On the Spectra of General Random Graphs

Transcription

1 On the Spectra of General Random Graphs Fan Chung Mary Radcliffe University of California, San Diego La Jolla, CA Abstract We consider random graphs such that each edge is determined by an independent random variable, where the probability of each edge is not assumed to be equal. We use a Chernoff inequality for matrices to show that the eigenvalues of the adjacency matrix and the normalized Laplacian of such a random graph can be approximated by those of the weighted expectation graph, with error bounds dependent upon the minimum and maximum expected degrees. In particular, we use these results to bound the spectra of random graphs with given expected degree sequences, including random power law graphs. Introduction The spectra of random matrices and random graphs have been extensively studied in the literature see, for example, [3], [4], [6], [8], [3]. We here focus on matrices with entries as independent random variables. Throughout, we will consider G to be a random graph, where prv i v j = p ij, and each edge independent of each other edge. For random graphs with such general distributions, we derive several bounds for the spectrum of the corresponding adjacency matrix and normalized Laplacian matrix complete definitions and notation are in section 2. Eigenvalues of the adjacency matrix have many applications in graph theory, such as describing certain topological features of a graph, such as connectivity and enumerating the occurrences of subgraphs [6], [3]. Eigenvalues of the Laplacian matrix provide information about diffusion, and have many applications in studying random walks on graphs and approximation algorithms [6]. Research supported in part by ONR MURI N , and AFOR s complex networks program.

2 Before we proceed to examine eigenvalues of random graphs with given expected degree sequences and random power law graphs, we will first prove the following two general theorems. Previously, Oliveira [8] considered the same problem of approximating the spectra of the adjacency matrix and the Laplacian of random graphs. The following two theorems improve the results in [8] since the assumptions in our theorems are weaker. In addition, we improve the bound for the eigenvalues of the adjacency matrix by a factor of 2, and we improve those for the Laplacian by a factor of 7/ 3. Theorem. Let G be a random graph, where prv i v j = p ij, and each edge is independent of each other edge. Let A be the adjacency matrix of G, so A ij = if v i v j and 0 otherwise, and Ā = EA, so Āij = p ij. Let denote the maximum expected degree of G. Let ɛ > 0, and suppose that for n sufficiently large, > 4 9 ln2n/ɛ. Then with probability at least ɛ, for n sufficiently large, the eigenvalues of A and Ā satisfy for all i n. λ i A λ i Ā 4 ln2n/ɛ Theorem 2. Let G be a random graph, where prv i v j = p ij, and each edge is independent of each other edge. Let A be the adjacency matrix of G, as in Theorem. Let D be the diagonal matrix with D ii = degv i, and D = ED. Let δ be the minimum expected degree of G, and L = I D /2 AD /2 the normalized Laplacian matrix for G. Choose ɛ > 0. Then there exists a constant k = kɛ such that if δ > k ln n, then with probability at least ɛ, the eigenvalues of L and L satisfy 3 ln4n/ɛ λ j L λ j L 2 δ for all j n, where L = I D /2 Ā D /2. Here ln denotes the natural logarithm. We note that in these two theorems, the bound is simultaneously true for all eigenvalues with probability at least ɛ. As an example, we apply these results to the Gw model, first introduced in [7], which produces a random graph with a specified expected degree sequence w = w, w 2,..., w n. The spectrum of the adjacency matrix of this model has been studied in [9]. In that paper, it is proven that if m = w max is the maximum expected degree, then d 2m 2 ρ ln n λ max A d + 6 m ln n d + ln n + 3 m ln n, P w 2 i P wi where ρ = w i, and d = Theorem, we prove the following: is the second-order average degree. Using 2

3 Theorem 3. For the random graph Gw, if the maximum expected degree m satisfies m > 8 9 ln 2n, then with probability at least /n = o, we have The largest eigenvalue, λ max A, of the adjacency matrix of Gw satisfies d 8m ln 2n λ max A d + 8m ln 2n For all eigenvalues λ i A < λ max A, we have λ i A 8m ln 2n In particular, if d m ln n, then λ A = + o d a.a.s.. While the asymptotics in Theorem 3 for λ A are the same as those in, the bounds in Theorem 3 are a significant improvement upon these. Moreover, [9] does not provide bounds for λ i A for i > other than in the case that mi d ln 2 n. For the Laplacian spectrum of Gw, the best known bound for λ k L > λ min L = 0 is given in [0]. If we take w to be the average expected degree, and gn a function going to with n arbitrarily slowly, the result in [0] is that, for w min ln 2 n, max λ k L + o 4 + gn ln2 n. k w w min We not only improve upon this bound in Theorem 4, but we extend to the case that w min ln n, rather than ln 2 n. Theorem 4. For the random graph Gw, if the minimum expected degree w min satisfies w min ln n, then with probability at least /n = o, we have that for all eigenvalues λ k L > λ min L of the Laplacian of Gw, λ k L 2 6 ln2n w min = o. Using the Gw model, we can also build random power law graphs in the following way. Given a power law exponent β, maximum degree m, and average degree d, we take w i = ci β for each i with i0 i < n + i 0. The values of c and i 0 depend upon β, m and d, in particular, c = β 2 β dn β and i0 = n dβ 2 β. mβ One can easily verify that the number of vertices of degree k is proportional to k β see [8], [9]. In section 4, we show how Theorems 3 and 4 3

4 can be applied in this setting to provide bounds on the spectra of random power law graphs. The remainder of this paper is organized as follows: In section 2, we develop notation and key tools to prove Theorems and 2. Section 3 is devoted to the proofs of these two main theorems. Section 4 contains the details of the proofs of Theorems 3 and 4, applications of these theorems to random power law graphs, as well as a discussion of our main results as applied to G n,p. 2 Key Tools As to notation, throughout the paper, given an n n Hermitian matrix A, A denotes the spectral norm, so A = max λ, where the maximum is over all eigenvalues λ of A. We order the eigenvalues λ λ 2 λ n. Given two matrices A and B, we say A B if B A is positive semidefinite. We refer to this ordering as the semidefinite order. If A is a random n n matrix, we write EA to denote the coordinate-wise expectation of A, so EA ij = EA ij. Similarly, vara = EA EA 2. We shall use several applications of functions to matrices. In general, if f is a function with Taylor expansion fx = n=0 a nx n, we take fa = n=0 a na n. We note that notions of convergence are as in [6]. In particular, we will often use the matrix exponential, expa = n=0 n! An. We note that expa is always positive definite when A is Hermitian, and that expa converges for all choices of A. Moreover, we shall require brief use of the matrix logarithm. In general, if B = expa, we say that A is a logarithm of B. As our matrices will be Hermitian, it is sufficient for uniqueness of this function to require that the logarithm also be Hermitian. Any notation not mentioned here pertaining to matrices is as in [6]. Given a graph G, we will use A to denote the adjacency matrix for G, so A ij = if v i v j and 0 otherwise. We use D to denote the diagonal matrix with D ii = deg G v i. If G is a random graph, then Ā denotes the expectation of A, and D the expectation of D. The Laplacian of G is denoted by L = I D /2 AD /2, and L = I D /2 Ā D /2 is the Laplacian matrix for the weighted graph whose adjacency matrix is Ā. All other notation referring to graph properties is as in [6]. We shall require the following concentration inequality in order to prove our main theorems. Previously, various matrix concentration inequalities have been derived by many authors including Ahlswede-Winter [2], Cristofides-Markström [2], Oliveira [8], Gross [5], Recht [9], and Tropp [20]. Here we give a short proof for a simple version that is particularly suitable for random graphs. Theorem 5. Let X, X 2,..., X m be independent random n n Hermitian matrices. Moreover, assume that X i EX i M for all i, and put v 2 = 4

5 varx i. Let X = X i. Then for any a > 0, a 2 pr X EX > a 2n exp 2v Ma/3 For the proof, we will rely on the following results: Lemma see, for example, [20]. Let f, g : R R, and suppose there is a subset S R with fa ga for all a S. If A is a Hermitian matrix with all eigenvalues contained in S, then fa ga. Lemma 2 [7]. Given a fixed Hermitian matrix A, the function X TrexpA+ log X is concave on the set of positive definite X. We note that any real-valued function that is convex with respect to the semidefinite order admits an operator Jensen s inequality see, for example, [9]. That is to say, if f is convex with respect to the semidefinite order, then for a random matrix X, fex EfX. Given a fixed matrix A and a random Hermitian matrix X, we may apply the function in Lemma 2 to e X. By then applying the operator Jensen s inequality as stated, we obtain the following: Lemma 3. If A is a fixed matrix and X is a random Hermitian matrix, then ETrexpA + X Trexp[A + loge[exp X]]. 2 We shall use this result to overcome the difficulties presented by working with the semidefinite order, as opposed to real numbers. The primary problem that must be overcome is that unlike real numbers, the semidefinite order does not respect products. Proof of Theorem 5. We assume for the sake of the proof that EX k = 0 for all k. Clearly this yields the general case by simply replacing each X k by X k EX k. Let gx = 2 x e x x = 2 2 k=2 xk 2 k!. Notice that g is increasing, so in particular, if x M, gx gm. Given θ > 0, we have that θx k θm, and thus gθx k gθmi by Lemma. Therefore, Ee θx k = EI + θx k + 2 θ2 X 2 kgθx k 3 I + 2 gθmθ2 EX 2 k 4 e 2 gθmθ2 EX 2 k. 5 5

6 We use this to prove the following claim: Claim : For matrices X k as given, [ m ] m E Trexp θx k Trexp 2 gθmθ2 EXk 2 Proof of Claim : For a given k, let E k := E X, X 2,..., X k. Then we have [ m ] [ m ] E Trexp θx k = EE E 2... E m Trexp θx k + θx m As the X i are independent, each X k is fixed with respect to E m except X m, and E m exp X m = Eexp X m. Applying inequality 2 from Lemma 3, we have m ] EE E 2... E m [Trexp θx k + θx m EE E 2... E m 2 [Trexp m θx k + log EexpθX m ] Iteratively applying this process, we obtain [ m ] m E Trexp θx k Trexp log Eexp θx m.. As both log and Trexp are monotone with respect to the semidefinite order these facts can be easily proven with basic manipulations, inequality 5 implies that [ m ] m E Trexp θx k Trexp log e 2 gθmθ2 EX 2 k as desired. Trexp m 2 gθmθ2 EXk 2, 6

7 Now, given a > 0, for all θ > 0 we have prλ max X a e θa Eexpθλ max X e θa ETrexpθX [ ] = e θa E Trexp θxk e θa Trexp 2 gθmθ2 EXk 2 e θa nλ max exp 2 gθmθ2 EXk 2 n exp θa + 2 gθmθ2 v 2 Notice that if x < 3, we have gx = 2 k=2 xk 2 k! Take θ =. Clearly, θm 3, and thus we have a v 2 +Ma/3 prλ max X a n exp θa + 2 gθmθ2 v 2 a 2 = n exp 2v 2 + 2Ma/3 a Therefore, pr X a 2n exp 2 2v 2 +2Ma/3 k=2 xk 2 = 3 k 2 x/3. 3 Proofs of the Main Theorems In this section, we provide proofs of the two main theorems using Theorem 5. Proof of Theorem. Let G be a random graph as described in the statement of the theorem, where the edge v i v j appears with probability p ij. Given i, j n, let A ij be the matrix with a in the ij and ji positions and a 0 everywhere else. Let h ij = with probability p ij and 0 otherwise. Take X ij = h ij A ij, so A = X ij. Thus, we can apply Theorem 5 to A with M =. We need first to calculate v 2. Now, if i j, then varx ij = Eh ij p ij 2 A ij 2 = varh ij A ii + A jj = p ij p ij A ii + A jj 7

8 Similarly, varx ii = p ii p ii A ii. Therefore, varx ij = n n p ij p ij A ii i= j= n = max p ij p ij i=,...,n j= max i=,...,n j= n p ij =. Take a = 4 ln2n/ɛ. By the assumption on, we have a < 3, and thus we obtain pr A Ā > a 2n exp a 2 2v 2 + 2Ma/3 4 ln2n/ɛ 2n exp 4 = ɛ. To complete the proof, we recall Weyl s Theorem see, for example, [6], which states that for Hermitian matrices M and N, max λ k M λ k N k M N. Thus, with probability at least ɛ, we have that for all i n, λ i A λ i Ā < 4 ln2n/ɛ. Proof of Theorem 2. We will again use Weyl s Theorem, as in the proof of Theorem, so we need only bound L L. For each vertex v i, put d i = degv i and t i = Ed i, the expected degree of the ith vertex. Let C = I D /2 A D /2. Then L L C L + L C. We consider each term separately. Now, C L = D /2 A Ā D /2. Using notation as in the proof of Theorem, let Y ij = D /2 h ij p ij A ij /2 D = h ij p ij A ij ti t j Then C L = Y ij, so we can apply Theorem 5 to bound C L. Notice Y ij t i t j /2 δ. Moreover, EY 2 ij = { t it j p ij p 2 ij Aii + A jj i j p ii p 2 ii Aii i = j t 2 i 8

9 Thus, we obtain v 2 = EYij 2 n n = p ij p 2 t i= j= i t ija ii j n = max n p ij p 2 ij i=,...,n t j= i t j t j= i t j max n p ij = i=,...,n δ t i δ 3 ln4n/ɛ Take a = δ. Take k to be large enough so that δ > k ln n implies a < in particular, choosing k > 3+lnɛ/4 is sufficient. Applying Theorem 5, we have pr C L > a 2n exp ɛ/2 j= 3 ln4n/ɛ δ 2/δ + 2a/3δ 2n exp 3 ln4n/ɛ 3 For the second term, note that by the Chernoff bound see, for example, [], for each i, pr d i t i > bt i ɛ 2n if b ln4n/ɛ t i ln4n/ɛ Take b = δ, so that for all i, we have pr d i t i > bt i ɛ 2n. Then we obtain D /2 D /2 di I = max i=,...,n t i. Note that for 0 < x <, we have x x. Taking x = di t i > 0, we have that with probability at least ɛ 2, this is at most b = ln4n/ɛ δ = 3 a < for all i. Thus we obtain D /2 D /2 di ln4n/ɛ I = max i=,...,n t i δ with probability at least ɛ 2. 9

10 We note that as the Laplacian spectrum is in [0, 2], we have I L. Therefore, with probability at least ɛ 2, we have L C = I D /2 AD /2 I + D /2 A D /2 = D /2 AD /2 D /2 D /2 D /2 AD /2 D /2 D /2 = I L D /2 D /2 I LD /2 D /2 = D /2 D /2 II LD /2 D /2 + I LI D /2 D /2 D /2 D /2 I D /2 D /2 + I D /2 D /2 bb + + b = b 2 + 2b Finally, as b = 3 a and a <, we have that with probability at least ɛ, completing the proof. L L C L + L C a + 3 a2 + 2a 2a, 3 4 Applications to Several Graph Models The above theorems apply in a very general random graph setting. Here we discuss the applications of Theorems and 2 for the Erdös-Rényi graph and for the Gw model, as discussed in section above. We begin by examining the Erdös-Rényi graph. The Erdös-Rényi graph is a well studied random graph see, for example, [], [5] where p ij = p for i j and 0 for i = j for some p 0,. We denote this graph by G n,p. If J represents the n n matrix with a in every entry, then for G n,p, Ā = pj I and D = n pi. An application of Theorem yields Theorem 6. For G n,p, if p > 8 9n ln 2n, then with probability at least /n = o, we have λ i A λ i pj I 8np ln 2n. We note that stronger results for the spectrum of the adjacency matrix of G n,p can be found in [2], [4]. Specifically, in [4], it is shown that for pn c ln n, λ A = pn+o pn, and all other eigenvalues satisfy λ i A = O pn. Here, we are at best able to show that λ i A λ i pj I Opn. The spectrum of pj I is {pn, p}, where p has multiplicity n, so if i >, we have only that λ i A = Opn. However, due to the very strong 0

11 symmetries in G n,p, it seems unlikely that the methods used to investigate this graph in detail will extend to general random graphs. For the Laplacian of G n,p, we obtain L = I Theorem 2 yields n J I. An application of Theorem 7. If pn ln n, then with probability at least /n = o, we have λ k L λ k I 6 ln2n J I 2 = o n pn for all k n. The spectrum of I n J I is { n, + n }, where + n has multiplicity n. Thus, we see that if pn ln n, then w.h.p. L has all eigenvalues other than λ min L close to. This result is not new see [0], [], and [] also considers the case where pn ln n. We now turn to the Gw model. We begin by precisely defining the model. Given a sequence w = w, w 2,..., w n, we define the random graph Gw to have vertex set {v,..., v n }, and edges are independently assigned to each pair v i, v j with probability w i w j ρ, where ρ = P wi. In this way, the expected degree of v i is w i for each i. Moreover, the matrix Ā with Āij = prv i v j is given by Ā = ρw w, and as such has eigenvalues ρ wi 2 and 0, where 0 has multiplicity n. Let d denote the expected second order average degree of Gw, so d P w 2 i = P wi = ρ wi 2. Applying Theorem with ɛ = /n, we immediately obtain Theorem 3. Similarly, we can apply Theorem 2 to obtain concentration results for the spectrum of the Laplacian matrix for Gw. Notice that the value of k given in Theorem will here be kɛ = k/n > 3 + ln/4n, so the requirement in Theorem 4 that w min ln n is sufficient to give that for n sufficiently large, w min > k ln n. This theorem improves on the bounds for the eigenvalues of the Laplacian given in [0], as seen in section above. Let x = w /2, w /2 2,..., wn /2. Then we have D /2 Ā D /2 ij = ρw iw j wi w j = ρ w i w j, so L = I ρx x. Thus, the eigenvalues of L are 0 and, where has multiplicity n. Applying Theorem 2 with ɛ = /n, we obtain Theorem 4. Recall, as described in section, we can use the Gw model to build random power law graphs. Given a power law exponent β, maximum degree m, and average degree d, we take w i = ci β for each i with i0 i < n + i 0, with c = β 2 β dn β and i0 = n dβ 2 mβ β. We obtain the following bounds for d see [9]:

12 d = d 2 d ln 2m d d β 22 β β 3 β 22 β 3 β + o if β > 3 + o if β = β + o if 2 < β < 3 mβ dβ 2 In [0], bounds are given for the largest eigenvalue of the adjacency matrix of a random power law graph as described. In particular, the authors show that for a random power law graph as above with adjacency matrix A,. If β 3 and m > d 2 log 3+ɛ n, then a.a.s. λ A = + o m. 2. If 2.5 < β < 3 and m > d β 2 β 2.5 ln 3 β 2.5 n, then a.a.s. λ A = +o m. 3. If 2 < β < 2.5 and m > ln β n, then a.a.s. λ A = + o d. We note that these theorems require specific relationships between m and d, as noted. Applying Theorem 3 to a random power law graph, we can eliminate such requirements, although the bounds are less clean: Theorem 8. Suppose Gw is a random power law graph as described above. If m > 8 9 ln 2n then with probability o, we have If β > 3, then λ β 2 2 A d + o β β 3 8m ln 2n If β = 3, then λ A 2 d ln 2m d + o If 2 < β < 3, then λ β 2 2 A d β 3 β mβ dβ 2 8m ln 2n 3 β + o 8m ln 2n From these bounds, one might be able to derive specific relationships between m and d that lead to particular values for λ A. Finally, we can apply Theorem 4 to the random power law graph model described here to obtain bounds on the Laplacian spectrum: Theorem 9. Suppose Gw is a random power law graph as described above. If w min = ci β 0 = mn β β ln n, then with probability o, we have 2

13 The smallest eigenvalue, λ min L of the Laplacian of Gw satisifies 6 ln2n λ min L 2 = o mn β β If λ k L > λ min L, then we have 6 ln2n λ k L 2 = o mn β β References [] N. Alon and J. Spencer, The Probabilistic Method 3rd edition, John Wiley & Sons, [2] R. Ahlswede and A. Winter, Strong converse for identification via quantum channels, IEEE Trans. Inform. Theory, 48 3, 2002, [3] G. W. Anderson, A. Guionnet, and O. Zeitouni, An Introduction to Random Matrices, Cambridge University Press, 200. [4] M. Aouchiche and P. Hansen, A survey of automated conjectures in spectral graph theory, Linear Algebra Appl. 432, 200, no. 9, [5] B. Bollobás, Random Graphs 2nd edition, Cambridge University Press, 200. [6] F. Chung, Spectral Graph Theory, AMS Publications, 997. [7] F. Chung and L. Lu, Connected components in random graphs with given expected degree sequences. Ann. Combin. 6, [8] F. Chung and L. Lu, Complex Graphs and Networks, AMS Publications, [9] F. Chung, L. Lu, and V. Vu, Eigenvalues of random power law graphs. Ann. Combin. 7, [0] F. Chung, L. Lu, and V. Vu, The spectra of random graphs with given expected degrees. Internet Math., [] A. Coja-Oghlan, On the laplacian eigenvalues of G n,p. Comb. Prob. Comp. 6, [2] D. Cristofides and K. Markström, Expansion properties of random Cayley graphs and vertex transitive graphs via matrix martingales, Random Structures Algs., 32 8, 2008,

14 [3] D. M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs, Theory and Applications, Academic Press, 980. [4] U. Feige and E. Ofek, Spectral techniques applied to sparse random graphs. Random Structures and Algorithms. 27 no. 2, [5] D. Gross, Recovering low-rank matrices from few coefficients in any basis, IEEE Trans. Inform. Theory, 57, [6] R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 985. [7] E. H. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture. Adv. Math., [8] R. Oliveira, Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges, [9] B. Recht, Simpler approach to matrix completion, J. Mach. Learn. Res., to appear. [20] J. Tropp, User-Friendly Tail Bounds for Sums of Random Matrices. Available at [2] Z. Füredi and J. Komlós, The eigenvalues of random symmetric matrices. Combinatorica, 98, no. 3,