KChung Author address University of Pennsylvania Philadelphia Pennsylvania 19104 Email address chungmathupennedu brPage 2br Contents Chapter 1 Eigenvalues and the Laplacian of a graph 1 11 Introduction 1 12 The Laplacian and eigenvalue ID: 29279 Download Pdf

KChung Author address University of Pennsylvania Philadelphia Pennsylvania 19104 Email address chungmathupennedu brPage 2br Contents Chapter 1 Eigenvalues and the Laplacian of a graph 1 11 Introduction 1 12 The Laplacian and eigenvalue

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Lectures on Spectral Graph Theory FanR.K.Chung Author address: University of Pennsylvania, Philadelphia, Pennsylvania 19104 E-mail address chung@math.upenn.edu

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Contents Chapter 1. Eigenvalues and the Laplacian of a graph 1 1.1. Introduction 1 1.2. The Laplacian and eigenvalues 2 1.3. Basic facts about the spectrum of a graph 6 1.4. Eigenvalues of weighted graphs 11 1.5. Eigenvalues and random walks 14 Chapter 2. Isoperimetric problems 23 2.1. History 23 2.2. The Cheeger constant of a graph 24 2.3. The edge expansion of a graph 25 2.4. The vertex expansion of a

graph 2( 2.5. A characterization of the Cheeger constant 32 2.6. Isoperimetric inequalities for cartesian products 36 Chapter 3. +iameters and eigenvalues 43 3.1. The diameter of a graph 43 3.2. Eigenvalues and distances between two subsets 45 3.3. Eigenvalues and distances among many subsets 4( 3.4. Eigenvalue upper bounds for manifolds 5, Chapter 4. -aths. ﬂows. and routing 5( 4.1. -aths and sets of paths 5( iii

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iv CONTENTS 4.2. 0lows and Cheeger constants 6, 4.3. Eigenvalues and routes with small congestion 62 4.4. 1outing in graphs 64 4.5. Comparison theorems 62

Chapter 5. Eigenvalues and quasi3randomness 43 5.1. 5uasi3randomness 43 5.2. The discrepancy property 45 5.3. The deviation of a graph 21 5.4. 5uasi3random graphs 25 Bibliography (1

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CHA-TE1 1 Eigenvalues and the Laplacian of a graph 1.1. Introduction 6pectral graph theory has a long history. In the early days. matrix theory and linear algebra were used to analyze adjacency matrices of graphs. Algebraic methodsareespeciallye8ective intreatinggraphswhich areregularand symmetric. 6ometimes.certaineigenvalueshavebeenreferredtoasthe9algebraicconnectivity: of a graph [ 126 ]. There

is a large literature on algebraic aspects of spectral graph theory.welldocumentedinseveralsurveysandbooks. suchasBiggs[ 25 ]. Cvetkovi= c. +oob and 6achs [ 90, 91 ]. and 6eidel [ 224 ]. In the past ten years. many developments in spectral graph theory have often had a geometric ﬂavor. 0or example. the explicit constructions of expander graphs. duetoLubotzky3-hillips36arnak[ 193 ] andMargulis[ 195 ]. arebasedoneigenvalues and isoperimetric properties of graphs. The discrete analogue of the Cheeger in3 equality has been heavily utilized in the study of random walks and rapidly mixing

Markov chains [ 224 ]. New spectral techniques have emerged and they are powerful and well3suited for dealing with general graphs. In a way. spectral graph theory has entered a new era. Just as astronomers study stellar spectra to determine the make3up of distant stars. one of the main goals in graph theory is to deduce the principal properties and structure of a graph from its graph spectrum Aor from a short list of easily computable invariantsB. The spectral approach for general graphs is a step in this direction. Ce will see that eigenvalues are closely related to almost all major

invariantsofagraph.linkingoneextremalpropertytoanother. Thereisnoquestion that eigenvalues play a central role in our fundamental understanding of graphs. The study of grapheigenvaluesrealizesincreasinglyrichconnectionswith many other areas of mathematics. A particularly important development is the interac3 tion between spectral graph theory and di8erential geometry. There is an interest3 ing analogy between spectral 1iemannian geometry and spectral graphtheory. The concepts and methods of spectral geometry bring useful tools and crucial insights to the study of graph eigenvalues. which in

turn lead to new directions and results in spectral geometry. Algebraic spectral methods are also very useful. especially for extremal examples and constructions. In this book. we take a broad approach with emphasis on the geometric aspects of graph eigenvalues. while including the algebraic aspects as well. The reader is not required to have special background in geometry. since this book is almost entirely graph3theoretic.

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2 1. EIGENVALUES AND THE LAPLACIAN OF A GRAPH 0rom the start. spectral graph theory has had applications to chemistry [ 27 ]. Eigenvalues were associated

with the stability of molecules. Also. graph spectra arise naturally in various problems of theoretical physics and quantum mechanics. for example. in minimizing energies of Hamiltonian systems. The recent progress on expander graphs and eigenvalues was initiated by problems in communication networks. The development of rapidly mixing Markov chains has intertwined with advances in randomized approximation algorithms. Applications of graph eigen3 values occur in numerous areas and in di8erent guises. However. the underlying mathematics of spectral graph theory through all its connections to the

pure and applied. the continuous and discrete. can be viewed as a single uniDed subject. It is this aspect that we intend to cover in this book. 1.2. The Laplacian and eigenvalues BeforewestarttodeDneeigenvalues.someexplanationsareinorder. Theeigen3 values we consider throughout this book are not exactly the same as those in Biggs 25 ]orCvetkovi= c. +oob and 6achs [ 90 ]. Basically. the eigenvalues are deDned here in a general and 9normalized: form. Although this might look a little complicated at Drst. our eigenvalues relate well to other graph invariants for general graphs in a way that

other deDnitions Asuch as the eigenvalues of adjacency matricesB often fail to do. The advantages of this deDnition are perhaps due to the fact that it is consistent with the eigenvalues in spectral geometry and in stochastic processes. Many results which were only known for regular graphs can be generalized to all graphs. Consequently. this provides a coherent treatment for a general graph. 0or deDnitions and standard graph3theoretic terminology. the reader is referred to [ 31 ]. In a graph .let denote the degree of the vertex . Ce Drst deDne the Laplacian for graphs without loops and

multiple edges Athe general weighted case with loops will be treated in 6ection 1.4B. To begin. we consider the matrix deDned as follows: u,v BE if 1if and are adjacent. ,otherwise. Let denote the diagonal matrix with the A v,v B3th entry having value .The Laplacian of is deDned to be the matrix u,v BE 1if and E, if and are adjacent. ,otherwise. Ce can write LT with the convention v,v BE,for E,. Cesay is an isolated vertex if E ,. A graph is said to be nontrivial if it contains at least one edge.

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1.2. THE LAPLACIAN AND EIGENVALUES 3 canbeviewedasanoperatoronthespaceoffunctions

which satisDes BE Chen is 3regular. it is easy to see that A, where is the adjacency matrix of .A i. e.. x,y BE1if is adjacent to .and ,otherwise.Band is an identity matrix. All matrices here are where is the number of vertices in 0or a general graph. we have LT AT Ce note that can be written as SS where is the matrix whose rows are indexed by the vertices and whose columns are indexed by the edges of such that each column corresponding to an edge u,v has an entry 1 in the row corresponding to .anentry in the row corresponding to . and has zero entries elsewhere. AAs it turns out. the choice

of signs can be arbitrary as long as one is positive and the other is negative.B Also. denotes the transpose of 0or readers who are familiar with terminology in homology theory. we remark that can be viewed as a 9boundary operator: mapping 913chains: deDned on edges Adenoted by B of a graph to 9,3chains: deDned on vertices Adenoted by B. Then. is the corresponding 9coboundary operator: and we have 6ince is symmetric. its eigenvalues are all real and non3negative. Ce can use the variational characterizations of those eigenvalues in terms of the 1ayleigh quotient of Asee. e.g. [ 162 ]B. Let

denote an arbitrary function which assigns to each vertex of arealvalue B. Ce can view as a column vector. Then g, g,g g,T LT g,g f,Lf f,T BB A1.1B

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4 1. EIGENVALUES AND THE LAPLACIAN OF A GRAPH where and denotes the sum over all unordered pairs u,v for which and areadjacent. Here f,g B denotesthestandardinnerproduct in .Thesum BB is sometimes called the Dirichlet sum of and the ratio on the left3hand side of A1.1B is often called the Rayleigh quotient .ACe notethatwecanalsousetheinnerproduct f,g Bforcomplex3valued functions.B 0rom equation A1.1B. we see that all eigenvalues

are non3negative. In fact. we can easily deduce from equation A1.1B that , is an eigenvalue of .Cedenotethe eigenvalues of by , E . The set of the Fs is usually called the spectrum of Aor the spectrum of the associated graph .B Let denote the constant function which assumes the value 1 on each vertex. Then is an eigenfunction of with eigenvalue ,. 0urthermore. Einf BB A1.2B Thecorrespondingeigenfunctionis asinA1.1B. Itissometimesconvenient to consider the nontrivial function achieving A1.2B. in which case we call harmonic eigenfunction of The above formulation for corresponds in a natural way

to the eigenvalues of the Laplace3Beltrami operator for 1iemannian manifolds: Einf | where ranges over functions satisfying E, Ce remark that the corresponding measure here for each edge is 1 although in the generalcasefor weightedgraphs the measure for anedge is associatedwith the edge weight Asee 6ection 1.4.B The measure for each vertex is the degree of the vertex. A more general notion of vertex weights will be considered in 6ection 2.5.

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1.2. THE LAPLACIAN AND EIGENVALUES 5 Ce note that A1.2B has several di8erent formulations: Einf sup BB A1.3B Einf BB A1.4B where vol and

vol denotes the volume of the graph .givenby vol By substituting for and using the fact that =1 i for =1 /N . we have the following expression Awhich generalizes the one in 126 ]B: Evol inf BB u,v BB A1.5B where u,v denotes the sum over all unordered pairs of vertices u,v in .Cecan characterize the other eigenvalues of in terms of the 1ayleigh quotient. The largest eigenvalue satisDes: Esup BB A1.6B 0or a general .wehave Einf sup BB BB A1.4B Einf TP BB A1.2B

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6 1. EIGENVALUES AND THE LAPLACIAN OF A GRAPH where is the subspace generated by the harmonic eigenfunctions

corresponding to .for 1. The di8erent formulations for eigenvalues given above are useful in di8erent settings and they will be used in later chapters. Here are some examples of special graphs and their eigenvalues. Example 1.1. 0or the complete graph on vertices. the eigenvalues are ,and n/ 1B Awith multiplicity 1B. Example 1.2. 0or the complete bipartite graph m,n on vertices. the eigenvalues are ,. 1 Awith multiplicity 2B. and 2. Example 1.3. 0or the star on vertices. the eigenvalues are , 1Awith multiplicity 2B. and 2. Example 1.4. 0or the path on vertices. the eigenvalues are 1 cos

πk for E, ,n 1. Example 1.5. 0or the cycle on vertices. the eigenvalues are 1 cos πk for E, ,n 1. Example 1.6. 0or the 3cube on 2 vertices. the eigenvalues are Awith multiplicity Bfor E, ,n More examples can be found in Chapter 6on explicit constructions. 1.3. Basic facts about the spectrum of a graph 1oughly speaking. half of the main problems of spectral theory lie in deriving bounds on the distributions of eigenvalues. The other half concern the impact and consequences of the eigenvalue bounds as well as their applications. In this section. we start with a few basic facts about

eigenvalues. 6ome simple upper bounds and lower bounds are stated. 0or example. we will see that the eigenvalues of any graph lie between , and 2. The problem of narrowing the range of the eigenvalues for special classes of graphs o8ers an open3ended challenge. Numerous questions can be asked either in terms of other graph invariants or under further assumptions imposed on the graphs. 6ome of these will be discussed in subsequent chapters. Lemma 1.7. For a graph on vertices, we have (i): with equality holding if and only if has no isolated vertices. (ii): For with equality holding if and only

if is the complete graph on vertices. Also, for a graph without isolated vertices, we have

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1.3. BASIC FACTS ABOUT THE SPECTRUM OF A GRAPH 7 (iii): For a graph which is not a complete graph, we have (iv): If is connected, then .If E, and +1 E, ,then has exactly H1 connected components. (v): For all , we have with E2 if and only if a connected component of is bipartite and nontrivial. (vi): The spectrum of a graph is the union of the spectra of its connected com- ponents. Proof. AiB follows from considering the trace of The inequalities in AiiB follow from AiB and E,. 6uppose

contains two nonadjacent vertices and . and consider BE if if ,if a,b. AiiiB then follows from A1.2B. If is connected. the eigenvalue , has multiplicity 1 since any harmonic eigen3 function with eigenvalue,assumesthe samevalueateachvertex. Thus. AivBfollows from the fact that the union of two disjoint graphs has as its spectrum the union of the spectra of the original graphs. AvB follows from equation A1.6B and the fact that BB 2A BH BB Therefore sup BB Equality holds for 1when BE B for every edge x,y in Therefore. since E,. has a bipartite connected component. In the other hand. if has a

connected component which is bipartite. we can choose the function so as to make E2. AviB follows from the deDnition. 0or bipartite graphs. the following slightly stronger result holds: Lemma 1.8. The following statements are equivalent: (i): is bipartite. (ii): has H1 connected components and E2 for (iii): For each ,thevalue is also an eigenvalue of

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8 1. EIGENVALUES AND THE LAPLACIAN OF A GRAPH Proof. It suJces to considera connected graph. 6uppose is bipartite graph with vertex set consisting of two parts and . 0or any harmonic eigenfunction with eigenvalue . we consider

the function BE Bif A, Bif B. It is easy to check that is a harmonic eigenfunction with eigenvalue 2 0or a connected graph. we can immediately improve the lower bound of in Lemma 1.4. 0or two vertices and .the distance between and is the number of edges in a shortest path joining and .The diameter of a graph is the maximum distance between any two vertices of . Here we will give a simple eigenvalue lower bound in terms of the diameter of a graph. More discussion on the relationship between eigenvalues and diameter will be given in Chapter 3. Lemma 1.9. For a connected graph with diameter , we

have vol Proof. 6uppose is a harmonic eigenfunction achieving in A1 2B. Let denote a vertex with Emax .6ince B E ,. there exists a vertex satisfying ,. Let denote a shortest path in joining and Then by A1.2B we have BB x,y } BB vol Gf BB vol Gf vol by using the Cauchy36chwarz inequality. Lemma 1.10. Let denote a harmonic eigenfunction achieving in (1.2). Then, for any vertex , we have BB E

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1.3. BASIC FACTS ABOUT THE SPECTRUM OF A GRAPH 9 Proof. Ce use a variational argument. 0or a Dxed .weconsider such that BE BH if vol otherwise. Ce have x,y BB x,y BB BB BB vol H2 %f vol x,y

BB BB BB vol H2 %f BH %f vol since E,.and BB E ,. The deDnition in A1.2B implies that x,y BB x,y BB If we consider what happens to the 1ayleigh quotient for as ,fromabove. or from below. we can conclude that BB E and the Lemma is proved.

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10 1. EIGENVALUES AND THE LAPLACIAN OF A GRAPH Ine can also prove the statement in Lemma 1.1, by recalling that where .Then Lf BA BE f, and examining the entries gives the desired result. Cith a little linear algebra. we can improve the bounds on eigenvalues in terms of the degrees of the vertices. Ce consider the trace of A −L .Cehave

Tr −L A1 1HA 1B A1.(B where Emax =0 In the other hand. Tr −L Tr AT AT A1.1,B x,y x,y y,x where is the adjacency matrix. 0rom this. we immediately deduce Lemma 1.11. For a -regular graph on vertices, we have max =0 | 1B A1.11B This follows from the fact that max =0 tr −L 1B Let denote the harmonic mean of the Fs. i.e.. It is tempting to consider generalizingA1.11B with replaced by . This. however. is not true as shown by the following example due to Elizabeth Cilmer. Example 1.12. Consider the 3petal graphon E2 H1vertices. ,v with edges ,v and ,v .for 1. This graph has

eigenvalues 2 Awith multiplicity 1B. and 3 2 Awith multiplicity H 1B. 6o we have max =0 E1 2. However. 1B as

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1.4. EIGENVALUES OF WEIGHTED GRAPHS 11 6till. for a general graph. we can use the fact that 1H λ. A1.12B Combining A1.(B. A1.1,B and A1.12B. we obtain the following: Lemma 1.13. For a graph on vertices, Emax =0 satisﬁes 1HA 1B A1 A1H BA 1BB where denotes the average degree of There are relatively easy ways to improve the upper bound for .0romthe characterizationintheprecedingsection. wecanchooseanyfunction and its 1ayleigh quotient will serve as an upper

bound for . Here we describe an upper bound for Asee [ 204 ]B. Lemma 1.14. Let be a graph with diameter ,andlet denote the maximum degree of .Then Ine way to bound eigenvalues from above is to consider 9contracting: the graph intoaweightedgraph Awhich will be deDned in the next sectionB. Then the eigenvalues of can be upper3bounded by the eigenvalues of or by various upper bounds on them. which might be easier to obtain. Ce remark that the proof of Lemma 1.14 proceeds by basically contracting the graph into a weighted path. Ce will prove Lemma 1.14 in the next section. Ce note that Lemma 1.14

gives a proof Asee [ ]B that for any Dxed and for any inDnite family of regular graphs with degree limsup This bound is the best possible since it is sharp for the 1amanujan graphs Awhich will be discussed in Chapter ?? B. Ce note that the cleaner version of /k is not true for certain graphs Ae.g.. 43cycles or complete bipar3 tite graphsB. This example also illustrates that the assumption in Lemma 1.14 concerning 4 is essential. 1.4. Eigenvalues of weighted graphs Before deDning weighted graphs. we will say a few words about two di8erent approaches for giving deDnitions. Ce could have started

from the very beginning with weighted graphs. from which simple graphs arise as a special case in which the weights are , or 1. However. the unique characteristics and special strength of

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12 1. EIGENVALUES AND THE LAPLACIAN OF A GRAPH graph theory is its ability to deal with the 3problems arising in many natural situations. The clean formulation of a simple graph has conceptual advantages. 0urthermore. as we shall see. all deDnitions and subsequent theorems for simple graphscan usuallybe easilycarriedoutforweightedgraphs. Aweightedundirected graph Apossibly with loopsB has

associated with it a weight function satisfying u,v BE v,u and u,v Ce note that if u,v B.then u,v B E ,. Unweighted graphs are just the special case where all the weights are , or 1. In the present context. the degree of a vertex is deDned to be: u,v vol Ce generalize the deDnitions of previous sections. so that u,v BE v,v Bif u,v Bif and are adjacent. ,otherwise. In particular. for a function .wehave Lf BE BB x,y Let denote the diagonal matrix with the A v,v B3th entry having value .The Laplacian of is deDned to be LT In other words. we have u,v BE v,v if .and E,. u,v if and are adjacent.

,otherwise.

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1.4. EIGENVALUES OF WEIGHTED GRAPHS 13 Ce can still use the same characterizations for the eigenvalues of the generalized versions of . 0or example. :E Einf g, g, g A1.13B Einf =0 Lf Einf =0 BB x,y A contraction of a graph is formed by identifying two distinct vertices. say and .intoasinglevertex . The weights of edges incident to are deDned as follows: x,v BE x,u BH x,v ,v BE u,u BH v,v BH2 u,v Lemma 1.15. If is formed by contractions from a graph ,then The proof follows from the fact that an eigenfunction which achieves for can be lifted to a function deDned on

B such that all vertices in that contract to the same vertex in share the same value. Now we return to Lemma 1.14. KETCHEDPROOFOFLEMMA 1.14: Let and denote two vertices that are at distance H2in . Ce contract into a path with 2 H 2 edges. with vertices ,x ,...x ,z,y ,...,y ,y ,y such that vertices at distance from u, . are contracted to . and vertices at distance from ., . are contracted to . The remaining vertices are contracted to . To establish an upper bound for . it is enough to choose a suitable function . deDned as follows: BE 1B i/ BE 1B j/ BE, where the constants and are chosen so

that E,

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14 1. EIGENVALUES AND THE LAPLACIAN OF A GRAPH It can be checked that the 1ayleigh quotient satisDes BB u,v H1 H1 since the ratio is maximized when ,x +1 BE 1B ,y +1 B. This completes the proof of the lemma. 1.5. Eigenvalues and random walks In a graph . a walk is just a sequence of vertices A ,v ,v Bwith ,v } B for all 1 . A random walk is determined by the transition probabilities u,v BE Prob +1 B. which are independent of . Clearly. for each vertex u,v BE1 0or any initial distribution with B E 1. the distribution after steps is just fP Ai.e.. a matrix

multiplication with viewed as a row vector where is the matrix of transition probabilitiesB. The random walk is said to be ergodic if there is a unique stationary distribution B satisfying lim fP BE It is easy to see that necessary conditions for the ergodicity of are AiB irre- ducibility i.e. . for any u,v .thereexistssome such that u,v , AiiB aperiodicity i.e. . g.c.d. u,v E 1. As it turns out. these are also suJcient conditions. A major problem of interest is to determine the number of steps required for to be close to its stationary distribution. given an arbitrary initial distribution. Ce

say a random walk is reversible if u,v BE v,u An alternative description for a reversiblerandom walk can be given by considering a weighted connected graph with edge weights satisfying u,v BE v,u BE v,u // where can be any constant chosen for the purpose of simplifying the values. A0or example. we can take to be the average of v,u BoverallA v,u Bwith v,u E ,. so that the values for v,u B are either , or 1 for a simple graph.B The random walk on a weighted graph has as its transition probabilities u,v BE u,v where u,z B is the AweightedB degree of . The two conditions for ergodicity are

equivalent to the conditions that the graph be AiB connected and AiiB non3bipartite. 0rom Lemma 1.4. we see that AiB is equivalent to ,and

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1.5. EIGENVALUES AND RANDOM WALKS 15 AiiB implies 2. As we will see later in A1.15B. together AiB and AiiB deduce ergodicity. Ce remind the reader that an unweighted graph has u,v B equal to either , or 1. The usual random walk on an unweighted graph has transition probability /d of moving from a vertex to any one of its neighbors. The transition matrix then satisDes u,v BE /d if and are adjacent. ,otherwise. In other words. fP BE for any

It is easy to check that −L where is the adjacency matrix. In a random walk with an associated weighted connected graph .thetransi3 tion matrix satisDes TP where is the vector with all coordinates 1. Therefore the stationary distribution is exactly T/ vol G, Ce want to show that when is large enough. for any initial distribution fP converges to the stationary distribution. 0irst we consider convergencein the Aor EuclideanB norm. 6uppose we write fT where denotes the orthonormal eigenfunction associated with 1ecall that vol and denotes the 3norm. so fT vol since f, E1.Cethenhave fP fP T/

vol fP fT −L =0 A1 A1 max min s max min A1.14B

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16 1. EIGENVALUES AND THE LAPLACIAN OF A GRAPH where if 1 otherwise. 6o. after / logAmax /% min B steps. the distance between fP and its stationary distribution is at most Although occurs in the above upper bound for the distance between the stationary distribution and the 3step distribution. in fact. only is crucial in the following sense. Note that is either or 2 . 6uppose the latter holds. i.e., . Ce can consider a modiDed random walk. called the lazy walk. on the graph formed by adding a loop of weight to each vertex

.The new graph has Laplacian eigenvalues 1. which follows from equation A1.13B. Therefore. and the convergence bound in distance in A1.14B for the modiDed random walk becomes / logA max min In general. suppose a weighted graph with edge weights u,v B has eigenvalues with . Ce can then modify the weights by choosing. for some constant u,v BE v,v BH /d if u,v Botherwise. A1.15B The resulting weighted graph has eigenvalues 1H where Then we have 1E 6ince 2 and we have A2H 3for 1. In particular we set Therefore the modiDed random walk corresponding to the weight function has an improved bound for

the convergence rate in distance: log max min

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1.5. EIGENVALUES AND RANDOM WALKS 17 Ce remark that for many applications in sampling. the convergence in distance seems to be too weak since it does not require convergence at each vertex. There are several stronger notions of distance several of which we will mention. A strong notion of convergence that is often used is measured by the relative pointwise distance Asee [ 224 ]B: After steps. the relative pointwise distance Ar.p.d.B of to the stationary distribution Bisgivenby MA BEmax x,y y,x Let denote the characteristic

function of deDned by: BE 1if x, ,otherwise. 6uppose where Fs denote the eigenfunction of the Laplacian of the weighted graph asso3 ciated with the random walk. In particular. vol vol Let denote the transpose of .Cehave MA BEmax x,y Emax x,y −L max x,y =0 A1 vol max x,y =0 vol max x,y vol vol min x,y (1 vol min

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18 1. EIGENVALUES AND THE LAPLACIAN OF A GRAPH where Emax =0 .6oifwechoose such that log vol min then. after steps. we have MA Chen 1 . we can improve the above bound by using a lazy walk as described in A1.15B. The proof is almost identical to the above

calculation except for using the Laplacian of the modiDed weighted graph associated with the lazy walk. This can be summarized by the following theorem: Theorem 1.16. For a weighted graph , we can choose a modiﬁed random walk so that the relative pairwise distance MA is bounded above by: MA t vol min exp t (2+ vol min where if and E2 otherwise. Corollary 1.17. For a weighted graph , we can choose a modiﬁed random walk so that we have MA if log vol min where if and E2 otherwise. Ce remark that for any initial distribution with f, E1and ,. we have. for any fP y,x BMA MA Another

notion of distance for measuring convergence is the so3called total variation distance . which is just half of the distance: TV BE max max y,x BB max y,x

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1.5. EIGENVALUES AND RANDOM WALKS 19 The total variation distance is bounded above by the relative pointwise distance. since TV BE max vol vol max y,x BB max vol vol BMA MA Therefore. any convergence bound using relative pointwise distance implies the same convergencebound using total variationdistance. There is yet another notion of distance. sometimes called squared distance . denoted by M B and deDned by: BE max y,x BB

max y,x E2M TV using the Cauchy36chwarzinequality. M B is also dominated by the relativepoint3 wise distance Awhich we will mainly use in this bookB. BE max x,y BB max AMA BB BB MA Ce note that x,y BB AA −L where denotes the projection onto the eigenfunction denotes the 3th orthonormal eigenfunction of and denotes the characteristic function of 6ince

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20 1. EIGENVALUES AND THE LAPLACIAN OF A GRAPH we have AA −L A1.16B BAA −L BA =0 BA1 =0 BA1 =0 A1 Equality in A1.16B holds if. for example. is vertex3transitive. i.e.. there is an automorphism mapping to for any

two vertices in . Afor more discussions. see Chapter 4 on symmetrical graphsB. Therefore. we conclude Theorem 1.18. Suppose is a vertex transitive graph. Then a random walk after steps converges to the uniform distribution under total variation distance or -squared distance in a number of steps bounded by the sum of A1 ,where ranges over the non-trivial eigenvalues of the Laplacian: TV BE =0 A1 A1.14B The above theorem is often derived from the Plancherel formula .Herewehave employed a direct proof. Ce remark that for some graphs which are not vertex3 transitive. a somewhat weaker version of

A1.14B can still be used with additional work Asee [ 81 ] and the remarks in 6ection 4.6B. Here we will use Theorem 1.12 to consider random walks on an 3cube. Example 1.19. 0or the 3cube . our AlazyB random walk Aas deDned in A1.15BB converges to the uniform distribution under the total variation distance. as estimated as follows: 0rom Example A1.6B. the eigenvalues of the are 2 k/n of multiplicity for E, ,n . The adjusted eigenvalues for the weighted graph corresponding to the lazy walk are E2 BE n/ H1B By using Theorem 1.12 Aalso see [ 104 ]B. we have TV =1 A1 H1 =1 log ks +1 if log /n. Ce

can also compute the rate of convergence of the lazy walk under the rela3 tive pointwise distance. 6uppose we denote vertices of by subsets of an 3set

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1.5. EIGENVALUES AND RANDOM WALKS 21 ,n . The orthonormal eigenfunctions are for ⊂{ ,n where BE 1B n/ for any ⊂{ ,n . 0or a vertex indexed by the subset . the characteristic function is denoted by BE 1if ,otherwise. Clearly. 1B n/ Therefore. X,Y ≤| A1 H1 =1 A1 H1 This implies MA BE =1 A1 H1 =1 log ks +1 if log /n. 6o. the rate of convergence under relative pointwise distance is about twice that under the total

variation distance for In general. M TV BandMA B can be quite di8erent [ 81 ]. Nevertheless. a convergence lower bound for any of these notions of distance Aand the 3normB is . This we will leave as an exercise. Ce remark that Aldous [ ]hasshownthat if M TV .then y,x B for all vertices .where depends only on Notes 0oraninduced subgraphofagraph. wecan deDne the Laplacianwith boundary conditions. Ce will leave the deDnitions for eigenvalues with Neumann boundary conditions and +irichlet boundary conditions for Chapter ??

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22 1. EIGENVALUES AND THE LAPLACIAN OF A GRAPH The

Laplacian for a directed graph is also very interesting. The Laplacian for a hypergraph has very rich structures. However. in this book we mainly focus on the Laplacian of a graph since the theory on these generalizations and extensions is still being developed. In some cases. the factor log vol min in the upper bound for MA Bcanbefurther reduced. 1ecently. -. +iaconis and L. 6alo83Coste [ 100 ] introduced a discrete ver3 sion of the logarithmic 6obolev inequalities which can reduce this factor further for certaingraphs Afor M BB. In Chapter 12. we will discuss some advanced techniques for

further bounding the convergence rate under the relative pointwise distance.

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