Download Random and Quantum Walks on Graphs: an Introduction

Random and Quantum Walks on Graphs:
an Introduction
Simone Severini ([email protected])
AIMS
June 12, 2012
References
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A. Ambainis, Quantum walks and their algorithmic applications,
arXiv:quant-ph/0403120v3
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R. Diestel, Graph Theory, Graduate Texts in Mathematics,
Springer-Verlag, Heidelberg, 2010.
http://diestel-graph-theory.com/
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N. Konno, Quantum Walks: Lecture at the School "Quantum
Potential Theory: Structure and Applications to Physics" Held at
the Alfried-Krupp-Wissenskolleg, Greifswald, 26 February - 9 March
2007.
I
L. Lovász, Random walks on graphs: a survey.
http://www.cs.elte.hu/~lovasz/erdos.pdf
M. Santha, Quantum walk based search algorithms.
arXiv:0808.0059v1 [quant-ph]
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R. de Wolf, Quantum Computing: Lecture Notes, CWI Amsterdam.
http://homepages.cwi.nl/~rdewolf/qwalksintro.pdf (*)
Plan
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Graphs
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Random walks
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Quantum walks
Syllabus
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Natural networks, Technological networks, Social networks, Graphs,
Subgraphs, Isomorphisms, Degrees, Degree distributions, Regular
graphs, Complete graphs, Clique number, Independence number,
Chromatic number, Paths and cycles, Connectedness, Regularity
Lemma, Ramsey numbers, Max-Flow Min-Cut, Visibility graphs,
Graph widths, Adjacency matrix, Adjacency spectrum, Expansion,
Random walks, Mixing/hitting, PageRank, Grover matrix, A special
case: regular graphs, Coined quantum walks, Mixing,
Continuous-time quantum walks, Mixing matrices, Applications:
keywords, Find the missing link, A general framework (1/2), A
general framework (2/2), A mathematical open problem
Natural networks
Technological networks
Social networks
Graphs
Graphs
De…nition
A graph G = (V , E ) is a pair of two sets: the elements of V are called
vertices; E
V V ffi, i g : i 2 V gg and its elements are called
edges.
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Generalizations: digraphs (directed graphs); weighted graphs.
Subgraphs
De…nition
A graph H = (W , F ) is a subgraph of a graph G = (V , E ) if W
and F
E.
V
Isomorphism
De…nition
Two graphs G = (V , E ) and H = (W , F ) are isomorphic (G = H) if
there is a bijection f : V ! W such that fi, j g 2 E i¤
ff (i ), f (j )g 2 F . The number of isomorphism classes on n vertices is at
most (1 + o (1)) 2n (n 1 )/2 /n!.
De…nition
A map on graph is called a (complete) graph invariant (i¤) if it assigns
equal values to isomorphic graphs.
Degrees
De…nition
The neighbourhood of a vertex i is N (i ) := fj : fi, j g 2 E g. The
nonnegative integer dG (i ) = jN (i )j is the degree of a vertex i. The
maximum degree is denoted by ∆ (G ); the minimum degree by δ (G ).
Lemma (Degree sum formula)
In a graph G , ∑i 2V dG (i ) = 2 jE j.
Lemma
In any graph, there are two vertices of equal degree.
Lemma (Handshaking lemma)
The number of vertices of odd degree in a graph is even.
Degree distributions
De…nition
The degree sequence of a graph G is the list of its degrees
f dG ( i ) : i 2 V g .
De…nition
The degree distribution of a graph G on n vertices is the list
fPG (k ) : k = 0, 1, ..., n 1g, where PG (k ) = nk /n and
nk = jfi : dG (i ) = k gj.
De…nition
A unigraph is a graph characterized by its degree sequence.
Degree distributions: example
Regular graphs
De…nition
A graph G is k-regular if dG (i ) = k for every vertex i.
Complete graphs
De…nition
A complete graph Kn is an (n
1)-regular graph on n vertices.
Clique number
De…nition
A clique is a complete induced subgraph.
De…nition
The clique number ω (G ) of a graph G is the order of its largest clique.
Independence number
De…nition
A set S V is an independent set in a graph G = (V , E ) if fi, j g 2
/E
for every i, j 2 S.
De…nition
The independence number, α (G ), is the size of the largest
independent set.
Chromatic number
De…nition
The chromatic number χ (G ) of a graph G is the minimum number of
colours for which every two adjacent vertices have a di¤erent colour.
Lemma
ω (G )
χ (G )
∆(G ) + 1.
Paths and cycles
De…nition
A path Pn = (V , E ) of length n is a graph of the form V = f1, 2, ..., n g
and E = ff1, 2g, f2, 3g, ..., fn 1, n gg.
De…nition
A cycle Cn = (V , E ) of length n is a graph of the form
V = f1, 2, ..., n g and E = ff1, 2g, f2, 3g, ..., fn 1, n g, f1, n gg.
Connectedness
De…nition
A graph is connected if there is a path between any two of its vertices.
De…nition
A maximal connected subgraph of a graph G is said to be a connected
component.
Regularity Lemma (1)
De…nition
Given a graph G = (V , E ), let X , Y
V with X \ Y = ∅. Let
kX , Y k = jffi, j g 2 E : i 2 X , j 2 Y gj. The density of the pair (X , Y )
kX ,Y k
is de…ned as d (X , Y ) = jX jjY j .
De…nition
Given a graph G = (V , E ), let A, B V with A \ B = ∅. Then (A, B )
is an e-regular pair if for all X
A and Y
B with jX j e jAj and
jY j e jB j we have jd (X , Y ) d (A, B )j e.
De…nition
Let fV0 , V1 , ..., Vk g be a partition of V . The set V0 (possibly empty) is
called an exceptional set. The partition is called an e-regular partition
if: (i) jV0 j e jV j; (ii) jV1 j = ... = jVk j; (iii) all but at most ek 2 of the
pairs Vi , Vj with 1 i < j k are e-regular.
Regularity Lemma (2)
Theorem (Szemerédi, 1978)
For every e > 0 and every integer n 1 there exists an integer N such
that every graph on at least m vertices has an e-regular partition
fV0 , V1 , ..., Vk g with n k N.
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The intuition behind Szemerédi regularity lemma is that every large
enough graph can be divided into subsets of about the same size so
that the edges between di¤erent subsets behave almost randomly.
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see J. Komlós, A. Shokoufandeh, M. Simonovits, E. Szemerédi,
LNCS, 2292 (2002), 84–112.
Ramsey numbers
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Ramsey Theory studies conditions when a combinatorial object
contains necessarily some smaller given objects.
De…nition
The Ramsey number R (k; l ), where k and l are arbitrary integers, is
de…ned as the smallest n such that for a graph G with n or more vertices
ω (G ) = k or α (G ) = l. (Ramsey’s theorem guarantees this.)
Example
In any group of six people, it is possible to …nd three persons who all
know each other or three persons who do not know each, i.e.,
2
4k .
p
R (3, 3) = 6; R (k, k ) (2k
k 1)
k
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F. P. Ramsey, Proceedings of the London Mathematical Society, 30
(1930), pp. 264–286.
Max-Flow Min-Cut
De…nition
In a digraph D = (V , E ), a vertex of in-degree 0 (resp. out-degree 0) is
called a source (resp. sink). Each arc (i, j ) has a capacity ci ,j
0. A
0
‡ow is a map f : E ! R such that: (i) f (i, j ) ci ,j ; (ii)
∑j f (i, j ) = ∑j f (j, i ), for every i 2 V .
Problem
Maximize ftot = ∑j f (1, j ) = ∑j f (j, n ), where 1 is the source and n is
the sink.
De…nition
A cut in a digraph is a set of arcs such that if they are deleted, there is
not path from the source to the sink (see edge cut). The capacity of a
cut is de…ned to be the sum of the capacities of every arc in the cut.
Theorem (Max-Flow Min-Cut)
In every digraph, the maximum ‡ow equals the minimum capacity of a
cut.
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Ford–Fulkerson, Elias–Feinstein–Shannon, 1956.
Visibility graphs
De…nition
Let X = (x1 , x2 , ..., xn ) be a sequence with xi 2 R 0 for i = 1, 2, ..., n.
Each element xi 2 X is associated to a line segment in R2 with end
points (i, 0) and (i, xi ). The visibility graph of the sequence X , denoted
by G (X ), is the graph with set of vertices f1, 2, ..., n g and an edge i j
if, for any i < k < j, we have xk < xj + (xi xj ) jj ki .
Graph widths
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Most NP-hard problems are e¢ ciently solvable for graphs with
bounded widths. Treewidth quanti…es how “close” a graph is to
being a tree.
De…nition
A tree decomposition (T , X ) of a graph G = (V , E ) S
consists of a tree
T = (I , F ) and a function X : I ! 2V such that: (i) i 2I X (i ) = V ;
(ii) for every fi, j g 2 E there is u 2 I such that fi, j g 2 X (u ); (iii)
X (i ) \ X (k ) X (j ), for every j on the path between i and k. The
width of a tree decomposition is maxi jX (i )j 1. The treewidth,
tw (G ), is the minimum width over all tree decompositions.
Random walks
Adjacency matrix
De…nition
The adjacency matrix of a graph G = (V , E ) on n vertices is de…ned as
follows:
1, if fi, j g 2 E ;
[A (G )]i ,j =
0, otherwise.
Lemma
The number of walks of length l from vertex i to vertex j in a graph G is
given by [A (G )l ]i ,j .
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Canonical labeling (lexicographic smallest adjacency matrix).
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Adjacency list.
(Adjacency) spectrum
De…nition
The spectrum of a graph G is the multiset of the eigenvalues of A (G ):
λ1 λ2
λn .
Theorem
(i) If diam (G ) = d then G has at least d + 1 distinct eigenvalues. (ii) A
(connected) graph G is bipartite i¤ λ1 = λn . (iii) χ (G ) 1 + λn /λ1 .
(iv) For any graph with m edges, ∑i λ2i = 2m. (v) (Ho¤man’s bound)
For a d-regular graph G , α (G ) n/(1 d /λn ). (vi) If G is d-regular
then (1, 1, ..., 1) is an eigenvector and d is an eigenvalue.
Theorem (Babai-Grigoryev-Mount, 1982)
If each eigenvalue has multiplicity O (1), isomorphism can be tested in
polynomial time.
Fact
If two graphs are isomorphism then they are cospectral. The converse
does not always hold. [Example]
Expansion
De…nition
An ε-expander is a graph G = (V , E ) in which, for every S V with
jS j jV j /2, the number of vertices in V S adjacent to some vertex
in S is at least ε jS j.
Theorem (Alon, 2001)
Le G be a d-regular graph. (i) If d λ2 2εd then G is an ε-expander.
(ii) If G is an ε-expander then d λ2 ε2 /5.
Example
V = Zp , p prime; fi, j g 2 E i¤ j = i + 1 and j = i 1 , for each i 2 Zp .
De…nition
A d-regular graph is said to be Ramanujan if λ2
Theorem (Friedman, 2003)
For almost every d-regular graph λ2
tends to 0 as n ! ∞).
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p
2 d
p
2 d
1.
1 + o (1) (the o (1)-term
S. Hoory, N. Linial, and A. Wigderson, Bull. Amer. Math. Soc. 43
(2006), no. 4, 439–561.
Random walks
De…nition
The transition matrix of a graph G = (V , E ) on n vertices is de…ned as
follows:
1/dG (i ) , if fi, j g 2 E ;
[T (G )]i ,j =
0,
otherwise.
The matrix T (G ) is stochastic (i.e., nonnegative with row sums 1).
Then, 1 λi (T ) 1, for every i.
De…nition
A (simple) random walk on a graph G = (V , E ) on n vertices is a
sequence of vectors fv(1 ) , v(2 ) , ...v(k ) g de…ned by the rule
T k (G ) ei = v(k ) , where ei is the i-th element of the standard basis of
Rn .
Theorem
If G is connected and χ (G ) 6= 2 then limk !∞ v(k ) π G
= 0,
TV
where π G is the (unique) stationary distribution given by
[π G ]i = dG (i ) /2 jE j.
Mixing/hitting
Theorem
The mixing rate is the rate of convergence of v(k ) towards π G and it is
determined by 1 λ2 (T ), the eigenvalues gap of the graph.
De…nition
In a random walk starting from vertex i, the hitting time Hi ,j is the
expected number of steps before vertex j.
Theorem
Let G be a (connected) graph with m edges. Then,
Hi ,j = 2m
∑
k =2
1
1
λk (T )
[λk ]2i
d (i )
[ λk ]i [ λk ]j
p
d (i )d (j )
!
,
where λk (T ) is the k-th eigenvalue of the transition matrix, [λk ]i is the
i-th component of its normalized eigenvector.
PageRank
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A digraph G = (E , V );
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Rank: p (j ) = ∑i :(i ,j )2E p (i )/dout (i ), p = (p (1), ..., p (n )).
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Wout = Aout Dout1 , where A := AT (G ) and Dout is out-degree
matrix.
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PageRank vector: pWout = p.
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Random jump: p = α n1 1 + (1 α)Wout p, with 0 < α < 1, so that
the probability of reaching every vertex is at least α/n.
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t 1.
Quick convergence: p = nα ∑t 0 (1 α)t Wout
P. Bonacich, American Journal of Sociology, 92:1170–1182, 1987;
L. Page, S. Brin, R. Motwani, and T. Winograd, TR, CS Dept.,
Stanford University, 1998; J. Kleinberg, Journal of the ACM,
48:604–632, 1999.
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Quantum walks
Grover matrix
De…nition
!
The line digraph of G = (V , E ) is a digraph D G = (V 0 , E 0 ),
(i, j ), (j, i ) 2 V 0 for each fi, j g 2 E and ((i, j ), (k, l )) 2 E 0 if j = k.
De…nition
Given a graph G = (V , E ), we denote by U (G ) the matrix de…ned as
!
follows: for al (i, j ), (k, l ) 2 V ( D G ),
[U (G )](i ,j ),(k ,l ) =
2/d (j )
0,
δi ,l ,
if j = k;
otherwise.
Fact
(i) We have U (G ) U T (G ) = I . (ii) Let [U p (G )]i ,j := 1 if
[U p (G )]i ,j > 0 and [U p (G )]i ,j := 0, otherwise. Then the spectrum of
U 3 (G ) is a complete graph invariant for regular graphs on n 13
vertices.
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Remark: "The secret of quantum algorithms is not entanglement
but minus signs." (Eddie Farhi, 11/06/2012)
D. Emms, et al., Pattern Recognition 42(9): 1988-2002 (2009).
A special case: regular graphs
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Let G be a k-regular graph on n vertices. The adjacency matrix
A(G ) can be decomposed into permutation matrices:
k
A(G ) =
∑ Pi .
i =1
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Let C be a k
k unitary matrix. Then the matrix
UC ( G ) =
k
M
Pi ( C
In )
i =1
is also unitary. In the previous case, [C ]i ,j = 2/k
C is often called the coin.
δi ,j . The matrix
Coined quantum walks
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We …x a correspondence between the vertices of a k-regular graph G
on n vertices and the elements of the standard basis
fj1i, j2i, ..., jnig for a space called vertex-space. Also, each
permutation corresponds to an element of fj1i, j2i, ..., jk ig for a
space called coin-space.
De…nition
A coined quantum walk on G starting from jk i with coin-state jl i is
de…ned by the rule
UC (G )t (jk i
jl i) = UC (G )UC (G )t
1
(jk i
jl i) = jψt i,
for any t.
De…nition
The probability of observing the walker at vertex j after t steps is given by
k
Pr[j jt] =
∑ j(hj j
i =1
hi j)jψt ij2 .
Mixing
Fact
A coined quantum walk does not mix. (In general, while random walks
are distribution-preserving, quantum walks are norm-preserving.
De…nition
Two notions of mixing: (i) Instantaneous mixing time (choose a time t
and measure); Average mixing (take the Cesaro mean)
M (1, t ) = 1s ∑si=1 Pr[j jti ].
Theorem (Moore-Russell 2001)
The mixing time of a random walk in the hypercube is Θ(n log n ). The
mixing time of a coined (with Grover coin) quantum walks in the
hypercube is (π/4)n.
Continuous-time quantum walks
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We …x a correspondence between the vertices of a graph G on n
vertices and the elements of the standard basis fj1i, j2i, ..., jn ig for
a space of dimension n.
De…nition
A continous-time quantum walk on G starting from jj i with is de…ned
by the rule
e iA (G )t jj i = jψct i,
for any t.
De…nition
The probability of observing the walker at vertex j after a walk of time t
is given by
Pr[j jt] = jhj jψct ij2 .
Mixing matrices
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De…ne a uni-stochastic matrix M (t ) := U (t ) U ( t ), where
U (t ) := e iA (G )t .
The average mixing matrix is
R
b (G ) := limT !∞ 1 T M (t )dt = ∑r Er 2 , where each Er is an
M
T 0
idempotent in the spectral decomposition of the adjacency matrix.
b (Cn ) = n 21 J + 1 I , where J is
E.g., for the n-cycle with n odd M
n
n
the all-ones matrix.
Theorem
The average mixing matrix of a graph is rational.
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We have uniform mixing at time t if U (t ) is ‡at, i.e., all entries of
U (t ) have the same absolute value.
Theorem
If uniform mixing occurs on Cn then 4jn and n is the sum of two squares.
Problem
Determine which graphs have uniform average mixing.
Applications: keywords
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quantum algorithms
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universal quantum computation
models of quantum information transport in spin systems
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models of energy transfer in large molecules
Find the missing link
De…nition
The (combinatorial) Laplacian matrix
vertices is de…ned as follows:
8
< dG ( i ) ,
1,
[L (G )]i ,j =
:
0,
of a graph G = (V , E ) on n
if i = j;
if fi, j g 2 E ;
otherwise.
Theorem
Let Kn be a complete graph on n = 4m vertices. A deleted edge from Kn
can be found with certainty by the use of O (n 1) continuous-time
quantum walks induced by the Laplacian, each one of time π/2.
A general framework (1/2)
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Let T (G ) be the transition matrix of a regular graph G on n
vertices. Then λ1 = 1 and 1 λi < 1, for i 6= 1. Let
u := π G = ( n1 , ..., n1 )T be the eigenvector of λ1 . Let jλi j 1
The number δ is called the spectral gap.
δ.
2
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T k (G ) v u
kv k2 (1 δ)2k . If δ is not too small then we
get quick convergence independently of v .
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Let us mark an ε-fraction of the vertices. Once we are close to the
uniform distribution, we have probability roughly ε of hitting a
marked vertex.
Suppose it costs S to set up an initial distribution v , it costs U to
perform one step of the random walk, and it costs C to check
whether a given vertex is marked.
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Algorithm: Check if the current vertex is marked; if not run a
random walk for roughly 1/δ steps.
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The total cost of the algorithm is S + 1ε (C + 1δ U ).
A general framework (2/2)
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A procedure based on quantum walks gives S + p1ε C + p1 U .
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The choice of the graph is important to encode the problem.
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Triangle …nding, element distinctness, etc.
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Lots of potential: e.g., graph isomorphism problem?
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[See Lecture 7 in Ronald de Wolf’s Course]
δ
Mathematical open problem
Problem
Given a graph G . Find necessary and su¢ cient conditions such that there
is a unitary matrix U for which [U ]i ,j 6= 0 if and only if [A(G )]i ,j = 1.
[I.e., obtain a matrix with the same structure of A(G ) and with max.
rank] [Open since the 1970s]