Spectral Graph Theory Lecture 2 Egenvalues of Random Graphs Danel A. Spelman November 4, 202 2. Introducton In ths lecture, we consder a random graph on n vertces n whch each edge s chosen to be n the graph wth probablty one-half, ndependently of course. We wll show that the egenvalues of the adjacency matrx of such a graph are tghtly concentrated. Curously, the adjacency matrx egenvalues are much more tghtly concentrated than the Laplacan matrx egenvalues. The adjacency matrx of such a random graph may be descrbed by choosng the values of A(, j) to be zero wth probablty /2 and wth probablty /2, subject to A(, j) = A(j, ). Of course, we fx A(, ) = 0 for all. The expectaton of every off-dagonal entry of the matrx s /2. Let M denote ths expected matrx, and observe that M = 2 A K n = 2 (J n I n ), where A Kn s the adjacency matrx of the compelete graph on n vertces, J n s the all-s matrx and I n s of course the dentty. From ths formula, we see that M has one egenvalue of (n )/2 and n egenvalues of /2. We wll show that the egenvalues of A are very close to ths. In partcular, we wll prove that A M.34 n, wth exponentally hgh probablty. So, we wll really focus on boundng the norm of A M. As A M s a symmetrc matrx, we have A M = max λ (A M) = max x T Ax x x T x. Our analyss wll focus on ths last term. Set R = A M, and let r,j = R(, j), for < j. Each r,j s a random varable that s ndependently and unformly dstrbuted n ±/2. 2.2 One Raylegh Quotent To begn, we fx any unt vector x, and consder x T Rx = <j 2r,j x ()x (j). 2-
Lecture 2: November 4, 202 2-2 Ths s a sum of ndependent random varables, and so may be proved to be tghly concentrated around ts expectaton, whch n ths case s zero. There are many types of concentraton bounds, wth the most popular beng the Chernoff and Hoeffdng bounds. In ths case we wll apply Hoeffdng s nequalty. Theorem 2.2. (Hoeffdng s Inequalty). Let a,..., a m and b,..., b m be real numbers and let X,..., X m be ndependent random varables such that X takes values between a and b. Let µ = E X ]. Then, for every t > 0, To apply ths theorem, we vew ] ( Pr X µ + t exp X,j = 2r,j x ()x (j) 2t 2 ) (b a ) 2. as our random varables. As r,j takes values n ±/2, we can set We then compute a,j = x ()x (j) and b,j = x ()x (j). (b a ) 2 = 4x () 2 x (j) 2 = 2 ( ) ( x () 2 x (j) 2 2 x () 2 <j <j j x (j) 2 ) = 2, as x s a unt vector. We thereby obtan the followng bound on x T Rx. Lemma 2.2.2. For every unt vector x, Pr R x T Rx t ] 2e t 2. Proof. The expectaton of x T Rx s 0. The preceedng argument tells us that Pr x T Rx t ] Pr x T Rx t ] + Pr x T Rx t ] Pr x T Rx t ] + Pr x T ( R)x t ] 2e t2, where we have exploted the fact that R and R are dentcally dstrbuted. 2.3 Vectors near v You mght be wonderng what good the prevous argument wll do us. We have shown that t s unlkely that the Raylegh quotent of any gven x s large. But, we have to reason about all x of unt norm.
Lecture 2: November 4, 202 2-3 Lemma 2.3.. Let R be a symmetrc matrx and let v be a unt egenvector of R whose egenvalue has absolute value R. If x s another unt vector such that v T x 3/2, then x T Rx 2 R. Proof. Let λ λ 2 λ n be the egenvalues of R and let v,..., v n be a correspondng set of orthonormal egenvectors. Assume wthout loss of generalty that λ λ n and that v = v. Expand x n the egenbass as x = c v. We know that c 3/2 and c2 x T Rx = c 2 λ c 2 λ 2 =. Ths mples that c 2 λ = λ c 2 2 c 2 = λ (2c 2 ) λ /2. We wll bound the probablty that R s large by takng Raylegh quotents wth random unt vectors. Let s examne the probablty that a random unt vector x satsfes the condtons of Lemma 2.3.. Lemma 2.3.2. Let v be an arbtrary unt vector, and let x be a random unt vector. Then, Pr v T x ] 3/2 πn2 n Proof. Let B n denote the unt ball n IR n, and let C denote the cap on the surface of B n contanng all vectors x such that v T x 3/2. We need to lower bound the rato of the surface area of the cap C to the surface area of B n. Recall that the surface area of B n s where I recall that for postve ntegers n nπ n/2 Γ( n 2 + ), Γ(n) = (n )!, and that Γ(x) s an ncreasng functon for real x. Now, consder the (n )-dmensonal hypersphere whose boundary s the boundary of the cap C. As the cap C les above ths hypersphere, the (n )-dmensonal volume of ths hypersphere s a
Lecture 2: November 4, 202 2-4 lower bound on the surface area of the cap C. Recall that the volume of a sphere n IR n of radus r s r n π n/2 Γ( n 2 + ). In our case, the radus of the hypersphere s r = sn(acos 3/2) = /2. So, the rato of the (n )-dmensonal volume of the hypersphere to the surface area of B n s at least rn π(n )/2 Γ( n 2 +) = rn Γ( n 2 + ) nπ n/2 πn Γ( n Γ( n 2 +) 2 + ) rn πn πn2 n. 2.4 The Probablstc Argument I m gong to do the followng argument very slowly, because t s both very powerful and very subtle. Theorem 2.4.. Let R be a symmetrc matrx wth zero dagonal and off-dagonal entres unformly chose from ±/2. Then, Pr R t] πn2 n e t2 /4. Proof. Let R be a fxed symmetrc matrx. By applyng Lemma 2.3.2 to any egenvector of R whose egenvalue has maxmal absolute value, we fnd Pr x x T Rx 2 R ] πn2 n. Thus, for a random R we fnd Pr R,x R t and x T Rx ] 2 R = Pr R R t] Pr R,x x T Rx ] 2 R R t Pr R R t] πn2 n. On the other hand, Pr R,x R t and x T Rx 2 R ] Pr R,x R t and x T Rx t/2 ] Pr R,x x T Rx t/2 ] = E x PrR x T Rx t/2 ]] 2e (t/2)2,
Lecture 2: November 4, 202 2-5 where the last nequalty follows from Lemma 2.2.2. Combnng these nequaltes, we obtan whch mples the clamed result. Pr R R t] πn2 n e (t/2)2, The probablty n Theorem 2.4. becomes small once e t2 /4 exceeds πn2 n. As n grows large, ths happens for t > 2 ln 2 n (5/3) n. Ths s a lttle worse than the bound that I clamed at the begnnng of the lecture. The reason s that I have optmzed ths proof so that all the numbers that appear are nce. To get the tghtest bound possble, we should choose an nner product other than 3/2 n Lemma 2.3., and then propagatng the change throught the proof. If we replace 3/2 by 0.957, we obtan an upper bound on the norm of.34 n. It s known that the norm of R s unlkely to be much more than n. Ths s proved by Füred and Komlós FK8] and Vu Vu07], usng a very dfferent technque. The dea behnd these papers s to consder Tr ( R k) for a hgh power of k. They show that the expectaton of ths varable s unlkely to be large, and explot the fact that R ( ( Tr R k)) /k. 2.5 Queston Is there a varant of ths proof that yelds good bounds when the entres of A have a lower probablty of beng? For example, consder the case n whch there s a number p < 0.5 such that each entry of A s wth probablty p. To get a good proof n ths regme, one needs a concentraton nequalty that s stronger than Hoeffdng s. References FK8] Z. Füred and J. Komlós. The egenvalues of random symmetrc matrces. Combnatorca, (3):233 24, 98. Vu07] Van Vu. Spectral norm of random matrces. Combnatorca, 27(6):72 736, 2007.