 2ǫ.
ǫ+ (ǫ− dm) + · · · (ǫ− (l − 1)d
m) ≥ 1
with
ǫ− (l − 1)dm
≥ 0
Solving this, we have
l ≤ 2ǫ.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 3 - proof 4
Take Q(x , y) as a polynomial of y with coefficients beingpolynomials of x .Divide Q(x , y) by the linear function y − P(x), linear in variabley , giving
Q(x , y) = (y − P(x))A(x , y) + R(x)
By the assumption,
Q(x ,P(x)) = R(x) ≡ 0.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 4
Linear codes We say that an ECC E : 0,1n → 0,1m islinear, if for every x , x ′ ∈ 0,1n, E(x + x ′) = E(x) + E(x ′)(componentwise addition modulo 2). A linear ECC can be seenan m × n matrix A such that E(x) = Ax , thinking of x as acolumn vector.
1. Prove that the distance of a linear ECC is equal to theminimum over all nonzero x ∈ 0,1n of the fraction of 1’sin E(x).
2. Prove that for every δ > 0, there exists a linear ECCE : 0,1n → 0,1m for m = Ω(n)/(1 − H(δ)) withdistance δ.
3. Prove that for some δ > 0, there is an ECCE : 0,1n → 0,1poly(n) of distance δ with poly timeencoding, and decoding algorithms.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 4 - proof 1
Let A be an m × n 0,1 matrix which defines a linear ECC.The distance of A is:
δ = minx 6=x ′
1m· |i | yi 6= y ′
i |
where
yi = ai ,1x1 + ai ,2x2 + · · · ai ,nxn
and
y ′i = ai ,1x ′
1 + ai ,2x ′2 + · · · ai ,nx ′
n
This is
δ = minx 6=0
1m· |i |yi = 1|.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 4 - proof 2
Remove the condition that E is linear.Given a vector y ∈ 0,1m, define the δ-ball of y to be the set ofthe vectors z ∈ 0,1m such that the distance between y and zis less than δ.Denoted by Bδ
y . Then
|Bδy | ≤
(
mδ ·m
)
= o(1) · 2H(δ)·m
In increasing order, for each x ∈ 0,1n, we define E(x) to be ay ∈ 0,1m such that Bδ
y disjoins all the δ-balls associated withthe codewords of x ′ < x .Suppose that m ≥ n
1−H(δ) . Then the definition above neverstops, since there are at least 2n many disjoint δ-balls in0,1m.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 4 - proof 3
Consider now the linear ECC.Each linear ECC is given by an m × n matrix A.Two approaches:Case 1. Consider the random matrix A.With nonzero probability that A is such an ECC.Case 2. Counting the number of linear ECC that have distance< δ.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 4 - proof 4Consider the first approach.Let A be a random m × n matrix.We say that x = (x1, x2, · · · , xn) is a witness showing that A hasdistance < δ, if x 6= 0 and there are < δm many j such thatyj = 1, where
yj = aj1x1 + · · ·+ ajnxn.
For each j , define
Yj =
1, if yj = 1,
0, otherwise.
Let
Y =m∑
j=1
Yj .
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 4 - proof 5Then for each j ,
E [Yj ] =12
E [Y ] = µ =m2
Clearly, all Yj ’s are independent.By the Chernoff bound, for ǫ = 1− 2δ,
Pr[Y < δm] = Pr[Y < (1 − ǫ)µ]
≤ [e−ǫ
(1− ǫ)(1−ǫ)]
m2
≤ 12c·m ,
for some constant c.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 4 - proof 6
By the union bound,the probability that A has a witness for distance < δ is
12c·m−n
which is ≈ 0 if
m = Ω(n).
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 4 - proof 7
Consider the Reed-Solomon code
RS : Fn → Fm
It is an ECC with distance δ1 = 1− nm .
For every x = (a0,a1, · · · ,an−1) ∈ Fn,
RS(x) = (z0, z1, · · · , zm−1)
where
zj =
n−1∑
i=0
ai ji
j ∈ F.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 4 - proof 8
Let |F| = 2k .Then each element f ∈ F is interpreted as an element inGF (2k ).For x ∈ F
n, we interpret it as an element in 0,1k ·n. Weencode RS(x) by
WH(z0),WH(z1), · · · ,WH(zm−1)
This is an ECC from 0,1k ·n to 0,1m·2k.
Choosing k such that m · 2k is a polynomial of k · n.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 5
1. Recall the spectral norm of a matrix A, written ‖A‖ to bethe maximum ‖Av‖2 for unit v . Let A be symmetricstochastic, i.e., A = AT, and every row and column of A hasnonnegative entries summing up to 1. Prove that ‖A‖ ≤ 1.
2. Let A, B be symmetric stochastic matrices. Prove thatλ(A + B) ≤ λ(A) + λ(B).
3. Let A, B be two n × n matrices.(a) Prove that ‖A + B‖ ≤ ‖A‖+ ‖B‖.(b) Prove that ‖AB‖ ≤ ‖A‖ · ‖B‖
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 5 - proof
For 1.First,
‖A‖ ≤ n2.
Second, for every such A,
• A2 is symmetric stochastic matrix
•‖A2‖ ≥ ‖A‖2.
• If there is an A such that ‖A‖ = 1+ α for α > 0. Then thereis such a B with ‖B‖ unbounded.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 6
Let G be an (n,d , λ)-expander graph, and B be a set of verticesof size at most βn for 0 < β < 1. Let X1,X2, · · · ,Xk be arandom walk of k steps in G from X1 that is randomly anduniformly chosen.
1. Prove that for every subset I ⊆ [k ],
Pr[(∀i ∈ I)[Xi ∈ B]] ≤ (1− λ)√
β + λ)|I|−1.
2. Conclude that if B < n/100 and λ < 1/100, then theprobability that there exists a subset I ⊆ [k ] such that|I| > k/10 and ∀i≤|I|Xi ∈ B is at most 2−k/100.
3. To show that every BPP algorithm that uses m coins anddecides a language L with probability 0.99 into analgorithm B that uses m + O(k) coins and decides thelanguage L with probability 1− 2−k .
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 6: Proof - I
For each i , 1 ≤ i ≤ k , letBi : the event Xi ∈ B. For I ⊆ [k ], let I = j1 < j2 < · · · ji. Then:
Pr[∧i∈IBi ]
= Pr[Bj1] · Pr[Bj2 |Bj1] · · · · · Pr[Bji |Bj1, · · · ,Bji−1]. (15)
Define B to be a linear transformation from Rn to R
n that keepsthe values indexed in B. That is, for (u1,u2, · · · ,un), define
(Bu)i =
ui , if i ∈ B,
0, otherwise.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 6: Proof - II
For every probability vector p,(i) Bp is the vector whose coordinates sum to the probabilitythat a vertex i is chosen according to p, is in B.(ii) The normalised Bp is the distribution of p conditioned to theevent that the vertex is in B.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 6: Proof - III
Let pj be the distribution of Xj conditioned on the eventsBj1 , · · · ,Bji . Then:
p1 =1
Pr[Bj1]· B1
p2 =1
Pr[Bj2 |Bj1 ]Pr[Bj1 ]BAB1
pi =1
Pr[Bji |Bji−1, · · ·Bj1] · · ·Pr[Bj1]
(BA)i−1B1.
Hence,
Pr[Bj1 ] · · ·Pr[Bji |Bji−1· · ·Bj1]p
i = (BA)i−1B1.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 6: Proof - IV
Pr[∧j∈IBj ] = Pr[B1] · · ·Pr[Bji |Bji−1· · ·Bj1 ] = ‖(BA)i−1B1‖1.
Let A = (1− λ)J + λC for some C with ‖C‖ ≤ 1.Then BA = (1− λ)BJ + λBC.Noting:(i) ‖B1‖2 ≤
√β‖1‖2
(ii) ‖BJ‖ ≤√β, ‖B‖ ≤ 1, ‖BC‖ ≤ 1.
(iii) ‖BA‖ ≤ (1− λ)√β + λ
Therefore,
|(BA)i−1B1|1 ≤ ‖(BA)i−1B1‖2 ·√
n
≤ ((1 − λ)√
β + λ)i−1. (16)
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 7
(1) Give a probabilistic polynomial time algorithm that given a3CNF formula φ with exactly three distinct variables ineach clause, outputs an assignment satisfying at least a 7
8fraction of φ’s clauses.
(2) Give a deterministic polynomial time algorithm with thesame approximation guarantee as Exercise 1 above.
(3) Show a polynomial time algorithm that given a satisfiable2CSP instance φ over binary alphabet with m clausesoutputs a satisfying assignment for φ.
(4) Show a deterministic poly (n,2q)-time algorithm that givena qCSP-instance φ over binary alphabet with m clausesoutputs an assignment satisfying m/2q of the constraintsof φ.
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 7 - proof
Easy
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 8
(5) Suppose that G = (V ,E) is an (n,d , λ)-expander. Showthat for any S ⊂ V of size ≤ n
2 , the following holds:
Pr(u,v)∈RE
[u ∈ S ∧ v ∈ S] ≤ |S|n
(12+
λ
2).
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 8 - proof- 1
Pre=(u,v)∈RE
[u ∈ S& v ∈ S]
= Pr[u ∈ S] · Pr[v ∈ S | u ∈ S].
Clearly,
Pr[u ∈ S] =sn,
where s = |S|.Recall the expander mixing lemma, for any X , and Y ,
|e(X ,Y )− vol X · vol Yvol G
| ≤ λ√
vol X · vol Y .
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Background Web graph Google’s matrix Teleportation 1 − α Personalised vector Sensitivity Proofs Local algorithms New
Exercise 8 - proof -2
For X = Y = S, using the lemma,
Pr[v ∈ S | u ∈ S] ≤ 12(1 + λ).