A = A U. U [n] P(A U ). n 1. 2 k(n k). k. k=1
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1 Lecture I jacques@ucsd.edu Notation: Throughout, P denotes probability and E denotes expectation. Denote (X) (r) = X(X 1)... (X r + 1) and let G n,p denote the Erdős-Rényi model of random graphs. 10 Random graphs For p [0, 1], n N and N = ( n ), let Gn,p denote the probability space (Ω, F, P) which is a product probability space whose components are (Ω ij, F ij, P ij ) : {i, j} [n] where Ω ij = {0, 1} and P(1) = p and P(0) = 1 p. In other words, we flip a coin with heads probability p independently for each pair {i, j} to decide whether {i, j} is an edge of a graph or not. Then for any graph G on [n] with v(g) vertices and e(g) edges, P(G) = p e(g) (1 p) N e(g). This is known as the Erdős-Rényi model or mean field model of random graphs. When p = 1/, each graph G is then sampled uniformly at random, and this allows us to count graphs with certain properties. For instance, let s consider again the connected graphs. Let A be the event that a graph G is disconnected in G n,1/. Then taking a union over proper non-empty subsets U of [n], we get A = U [n] where A U is the event that there are no edges between U [n] and [n]\u. By the union bound from probability, P(A) P(A U ). Now clearly and therefore U [n] A U P(A U ) = U (n U ) P(A) 1 n 1 k=1 ( ) n k(n k). k This is at most n 1 n and therefore a random graph has probability at most 1 n of being disconnected. So the number of connected labelled graphs on [n] is at least (n ) ( 1 n n ) in agreement with the results we obtained using generating functions. 1
2 Let us say that almost all graphs on n vertices have property P as n if o( (n ) ) graphs on [n] do not have property P as n. So above, almost all graphs on n vertices are connected as n. Consider another example: a graph has diameter two (a metric property) if every pair of vertices is connected by at least one path of length at most two. We prove that almost all graphs have diameter two (even stronger than being connected). Fixing a pair {u, v} of vertices, if the diameter is not two, then they are not joined and none of the n paths of length two between those vertices occurs. Crucially, all those events are independent since distinct edges are independent, and so the probability that u and v are not at distance at most two is (1 1 ) (1 1 4 )n = 1 ( 3 4 )n. This means the probability that there exists a pair like that is at most ( ) n 1 ( 3 4 )n 0 as n. We conclude that almost all graphs have diameter two. Ramsey Theory Finally, we give a striking example from Ramsey Theory due to Erdős. Let A 1, A,..., A N be an enumeration of the N = ( n k) sets of k elements of [n] and let Pi be the property of containing a complete graph or an empty graph on A i. Then P(P i ) = 1 ( k ) and there are ( n k) possible values of i. If n > k/, then we see that almost all graphs do not have any of the properties P i. In other words, almost all graphs on n vertices have no independent set of size log n and no clique of size log n. This is despite the fact that there is no known explicit construction for all n N of a graph on n vertices in which the cliques and independent sets have size at most (log n) The best constructions, which are algebraic and due to Frankl and Wilson, have no independent sets and cliques of size larger than roughly n 1/ log log n Unlabelled connected graphs Let G be a graph on [n] and π a permutation of [n]. For v V (G) let Γ(v) be the neighborhood of v the set of vertices adjacent to v. Define the defect of G under π as D π (G) = max{ Γ(π(v)) π(γ(v)) : v V (G)} where we refer to the quantity D π (v) = Γ(π(v)) π(γ(π(v)) as the defect at v under π. Define the defect of G as D(G) = min{d π (G) : π Id}. Observe that the automorphism group of G is non-trivial if and only if D(G) = 0. For example, consider the cycle of length six:
3 (,1,3,4,5,6) ~ (,3,4,5,6,1) D D ~ 0 Figure 1 : Defect for the hexagon. Now consider D(G n,p ). We shall need the following inequality from probability. A function f : R n R is c-lipschitz if f(x) f(y) c whenever x and y differ in one co-ordinate. Theorem 1 (Concentration Theorem) Let X be a random variable that is a function f of independent Bernouilli random variables X 1, X,..., X N, such that f is c-lipschitz and let P (X i = 1) = p i and σ = c N i=1 p i(1 p i ). Then P( X E(X) λσ) e λ /4. In the present situation, if π is a permutation of V (G) and U is the set of non-fixed points of π, then X π = v U D π (v) is a function of the edges of G containing a vertex of U. expectation E(X) p(1 p)n U Observe also by linearity of since D π (v) is a binomial random variable with mean asymptotic to np(1 p): if π(v) = u v, then a vertex w {u, v} contributes to D π(v) with probability p(1 p) since w must be adjacent to u but not v, and π 1 (w) must be adjacent to v but not u, and these are independent events for different vertices w. If π(π(v)) = v, this shows E(D π(v) ) = 3
4 (n )p(1 p), and if π(π(v)) v, then E(D π (v)) = (n 1)p(1 p). Furthermore X π changes by at most four when an edge of G is added or removed, for an edge {x, y} only D π (x), D π (y), D π (π 1 (x)) and D π (π 1 (y)) can change, each by at most one. Applying the concentration theorem with parameters λσ = E(X), σ = λ and c = 4 and N = ( ) U + U (n U ) U n, the probability that X is zero is at most P( X E(X) λσ) e λ /4 e U np(1 p)/. Now the number of π with U = k is at most k! ( n k) and so the probability that the defect is zero is at most is at most n ( ) n k!e kp(1 p)n/ ne p(1 p)n/. k k=1 Provided pn > ( + δ) log n and (1 p)n > ( + δ) log n for some δ > 0, this goes to zero as n. So almost surely as n, the defect of the random graph G n,p is non-zero. In particular, when p = 1, we obtain: Theorem Almost all n-vertex graphs have a trivial automorphism group as n. In particular, there are indeed asymptotic to 1 n! (n ) ( 1 n n ) unlabelled connected graphs on n vertices as n. 10. Generating functions in probability Let X be a random variable on some probability space (Ω, F, P). The probability generating function of X is defined by G X (t) = E(t X ) = t x f(x) where f is a density function for X, for t 0. In the discrete case, the coefficient of t x is P(X = x). The moment generating function of X, when it exists, is M X (t) = E(e tx ) = e tx f(x) and we observe that M X (0) = 1 and M X (n) (0) = E(X) and in general M X (0) = E(Xn ). A random variable has Poisson distribution if Ω Ω P(X = k) = e λ λ k 4 k!
5 for k 0. One computes The characteristic function of X is M X (t) = e λ(et 1). φ X (t) = M X (it). The boundedness of φ X is why it is often a preferable choice to moment generating functions. One of the key theorems in probability, Lévy s Convergence Theorem, is that convergence of characteristic functions of a sequence X n of random variables to the characteristic function of a random variable X implies convergence in the distribution of X n to the distribution of X, under some conditions on moments. This reduces checking convergence in distribution to checking convergence of moments. This theorem for our purposes is as follows when X has a Poisson distribution: Theorem 3 (Poisson Convergence Theorem) Let X 1, X,..., X k be random variables depending on some parameter n. Suppose that each sequence (r 1, r,..., r k ) of positive integers, the random variables Y i = (X i ) (ri ) = X i (X i 1)... (X i r i + 1) satisfy lim E(Y 1Y... Y k ) n k i=1 λ r i i. Then as n, the random variables X 1, X,..., X k random variables with means λ i respectively, namely are jointly independent Poisson lim P(X 1 = x 1, X = x,..., X k = x k ) = n for any non-negative integers x 1, x,..., x k. k i=1 e λ i λ x i i x i! If X has Poisson distribution with parameter λ, then one quickly verifies that E(X(X 1)... (X r + 1)) = e λ k=0 k (r) λ k k! = λr. The convergence theorem here is a multivariable form of the usual convergence theorems for characteristic functions. We show how to use this to count regular graphs Counting Regular Graphs To attack the enumeration of labelled d-regular graphs, namely graphs all of whose vertices have degree d, we introduce a suitable probability space i.e. model of random graphs. Suppose we want to count graphs which have a fixed degree sequence d 1, d,..., d n on n vertices. Not all sequences of positive integers are degree sequences of a graph for instance 5
6 they must add up to an even number. Necessary and sufficient conditions have long been known to graph theorists. However, all sequences of positive integers adding up to an even number are the degrees of some pseudograph a graph with loops and multiple edges. To create a random instance, consider disjoint sets D 1, D,..., D n of size D i = d i for i [n]. Then uniformly and randomly place a matching on D 1 D D n. Finally contract each set D i to a single vertex to obtain a pseudograph with degrees d 1, d,..., d n. We consider the special case of counting d-regular graphs graphs with d 1 = d = = d n = d. We must estimate the probability that loops or multiple edges are produced in a random d-regular graph. We use of the key facts from example sheet 1 is ( ) k (k 1)!! = k! k k! πk 4 k where (k 1)!! is the product of all odd positive integers in [k 1]. Theorem 4 Let d N. Then as n, the number of d-regular n-vertex graphs with dn even is asymptotic to ( exp( d 1 d d/ ) nn ) dn/. 4 e d/ d! Proof Let Z be the number of subgraphs with at most K vertices consisting of two cycles sharing at least one vertex (so the number of edges is at least K + 1). Then ( ) n K+ (dn K 3)!! n K K!d K+ (Kd)K+1 E(Z) < (K + 1)! d < <. K (dn 1)!! (dn K 1) K+1 n Therefore E(Z) 0 as n. In particular, as n, almost every d-regular graph on n vertices has no subgraphs on constantly many vertices with more edges than vertices. The number of cycles of length k in the complete graph on n vertices is ( ) n (k 1)!. Therefore k if X k is the number of cycles of length k in a random d-regular graph, ( ) n (k 1)! E(X k ) = d k (d 1) k (dn k 1)!! k (dn 1)!! ( = 1 dn k ) k dk (d 1) k (dn/ k)! (dn k)/ n (k) ) (dn/)! (d 1)k k := λ k. ( dn dn/ Furthermore, fixing r i N for i = 1,,..., k, if Y i = X i (X i 1)... (X i r i + 1) then Y 1 Y... Y k is the number of choices of r 1 + r + + r k distinct cycles in the random d- regular graph, where r i cycles of length i are chosen for i [k]. With probability tending to 1, none of these cycles share any vertices. Thus a similar computation to the one given above for E(X k ) shows E(Y 1 Y... Y k ) 6 k i=1 λ r i i
7 for each fixed sequence (r 1, r,..., r k ) N k. So considering just Y 1 and Y, we see P (X 1 = 0, X = 0) exp( λ 1 λ ) = exp( 1 (d 1) 1 4 (d 1) ) = exp( d 1 4 ). This is the proportion of the (dn 1)!!/d! n possible d-regular pseudographs on n vertices that have no loops and no multiple edges. Now by Stirling s Formula, as n, f(n, d) = (dn 1)!! ( d d/ ) nn dn/ d! n e d/ d! and combining this with the preceding computation gives the result. For instance, the number of labelled 3-regular graphs for even n is asymptotic to ( 7 ) nn 3n/ e 6e 3/ and the number of labelled -regular graphs for all n is asymptotic to ( n ) n. e 3/4 e This can be derived by an expansion around z = 1 of the exponential generating function Ψ(z) for labelled -regular graphs, which is Ψ(X) = exp( X/ X /4) 1 X. Note that z = 1 is the only pole, so we may use the meromorphic asymptotics theorem. In fact, the theorem above gives a lot more: it shows that the number of d-regular graphs with no cycles of length at most k is asymptotic to ( exp k j=1 ) (d 1) j f(n, d) j as n. In fact, the proof of the above theorem can be carried through for k log d 1 n as far as the requirement E(Z) 0 will go and so we obtain d-regular n-vertex graphs with no cycles of length less than about log d 1 n. Note that every d-regular graph has a cycle of length k the moment 1 + d + d(d 1) + + d(d 1) k 1 > n. This is called the Moore bound, and it shows that every d-regular n-vertex graph has a cycle of length at most log d 1 n. The best constructions have no cycles of length less than about 4 3 log d 1 n. The problem of determining whether there are d-regular n-vertex graphs for infinitely many n with no cycles of length less than c log d 1 n where c > 4/3 is a notorious and deep open problem with connections to many other areas of mathematics. 7
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