Quasi-random graphs with given degree sequences

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1 Quasi-random graphs with given degree sequences Fan Chung and Ron Graham y University of California, San Diego La Jolla, CA 9093 Abstract It is now known that many properties of the objects in certain combinatorial structures are equivalent, in the sense that any object possessing any of the properties must of necessity possess them all. These properties, termed quasirandom, have been described for a variety of structures such as graphs, hypergraphs, tournaments, Boolean functions, and subsets of Zn, and most recently, sparse graphs. In this paper, we extend these ideas to the more complex case of graphs which have a given degree sequence. Introduction During recent years there has been increasing interest in investigating the following phenomenon. For a given nite collection C of \objects", suppose we have some probability distribution given on C. Typically, there are many properties which are satised by most (or almost all) of the objects in C as seen in [4]. It turns out, however, that in many cases there is a large subclass Q of these properties which are strongly correlated, in the sense that any object in C which satises any of the properties in Q must in fact necessarily satisfy all the properties in Q. Such properties are called \quasi-random". Specic cases where this behavior is investigated can be found in [5, 6, ] (for graphs), [, 3, 7, 9] (for hypergraphs), [4] (for tournaments), [8] (for sequences), [5] (for permutations) and [0] (for sparse graphs), for example. In this paper we will take C to be the class G n (d) of all graphs on n vertices having some given degree sequence d. This is rather dierent from the classical model of a random graph, in which all vertices have the same expected degree. Special cases of such graph families include the so-called power law graphs in which the number of vertices of degree k is proportional to k for some positive Research supported in part by NSF Grants DMS 04575, ITR and ITR y Research supported in part by NSF Grant CCR 03099

2 real. Such graphs arise in a variety of applications such as Web connectivity [5, 6, 9, 4, 6, 8, 9], communication networks [, 3], biological networks [], collaboration graphs [7], etc. In this paper, we will introduce a class of quasi-random properties for G n (d) and establish quantitative bounds on the strength of correlation between these properties. In particular, these results generalize and strengthen those in [0, ]. Notation We will consider graphs G = (V; E) where V denotes the set of vertices of G and E denotes the set of edges of G. (For undened graph theory terminology, see [33].) Our graphs will be undirected, having no loops or multiple edges. We will let jv j, the cardinality of V, be denoted by n. If fx; yg E is an edge of G, we say that x and y are adjacent, and write this as x y. The neighborhood nd(x) of a vertex x V is dened by nd(x) := fy V : y x in Gg: For x V, the degree d x of x, denotes jnd(x)j. The degree sequence d = d G of G is given by d = (d x : x V ); or equivalently, d can be viewed as a mapping d : V! Z + [f0g. For ; Y V, dene e(; Y ) :=j f(x; y) : x ; y Y and x yg j : For V, dene vol(), the volume of, by vol() = x d x : A walk P = P t (x; y) from x to y is a sequence P = (x 0 ; x ; : : : ; x t ), where x 0 = x; x t = y and x i x i+ for 0 i < t. Such a walk is said to have length t. Here we do not require all x i 's to be distinct. If all x i 's are dierent, we say the walk is a path. In this paper, we consider graphs with every vertex having positive degree. The weight w(p ) of such a walk P is dened to be Y w(p ) = 0<i<t (thus, both endpoints are excluded in the product). If P has length (and therefore is an edge of G), then w(p ) is dened to be. d xi

3 A circuit C of length t is a sequence of t vertices (x ; x ; : : : ; x t ) where x i x i+ ; i < t, and x t x. (We remark that in this denition, a circuit can be viewed as a rooted closed walk.) The weight w(c) of such a circuit is dened by Y w(c) = : d xi it The weighted adjacency matrix M = M(G) is an n n matrix with rows and columns indexed by V, dened by ( p if x y; M(x; y) = dxd y () 0 otherwise. Note that M can be written as M = I L where I is the identity matrix and L denotes the (normalized) Laplacian (see []). The eigenvalues of M are denoted by i ; 0 i n, indexed so that = 0 j j j j : : : j n j using the Perron-Frobenius theorem. Note that 0 = has as its eigenvector ( p d x ) xv. Finally, dene for ; Y V, and t, e t (; Y ) = P P t(;y ) w(p ) where P t (; Y ) denotes the set of all walks of length t between x and y Y. This is a weighteersion of the number of walks of length t between and Y. Note that e (; Y ) = e(; Y ). In particular, e (V; V ) = P x d x =. It is not dicult to check that for t, we have e t (V; V ) =. 3 The quasi-random properties In this section we will state various properties that the G G n (d) might satisfy. Each of these properties will depend on a parameter, which we will always assume to satisfy 0 < <. The closer is to 0, the more the graph in question behaves like a random graph with respect to the property in question, that is, the more the value of the corresponding parameter is closer to its expectealue for a random graph in G n (d). DISC(): For all ; Y V, je(; Y ) vol()vol(y ) j : 3

4 DISC t (): For all ; Y V, je t (; Y ) vol()vol(y ) j : Note that DISC () is just DISC(). EIG(): With the matrix M = M(G) = (M(x; y)) x;yv eigenvalues satisfying as dened in () and with = 0 j j j j : : : j n j; we have j i j < for all i : TRACE t (): The eigenvalues of M satisfy i t i : CIRCUIT t (): The weighted sum of the t-circuits C t in G satises j C t:t circuit w(c t ) j : We will prove the following implications in Section 4: Theorem For t, the following implications hold. 4

5 Here the notation A ) B is shorthand for A() ) B(). We say A implies B, denoted by A ) B, if for every > 0, there exists > 0 such that A() ) B(). There are several one-way implications in the above gure. A natural question is which, if any, of the reverse directions hold for any of these implications. In Section 5, we will give counterexamples which show that EIG 6) Trace t for any t. In Section 6, we introduce an additional property U t. Then we show that if a graph satises U t for some t, then DISC ) CIRCUIT t. Using property U t, we will prove the following result. Theorem If G satises U t for some t, then CIRCUIT t, TRACE t, EIG, DISC, DISC, DISC t are all equivalent. 4 The implications Lemma EIG() =) DISC(). Proof: For S V, dene Then, for ; Y V, f S (x) = p dx if x S; 0 otherwise. e(; Y ) = hf ; Mf Y i P where hf; gi = xv f(x)g(x) denotes the usual inner product. Now, write f = i a i i where the i 's form an orthonormal basis of eigenvectors with 0 (v) = s ; 5

6 for all v V. Hence, a 0 = hf ; 0 i d = p x x = vol() p : Similarly, we write Thus, f Y = i b i i : hf ; Mf Y i = a 0 b 0 + i i a i b i = vol()vol(y ) + i i a i b i Therefore, je(; Y ) vol()vol(y ) j = j i a i b i j i max j i j( ja i j ) = ( jb i j ) = i i i kf k kf Y k = p vol()vol(y ) by using EIG and the Cauchy-Schwarz inequality where k k denotes the L - norm. Therefore, the proof is complete. In a similar way, we prove Lemma EIG() =) DISC t ( t ) for any t. Proof: In this case we observe that for ; Y V, e t (; Y ) = hf ; M t f Y i (using the notation of Lemma ). Thus, writing f = i a i i ; f Y = i b i i ; 6

7 we nd jhf ; M t f Y i vol()vol(y ) j = jhf ; M t f Y i t 0a 0 b 0 j max j i j t ja i b i j i i max j i j t kf k kf Y k i tp vol()vol(y ) t and Lemma is proved. Lemma 3 CIRCUIT t () () TRACE t (). Proof: Let C t(u) denote a rooted t-circuit with starting and ending point u. Then, M t (u; u) = C t (u) w(c t(u)): Thus, the trace of the matrix M t can be expressed as: Tr(M t ) = u = C t w(c t ): C t (u) w(c t(u)) On the other hand, the same trace can be evaluated using eigenvalues: Tr(M t ) = i t i = + i t i : Thus, we have j C t w(c t ) j = j Tr(M t ) j = i and Lemma 3 is proved. t i Lemma 4 TRACE t () =) EIG( =t ); for any t. 7

8 Proof: By hypothesis, we have j i t i j = t i : i Therefore max j i j =t : i Lemma 5 TRACE t () =) TRACE t+ (). Proof: Since j i j for all i, we have i t+ i i by hypothesis. t i Lemma 6 For t, DISC t () =) DISC t ( p ). Proof: For V, e t (; ) = = y x;x 0 y e t (y; ) d y : By applying DISC t () to e t (; ), we have Note that Therefore, y y e t (y; ) d y e t (x; y)e t (y; x 0 ) d y () vol() + : e t (y; ) = e t (V; ) = xv y = y (e t (y; ) e t (y; ) d y d y vol() ) d y e t (y; ) vol() = d y y : d x = vol(): e t (V; ) vol() + vol() 8

9 by () and DISC t (). But y (e t (y; ) yy (e t (y; ) = d y vol() ) d y d y vol() ) d y ) vol() (e t (y; ) d y yy e t (Y; ) vol(y )vol() by applying the Cauchy-Schwarz inequality. Thus, j e t (Y; ) vol(y )vol() A P yy d y =vol(y ) p j vol(y ) p : This is exactly DISC t ( p ). Lemma 7 For any t, DISC t () =) DISC t+ (6 p ): Proof: For ; Y V, Consider Dene Thus, j e t (; Y ) vol()vol(y ) e t+ (; Y ) = v S := fz V : e t (z; Y ) > j : (3) e(; v)e t (v; Y ) : d z (vol(y ) + p )g: e t (z; Y ) = e t (S ; Y ) > vol(s )vol(y ) zs Hence, by (3) applied to = S and Y, + p vol(s ): vol(s ) < p : (4) In the same way, if we dene S := fz V : e t (z; Y ) < dz (vol(y ) p )g, then vol(s ) < p : (5) 9

10 Now, e t+ (; Y ) = For the rst sum, we have e(; v)e t (v; Y ) v6s [S v6s [S + vs [S e(; v) v6s [S v A e(; v)e t(v; Y ) : (vol(y ) + p ) e(; v) (vol(y ) + p ) vol()vol(y ) + p and e(; v)e t (v; Y ) v6s [S e(; v) v6s [S (vol(y ) p ) e(; v) (vol(y ) p ) v6s [S (vol() vol(s ) vol(s )) p (vol(y ) ) (vol() p ) p (vol(y ) ) vol()vol(y ) 3 p ; by using (4) and (5). Thus, j e(; v)e t (v; Y ) vol()vol(y ) v6s [S j 3 p : For the second sum, we have e(; v)e t (v; Y ) vs [S e t (v; Y ) vs [S = e t (S [ S ; Y ) (vol(s ) + vol(s ))vol(y ) + by DISC t () p vol(y ) + = p vol(y ) + 3 p : 0

11 Putting these two estimates together, we obtain j e t+ (; Y ) vol()vol(y ) j 6 p which is DISC t+ (6 p ): Lemma 8 For any integers s and t, DISC s and DISC t are related as follows: (i) If s < t, then DISC s () =) DISC t (36 =t s ). As a special case, DISC() =) DISC t (36 =t ). (ii) If t < s k t for some k, then DISC s () =) DISC t (36 =k =k t+k s). Proof: (i) follows from Lemma 7, i.e., DISC s () ) DISC s (36) ) DISC s+ (36 = ) ) : : : ) DISC t (36 =t s ) To prove (ii), we have, from (i) that DISC s () ) DISC k t(36 =k t s) Now apply Lemma 6 k times to get the desired implication.. By combining Lemmas to 8, we have proved all the implications in Theorem 5 Separation of properties In this section we give an example showing that at least one of the implications in Theorem cannot be reversed. Whether this is true of the others is not known at this point. Fact For any t, for any = (). EIG() 6=) TRACE t ()

12 Proof: Choose t and let G = G(n) be a random regular graph with n vertices anertex degree n =t. Thus, M = M(G) has M(u; v) = =n =t if u v; 0 otherwise. It was shown in [3] that the eigenvalue distribution of M(G) for a random graph G with a given expected degree distribution satises the semi-circle law if the minimum degree is greater than a power of log n. As a consequence, if = 0 j j j j : : : j n j are the eigenvalues of M, then () = ( + o())=n =t, () If N(x) denotes the number of i with i x=n =t, then Z N(x) = ( + o()) x p u du: n In particular, for x = =, we have Thus, N(=) n i t = ( + o())( 3 + p 3 4 ) 0:8045 : : : : i (n N(=))( n =t )t 0:39: Hence, for any > 0, G satises EIG(), provided n n 0, but G does not satisfy TRACE t (0:39). It would be interesting to know if some of the other possible implications hold. For example, does DISC ) EIG? Recently, Bilu and Linial [7] proved the following partial implication for regular graphs: For a d-regular graph G on n vertices, if for all ; Y V, e(; Y ) then j j = O((log(=) + )). d n jj jy j d p jj jy j; (6) The above property in (6) was introduced by Thomason [3] in the context of what he called (p; )-jumbled graphs. Of course, this property is quite a bit stronger than DISC. Properties of random graphs based on this concept (without the equivalence relations) are often referred as the pseudo-random properties. The reader is referred to [30, 3] for discussions on pseudo-random graphs. Butler [0] combines the methods in [7] and [8] to prove the following :

13 For a graph G with no isolateertices, if for all ; Y V, e t(; Y ) then j j t 5 8( log ). vol() vol(y ) p vol()vol(y ); For t =, this is the best possible (up to a constant) by considering a class of regular graphs constructed by Bollobas and Nikiforov [8]. In their example, the graphs have = Cn =6 and j j c log n for some constants c and C. 6 Reversing the implications It is clear from the examples in the preceding section that in order to establish some of the reverse implications, e.g., DISC ) CIRCUIT t, we will have to make further assumptions for the G G n (d). One such condition is the following: For t, a graph satises U t (C) if for all x; y V; e t (x; y) C dxdy. We will think of C as a large positive real. We note that for t = and for G with minimum degree n, the property U (C) is automatically satised for C =. Note that for a d-regular graph, U t implies that n Cd t or, equivalently, the volume of the graph is of order at least n +=t. Lemma 9 For any t, U t (C) =) U t+ (C): Proof: Observe that e t+ (x; y) = z z e(x; z)e t (z; y) d z e(x; z) d z = C d y = C d xd y : z C d zd y e(x; z) The lemma is proved. 3

14 Theorem 3 If G satises U t (C) for some t, then DISC() =) CIRCUIT t () where = C 0 C = + C (C 0 + ) + 0 p + + 6C 04 3= + 8C 0 3=, with C 0 = dc= =4 e, and = maxf p ; 36 =t g. (Note that! 0 as! 0.) Proof: We are going to consider the sum u;vv where, as usual, V = V (G). (e t (u; v) ) Since G satises DISC() by hypothesis, then by Lemma 8, G also satises DISC t () where 36 =t, i.e., for all ; Y V. j e t (; Y ) vol()vol(y ) j We here choose = maxf p ; 36 =t g. For a xeertex u, we partition the vertex set V into the sets W i = W i (u); 0 i < C 0, as follows. (To simplify the notation, we use W i instead of W i (u) below.) and, in general, W 0 = fv : 0 e t (u; v) < =4 d u g; W = fv : d ud =4 v e t (u; v) < d ud =4 v g; W = fv : d ud =4 v e t (u; v) < 3 d ud =4 v g; W i = fv : i =4 d u e t (u; v) < (i + ) =4 d u g for 0 i < C 0 = dc= =4 e. Since e t (u; v) C = by U t (C), the W i form a partition of V. Since then W i fv : je t (u; v) i d ud =4 v j < d u =4 g; j (e t (u; v) i d ud =4 v ) j < vw i and j e t (u; W i ) i =4 d uvol(w i ) vw i =4 d u j < d uvol(w =4 i ) : (7) 4

15 Since P i e t (u; W i ) = e t (u; V ) = d u, then j i = j i =4 i i =4 d uvol(w i ) i =4 d uvol(w i ) d u vol(w i ) d u j i e t (u; W i ) j = =4 d u : (8) Now, for each i, if vol(w i ) p, then dene i = i (u) and 0 i = 0 i (u) as follows: i = fv : e(v; W i ) > vol(w i ) ( + p )g; i 0 = fv : e(v; W i ) < vol(w i ) p ( )g: If vol(w i ) < p then dene i = i 0 = ;. Also dene Thus, W u = [ fw i : vol(w i ) < p g: vol(w u) C 0p since there are just C 0 possible values of i. By DISC(), we have j e(w i ; i ) vol(w i )vol( i ) j ; but from the denition of i, we have j e(w i ; i ) vol(w i )vol( i ) j p vol( i)vol(w i ) p p vol(i ) = vol( i ): Therefore, vol( i ) = : A similar argument shows that vol(i) 0 = 5

16 as well. Consequently, for each u, we consider and we have u := [ f i [ 0 i : W i 6 W u g For v 6 u, we have, from the denition of u, e(w u ; v) = vol( u ) C 0 = : (9) e(w i ; v) ( p ) vol(w i ) p ( vol(w = ( ) u)) = p p vol(w + ( ) u) p + ( p p )C 0 (C 0 + ) p : (0) We now begin considering the sum, u v (e t (u; v) u ) = v u + v6 u (e t (u; v) ) : For the rst sum, we use property U t (C) and Lemma 9 to obtain the following estimate: u v u (e t (u; v) ) u v u C ( d u ) = C vol( u) C 0 C = () 6

17 by (9). For the second sum we have u = u = u u v6 u (e t (u; v) v6 u ( z v6 u (( v6 u zw u ( ) e t (u; z)e(z; v) d z + zw u z6w u ) ) e t (u; z)e(z; v) d z ) e t (u; z)e(z; v) e ) t (u; z)e(z; v) + ( d z d z6w z u! ) : For the rst sum just above (without a factor of ), we have e t (u; z)e(z; v) A d u u d z v6 u u = u u zw u v6 u v6 u v6 u zw u C du(e(w u ; v)) Cd u d z e(z; v) d z A by U t (C); C (C0 + ) by (0) C (C 0 + ) : () For the second sum above, we have d u u v6 u = u u z6w u v6 u v6 u ( + ( e t (u; z)e(z; v) d z e t (u; z)e(z; v) d z zw i i=4 d u d z e(z; v) d z zw i =4 d u d z e(z; v) d z zw i )! ) since ja aj b ) (A B) ((a B) + b ) and inequalities in (7). 7

18 For the second sum we have ( =4 d u e(z; v) d u u v6 u zw i = p u p u p u v6 u v6 u ( ( d u e(w i ; v) ) ) d u ( + p ) vol(w i ) v6 u ( + p ) d u ) by def. of u p + 3 (3) (upper bounded by the sum over all u an). Finally, for the rst sum we have u = u u u u;v v6 u v6 u v6 u + v6 u + P 4 W iw u i i=4 d u e(z; v) zw i i =4 d u e(w i ; v) i =4 vol(w i ) i =4 vol(w i ) i =4 vol(w i ) i =4 vol(w i ) + du =4 p + 4 C 0 =4 = by the def. of u, i =4 vol(w i ) p + C0 3=4 by (8), def. of W u and the fact that i < C 0, 4 p + 4C 04 3= + C 0 3= : (4) Now, we have to put everything together. 8

19 First observe that (e t (u; v) u;vv = = = u;vv C tt C tt e t (u; v) circuit circuit ) u;vv w(c t ) e t(v; V ) + w(c t ) e t (u; v) + u;vv (using e t (V; V ) = ) so that the preceding results, including inequalities (), (), (3) and (4), give j C tt = u;vv circuit w(c t ) (e t (u; v) j ) C 0 C = + C (C 0 + ) + 4( p + 3) + 4(4 p + 4C 04 3= + C 0 3= ) C 0 C = + C (C 0 + ) + 0 p + + 6C 04 3= + 8C 0 3= : This proves Theorem 3. Corollary If G has minimum degree n, then where depends only on, and t. DISC() =) CIRCUIT t () Theorem 4 If G has minimum degree n for some constant, then CIRCUIT t, TRACE t, EIG, DISC, DISC, DISC t are all equivalent for t. 7 Concluding remarks We can summarize the main theorems in the following: 9

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