The Evolution of Random Graphs—The Giant Component

TRANSACTIONS OF THE

AMERICAN MATHEMATICAL SOCIETY

Volume 286, Number 1, November 1984

THE EVOLUTION OF RANDOM GRAPHS

BY

BéLA BOLLOBáS1

ABSTRACT. According to a fundamental result of Erd?s and Rényi, the struc-

ture of a random graph Gm changes suddenly when M ~ n/2: if M = [cnj

and c < k then a.e. random graph of order n and since M is such that its

largest component has O(logn) vertices, but for c > ^ a.e. G m has a giant

component: a component of order (1 — a c + o(l))n where ac < 1. The aim of

this paper is to examine in detail the structure of a random graph G m when

M is close to n/2. Among others it is proved that if M = n/2 + s, s = o(n)

and s > (log nj'^n2/3 then the giant component has (4 + o(l))s vertices.

Furthermore, rather precise estimates are given for the order of the rth largest

component for every fixed r.

1. Introduction. Let n be a natural number and set N = (￡) and V = {1,2,... ,n}. A graph process on V is a sequence {Gt)o such that (i) each Gt is a graph on V with í edges, and (ii) G0 C Gi C ? ?? C Gyv- Let § be the set of all N\ graph processes. Turn Q into a probability space by giving all members of it the same probability and write G for random elements of Q. Furthermore, call Gt the state of the process G = {Gt)o at time t. Clearly a (random) graph process is a Markov chain whose states are graphs on V. This Markov chain is a model of the evolution of a random graph (r.g.) with vertex set V.

The evolution of random graphs was first studied by Erd?s and Rényi [5-7]. They investigated the least values of t for which certain properties are likely to appear, i.e. they studied the stage of the evolution of a r.g. at which a given prop-erty first appears. Erd?s and Rényi proved the surprising fact that most properties studied in graph theory appear rather suddenly: there are functions ti(n) < í2(n) rather close to each other such that almost no Gt, has the property and almost every Gtl has the property. (As customary in the theory of random graphs, the term 'almost every' (a.e.) means 'with probability tending to 1 as n —> oo'.) Perhaps the most interesting results of Erd?s and Rényi concern the sudden change in the structure of Gt around t = n/2.

They proved that if t ~ cn for some constant c, 0 < c < 1/2, then a.e. Gt is such that its largest component has O(logn) vertices: if t ~ cn and c > 1/2 then the largest component of a.e. Gt has (1 — q c + o(l))n vertices, where 0 < ac < 1; and if t = [n/2\ then the maximal size of a component of a.e. Gt has order n2'3. (In fact, Erd?s and Rényi [6, 7] asserted the last statement for t ~ n/2 but, as we

Received by the editors December 3, 1982 and, in revised form, August 15, 1983.

1980 M athematics Subject (Classification. Primary 05C99, 05C30; Secondary 60J99, 62P99.

1 R esearch supported by NSF Grant MCS-8104854.

?1984 American Mathematical Society

0002-9947/84 $1.00 + $ 25 per page

257

258BELA BOLLOBAS

shall see, that is not true.) Furthermore,

1 ~ k-k-i

it=i

so ace2c —> 1 as c —? oo.

The aim of this note is to prove considerably more precise results about the structure of Gt, especially when t is close to n/2. Among others, our results will shed light on the curious double jump described above.

Loosely speaking, we shall show that a.e. G is such that for t > n/2 + (logn)1/2n2/3 the graph Gt has a unique component of order at least n2/3 (the giant component of Gt) and all other components have fewer than n2/3/2 vertices. Furthermore, if n01 < s < n"2, where 2/3 < ?i < ?2 < 1, then Gn/2-s and Gn/2+s have remarkably similar structures. A.e. Gn/2-s has all its vertices on components of order at most n2 (log n)/s and so has Gn/2+s, except for its vertices on its giant component.

The giant component of G…/2+s has (4 + o(l))s vertices. Most vertices are on small components which are trees. The distributions of these tree-components of Gn/2-s and Gn/2+s are vei7 similar, except in Gn/2+s there are about 1 — 2s/n2 times as many of them as in Gn/2-s. As an easy corollary of our results, for t ~ cn, c > 1/2, we obtain precise information about the distribution of the order of the rth largest component of Gt for every fixed r.

2. Definitions and basic facts. In addition to graph processes we shall con-sider two other models of random graphs. The space $(n, M) consists of all graphs with M = M(n) edges and with vertex set V = {1,2,...,n}; the elements of ?(n,M) are equiprobable. The model $(n,p) consists of all 2N graphs with ver-tex set V, in which the probability of a graph with m edges is pmqN~m, where q — í — p. Thus ?(n,p) consists of all graphs with vertex set V in which the edges are chosen independently and with the same probability p = p(n), 0 < p < 1. For basic properties of these models see [1, Chapter 7 and 3]; for undefined terminology in graph theory see [2]. As customary, we shall talk of random graphs G m and Gp, meaning that we consider elements of Q(n,M) and Q(n,p). Note that the prob-ability that a r.g. Gm has Q is the same as the probability that a graph process G = {Gt)Q is such that Gt has Q for t = M. Therefore no confusion will arise from the two slightly different meanings of the symbol Gm- We say that almost every G m (or Gp) has a property Q if the probability that G m (or Gp) has Q tends to 1 as n —? oo. If M is close to pN then the models §(n,M) and ${n,p) are virtually interchangeable. In particular, if pqN —? oo, Q is a convex property (that is if F c G C H and F and H have Q then so does G) and a.e. Gp has Q then so does a.e. Gm, provided \M - pn\ = 0(pqN)1^2. These remarks imply that most of our results could be formulated for any of our models; nevertheless, there are advantages in considering all three.

Denote by C{k,d) the number of connected labelled graphs with k vertices and k + d edges. Thus C(k,d) = 0 if d < 2, C(k, -1) is the number of labelled trees of order k, so C{k, -1) = kk~2, and C(k,0) is the number of connected unicyclic graphs. For d > 0 the function C(k,d) is less simple. It was proved by Katz [9]

THE EVOLUTION OF RANDOM GRAPHS259 and Rényi [10] (and it is also a consequence of more general formulae in [8]) that

r=3j=l ^ '

For d > 1 Wright [11-13] proved a number of results about C(k,d). He showed that for d = oik1'3)

(2) C{k,d) = fakd+^d-^2{l + 0(d3/2 /fc)},

where f? depends only on d. In particular, /o = (tt/8)1/2, /i = 5/24 and /2 = 5\/2/128. We shall not need exact estimates in the vein (2), but we shall need a bound on C(k, d) which is valid for all values of d. To be precise, we need the following inequality from [4]: for every K > 0 there is a constant cc, = cq(K) such that

(3) C(k,d)

for every d, — 1 < d < (2) — fc. In fact, we could manage with considerably cruder bounds than (3) but the calculations become more pleasant if fck+f3?'-1)/2 is multiplied by a factor tending to 0 as d —? oo, rather than by one increasing with d. We could also use the fact that C(fc, d) is long concave as a function of d. This was proved by Odlyzko, answering a question of mine, by making use of the proof in [13].

A component of a graph is said to be a (k,d)-component if it has fc vertices and fc +d edges. We denote by X(k,d) the number of (k,d)-components of a random graph. Note that in §{n,p) the expectation of X(k,d) is

(4) Ep(X(k, d)) = (f\ C(k, d)pk+d(l - p)k(n-k)+(i)-k-d

In particular, the expected number of tree-components of Gp is

(5) Ep(X(k,-l)) = ^V-V-1(l-p)fcn~fc2/2~3fc/2+1-

For the sake of convenience we shall omit the integrality signs throughout the paper. It is easily seen that the validity of the arguments will remain unaffected. Furthermore, our inequalities are asserted to hold if n is sufficiently large. Finally, Ci,c2,... denote positive constants.

3. Gaps in the sequences of components. The key result in proving the sud-den emergence of the giant component and in estimating its order rather precisely is that from shortly after time n/2 most graph processes never have a component of order between n2/3/2 and n2/3. The restriction t < 2n/3 in the result below is only for the sake of convenience, it can be easily removed.

THEOREM 1. Let s0 = (f logn)1/2n2/3 and t0 = n/2 + s0. Then a.e. graph process G — (Gt)o° is such that ifto n2/3.

PROOF. Throughout the proof we shall assume that n2/3/2 < fc < n2/3. De-note by Et(k) the expected number of components of order fc in Gt: Et(k) =

260 BéLA BOLLOBáS

J2{Et(X{k,d)):

-1 < d < (*) - fc}- In order to prove our theorem it suffices to

show that 2n/3 n*'3

?2 ￡

Et(k) = 0(i).

t=t0 fc=n2/3/2For d > (j) - fc w e have X(k, d) = 0 and for fc + d < t and -1 < d < (k) - fc t he expected number of (fc, d )-components of Gt is

(6) WMt-^IMJ/^J/^).

Indeed, having chosen the k + d edges of a (fc, d )-component, the remaining t — k — d edges have to be chosen from a set of (n^k) edges. Set

lOfc ￡d=-l

E't(k) = J2 Et{X{k + d)) and E't'(k) = Et{k) = E't{k).We shall estimate separately the sums of the ?￡{(fc)'s and E't'(k)'s. In the first case we shall make use of (3), but in the second case it will suffice to bound C(k,d)by the total number of graphs with fc labelled vertices and k + d edges. In both cases the initial difficulty is that (6) is considerably more unpleasant than (4) with p = t/N.

(i) Suppose \n2>3 < fc < n2/3, t0 < t < 2n/3 and -1 < d < lOfc. Then

(7)

( W \;(N\ {t)k+d{n~2^-k-d

yt-k-d)'\tj

(N)t k+d ' (k + d)2\(n2k)t-k-d

s?(?r-{JHrt-ís^

(fcn - fc2/2)2

2A"2

, k+d ( 1_1.2

*C3 {h)k+d exp {-^sr^*+2

- ^r- -2kH/n2}( t \k+d f kn-k2/2 )

^C3U) eXP{-A^}-The last inequality holds since for fixed values of n, fc and d the minimum of (fc 4- d)2/2t + 2k2t/n2 is attained at t = (fc + d)n/2fc and the minimum is exactly 2(fc + d)fc/n.

THE EVOLUTION OF RANDOM GRAPHS 261

Relations (3), (6) and (7) imply that

lOfc / . N fc+d

Setting s = t - n/2 and e = 2a/n (so that t/N is well approximated by (1 + e)/n and e < ~) we find that

{fc2 ￡2 e3 fc2 efc21

-— +k + ek-—k+—k-k+--efc + -— >

2n 2 3 2n 2n J

< cenfc-^expí-^íl - e)fc| = c6nfc-5/2exp i~s2 (l - *\ fc/n2 j. Consequently

? 4(fc) < c7n-2/3exp{-iS2 (l - ??) …-V3J =nJ/3/2 ^ V n / J

2 -2 n a

fc=n3/3/2

and so

n/6 r,2/3

￡ ￡ f;i(fc)

.=aofc=…V3/2 I ? \ n J )

< ci0n2/3(logn)-3/2exp |-| (^ logn) J

-o(l).

(ii) Suppose ±n2/3 < fc

( (V) W"W n V^'^w'^w

t-k-dj' \tj - \t-k-d) '\tj- \NJ -\n)

and

Therefore by (6)

?***(?)'(Ar

er o'^-

This implies that ￡"(fc) = o(n-3) and so

2n/3 n2/3

? ￡ ￡?(*)= oí?'1).

t=t0 fc=n2/3/2

262 BELA BOLLOBAS

Let us call a component small if it has fewer than n2//3/2 vertices and large if it has more than n2//3 vertices. Theorem 1 states that a.e. graph process is such that for to < t < 2n/3 every component of Gt is either small or large. This will enable us to prove that a.e. graph process such that shortly after time n/2 it has a unique large component (a 'giant' component), and all other components are small. In the sequel we shall need a fairly good upper bound for the variance of the number of small components.

THEOREM 2. Suppose A C N x {NU {0,-1}}, -1 < e = e(n) < n/4 and p= (l + e)/n. Set

Xt = ￡{fclX(fc,d): (fc,d) G A}, m = Ep(Xl)

and k = max{fc: (fc,d) e A}. If -I < e < -(1 -I- e)2/n then

o2(Xi)

PROOF. By relation (4)

Ep(Xt) = Yl fcl(^G(fc,d)pfc+d(l-P)fc("-fc)+^)-^.

(fc,d)€A ^ '

Also, if (fci,di) and (fc2,d2) are distinct elements of A then

n\ (n — k\

￡p(X(fci,d1)(fc2,d2))=(fc )( fc2 MG(fci,di)G(fc2,d2)

(8) .pfci+fcs+di+djQ _p\(k?+k2)(n-kl-k2) + (kl?k2)-k1-k2-dí-d2

= Ep(x(ki,di))Ep(x(k2,d2))/")fc;+t2 (i-p)"fclfca-

(?)fcl(")fc2

Similarly, for (fc, d) € A,

^2^ ^ p /vit jh a xr tvtt, jh* (n)2* n _ ^-*a

(9) Ep(X(k, d)2) < ￡p(X(fc, d)) + ￡p(X(fc, d))a,_\/"v (1 - PY

(?)k(?)*

Relations (8) and (9) give

u+^Uki

ep(x2) < Ep(Xi) + > k\k2E(X(k1,di))E(x(k2,d2))

(10)

?/w+?' (l-p)-fclfca: (fc1,di),(fc2,d2)€A

(n)fc.(n)fc2 J Since 1 — y< (1 - x)eI-y whenever 0 < x < y < 1, we find that

(n)

< exp ^ ^ (-j > = exp{-fc!fc2/n}.

THE EVOLUTION OF RANDOM GRAPHS 263 Furthermore,

, 1 + e / 1 + e (1 + e)2)

1 _ p = i-> exp {-— - i-s-i- \ ,

n I n n J

so by (11)

Wfc.+fc? (1 i-fc.fc, < í fclfc2 | (! + ￡)fcifca ! (l + g)2Mfrl

(?)fci(?)fc2 I n n n2 J

ffcifc2 / (1 + ￡)2\1 , C/I , ,

= expj -i-i í￡ + --'—) > = l + <5(fci,fc2).

Now if ￡ + (1 + ￡)2/n < 0 then ?(fci, f c2) < 0 and so by (10)

o-2(Xt) < Ep(X2t) = p2l.

Finally, if -(1 + e)2/n < e and fc2(￡ +(1 + e)2/n) < n then

? fci,fc2 < —— l￡+-ZT-)-

n \ n /

Therefore (10) gives

cr2(Xt) < E(X2t) + (| + 2(1nh2￡)2) ^i^^^W*??*))*^*!?**))!

(fci,di),(fc2,d2)GA} /2￡ 2(l + ￡)2\ 2

Armed with Theorem 2, we can locate the maximal order of a component of Gp having fewer than n2^3 vertices, provided p is too close to the critical value 1/n. As for p > 1/n this will turn out to be the order of the second largest component, we denote it by S(GP):

S(GP) = max{fc: fc =0 or fc < n2'3 and Gp has a component of order fc}.

Theorem 3. Let -\ < e = e(n) <\,p = (\ + e)/n and define

g￡{k) =logn - - logfc + fc(log(l + ￡)-￡) + 21og(l/￡).

Let fco =fcn(n) and fc2 =fc2(n) < n2^3 be such that

?e(ko) —? oo and ge{k2) -? -co.

Then a.e. Gp is such that S(GP) < fc2 and i/n|￡|3(logn)~2 —> oo then a.e. Gp is such that fc0

PROOF, (i) As in the proof of Theorem 1, for fc

￡ Ep(X(k,d))

d>-l

264 BéLA BOLLOBáS

Hence the expected number of components of Gp with order between fc2 and n2^3 is at most

…2/3

c, ￡i?p(X(fc,-l))

k=k?

n2/3 , , fcn-fc2/2

< c2n ￡ fc~5/2 exP{fc -fc2/2n}(l + ￡)fc I 1 - — )

k=k3 \ n '

?2/3

< c3n ]T fc_5/2exp{fc(log(l +e) - ￡)}

k=k2

< c^nk^ exp{fc2(log(l + ￡) - ￡)}￡~2

= c4exp{i/￡(fc2)}.

By our choice of fc2 w e have g￡(fc2) —> — oo so almost no Gp has a component whose order is between fc2 a nd n2!3.

(ii) In the proof of the second inequality we may and shall assume that n|￡|3(logn)~2 —> oo and fco —? oo. Set ke =■- [8(logn)￡_2J, A = {(fc,-1): fco < fc < fc￡} and let Xr, be as in Theorem 2. Then g€(ke) —* —oo and

k k

p0 = E(X0) = ? E(Tk) ~ JL ? ^5/2 exp{fc(log(l + e) - ￡)}?

fc = fco fc = fco

Clearly fc￡ > fco +￡2 so by the choice of fco w e have po —* oo. Furthermore, we may suppose that p0 does not grow too fast, say po = o(n|￡|3(logn)~2). Then

po,ek2/n = O(/x0￡(logn)2￡"4n_1) = o(l).

Theorem 2 implies that

cr2(X0) zp\/n < p0 + 3￡/igfc2/n = /?0(1 + o(l)),

so by Chebyshev's inequality P(X > 0) —? 1. D

Let us state some explicit bounds for S(GP) implied by Theorem 3.

COROLLARY 4. Let p = (1 + e)/n.

(i)IfO<￡<\ is fixed then a.e. Gp satisfies S(GP) < 3(logn)￡-2,

(ii) If n '° < ￡ — o((logn)_1) ond w(n) —? oo then

\S(GP) - (2 logn + 6 log￡ - 5 loglogn)￡-2| < u(n)e~2

for a.e. Gp.

PROOF, (i) All we have to check is that g￡(3(logn)￡-2) —? -oo.

(ii) Straightforward calculations show that

fc0 =(2 logn + 6 log￡ - 5 loglogn - uj(n))e~2

THE EVOLUTION OF RANDOM GRAPHS265 and

k2 — (2 log n -I- 6 log ￡ - 5 log log n + u>(n))e~2

satisfy the conditions in Theorem 3. G

The corollary above enables us to prove an analog of Theorem 1 for t > |n.

THEOREM 5. A.e. graph process G = {Gt)o is such that for t > 5n/8 the graph Gt has no small component of order at least 100 logn.

PROOF. By Corollary 4 a.e. graph process is such that for some t satisfying 3n/5 < t < 5n/8 the graph Gt has no small component whose order is at least 75 logn. Since the union of two components of order at most a = [100 logn] has order at most 2a < n2/3, the assertion of the theorem will follow if we show that a.e. graph process is such that for t > 3n/5 the graph Gt has no component whose order is at least a and at most 2a. Furthermore, since for t > 2n logn a.e. Gt is connected, it suffices to prove this for t < 2n logn.

Let |n < í < 2n logn and set c = 2t/n. Then the expected number of compo-nents of Gt having order at least a and at most 2a is

Sc.gM^l)'-'^)'-'

2a

fc=a

^n^V^ce1-)",

fc=a

where Co,ci and c2 are absolute constants. Since for c = 1.2 we have c — 1 — l oge > 0.0176, the last expression is at most

canOogn)-3/^"1-7 = o{n~1'2). D

4. The emergence of the giant component. Given a graph process G —(Gt)o°, denote by wt the number of components of Gt and let

ft

V=[jUt(Gt)

t=i

be the partition of V into vertex sets of its components. Note that for every t either Gt and Gt+i define the same partition of V or else wt+i = wt — 1 and the partition defined by Gt is a refinement of the partition defined by Gt+i- Hence if G is such that for t > t0 the graph Gt does not have a component whose order is between n2!3 ?2 and n2/3 then for tr?

266BéLA BOLLOBáS

Moreover, if in such a G the graph Gt has a component which contains all large components of Gto then Gt has a unique component of order at least n2/3 and so has Gf for every f > 1. Therefore in order to establish the existence of the giant component, we have to show that the large components of Gt0 are contained in a single component of Gt for some t > to- Furthermore, we have to estimate the number of vertices on the giant component or, equivalently, the number of vertices on the small components. We start with the second task. For the sake of convenience we take ￡ — p n — 1 = n '.

THEOREM 6. Let 0 < -y < |, ￡ = n"1, p = (l + e)/n ond w(n) -? 00. For a graph G, denote by Yi(G) the number of vertices on the small (fc, d)-components with d > 1, by Yr,{G) the number of vertices on the small unicyclic components and by V_i(G) the number of vertices on the small tree-components. Then a.e. Gp is such that

Yi < uj(n)n5~<-1, Y0 < u{n)n2~<

and

F_i = n - 2nx^ + 0{u(n)nil+"l)/2 + (logn)n1"2^).

PROOF. Set fc3 =9(logn)n21. Then for fc3

￡2fc/2 - fc￡3/3 - fc2￡/(2n) > ￡2fc/3 > 3 logn,

so

n2'3 ("2') in2'3

Y, ￡ E(kX(k,d)) = O i Y, E(kX(k,-l))

k=fc3d=-l \k=k3

= 0\nY k'3/2 exp {-￡2fc/2 + fc￡3/3 + fc2￡/(2n)}

= Oln-2 Y k~3/A =o(n-2).

V k=k3 J

This shows that in our proof we may replace Y? by

Yi= ￡fcX(M), i = -1,0,1-

fc

(i) Note first that by (3)

\fcn-fc2/2

E(Yi) = O l Y k(nk)C^ l)Pk+1(l -V?

0\n-x ￡fc3/2exp{-￡2fc/2}

k

since for 1 < fc < fc3 w e have

fc￡3/3 + fc2￡/(2n) =0(l).

THE EVOLUTION OF RANDOM GRAPHS 267

Hence

￡(y1) = 0(n-1￡-5) = 0(n5'?-1),

implying the first assertion.

(ii) The expectation of Fo is easily estimated:

E(F0)- Y fc￡(*(M))

k

fcn-fc2/2+3fc/2

-L^^

exp{-fc￡2/2}

U°°exp{"fc

?a

Therefore, by Chebyshev's inequality, F0 < ui(n)n21 for a.e. random graph Gp.

(iii) We shall estimate E(Y-i) rather precisely and then we shall make use of Theorem 2 to conclude that for a.e. Gp the variable Y_i is rather close to its expectation.

Set p' = (1 — ￡)/n and write E' for the expectation in ${n,p'). We shall exploit the fact that E and E' are rather closely related.

First of all, calculations analogous to those in (i) and (ii) show that

E'{n-Y0-Y-i)=o(n2'>)

and E'{Y0) ~ ±n2^.

Hence E!(Y-i) = n - \n* - o{n2"i).

In order to pass from ￡"(F_i) to E[Y-i), note that for fc < fc3 w e have

/. . \ fc-l /, z, , \ / \ fcn-fc2/2+3fe/2-l E,x(*,-1,,Wx(t,-i… = (i±|) (!={￡$)

= exp {2(fc - 1)￡ - (2fcn - fc2)￡/n + 0((logn)n-2"')}

= exp { -2e + fc2￡/n + 0((logn)n-2^)}

= 1 - 2￡ + 0(fc2￡/n) + 0((logn)n-2^).

Consequently

(12)￡(F_i) - (1 - 2￡)E'(Y-i) + OUlognjn1-2^)

= n - 2n1""' + 0{n2-<) + 0({logn)nx-2~<)

+ 0(￡^fc1/2exp{-fc￡2/2})

= n - 2n^ + 0(n2^ + (logn)n1-2^).

268BéLA BOLLOBáS

Finally, in order to apply Theorem 2 we have to estimate the following expectation:

E ( Y k2X(k,-\) j = O I n Y k~1/2 e xp{-fc￡2/2} j- Oin1^).

\k

By Theorem 2 we have

(13) |F_i - ￡(Y_i)| < w(n)n<1+^/2.

Relations (12) and (13) imply that

Y-i = n - 2n1-'1 + 0(u)(n)n(l+~

as claimed. D

Let us establish now the emergence of the giant component shortly after time n/2.

THEOREM 7. A.e. graph process G = (Gt)o' is such that for every t > t\ = n/2 + (logn)1/2n2/3 the graph Gt has a unique component of order at least n2/3. The other components of Gt have at most n2/3/2 vertices each.

PROOF. As before, we call a component small if it has fewer than n2/,3/2 ver-tices, and large if it has at least n2/3 vertices. By Theorem 1 a.e. graph process G is such that for t > to — n/2 + so every component of Gt is either small or large, where to is as defined in Theorem 1. Let Ci, C2,..., C? be the large components of Gt0 in such a G and suppose in some Gt, t > to, all the Cj's are contained in the same component. Then this component of Gt is the unique large component and for V > t the graph Gf has also a unique large component. Indeed, as t increases from io, a vertex x GV can become a vertex of a large component only if that com-ponent contains a C?, for the union of two small components contains fewer than n2/3 vertices. Consequently our theorem will follow if we show that a.e. G is such that for some t between to and ti all large components of Gto are contained in the same component of Gt. Imitating the proof of Theorem 6, one can show that a.e. G is such that Gto has at least 2ao > (logn)1/2^/3 vertices on its large components. (In fact, the expected number of these vertices is about 4ao.) Therefore one can find disjoint subsets Vi, V2,..., Vm of V(Gto) = V such that

m>(logn)1/2/2, |Vi|>n2/3, ?=l,...,m,

each Vi is contained in some V(Cj) and

m (

\JVi~\JViCj),

t=l j=l

where Ci,..., C? are the large components of Gto.

Set p — n~4/3 and denote by Hp the random graph obtained from Gto by adding to it edges independently and with probability p. Then a.e. Hp has at most to + n2/3 < ?i edges so it suffices to show that (J^i Vj is contained in a single component in a.e. Hp.

THE EVOLUTION OF RANDOM GRAPHS 269 What is the probability that for a given pair {i,j), 1 < i < j < m, some edge of Hp joins Vi to Vj? It is clearly at least

l-(l-p)n"/3 >l-e-x > 1/2.

Hence the probability that in Hp all V?'s are contained in the same component is at least the probability that an element of G(m, 5) is connected. Since m —? 00, this probability tends to 1. (See [5 or 2, p. 140] for considerably stronger results.) This completes our proof. D

Combining Theorems 3, 6 and 7, we obtain rather precise information about the orders of the largest components. Since the property of having at most x vertices on the small tree components is monotone, and so are the properties of having at least y vertices on the large components and at most z vertices on the small components which are trees or unicyclic graphs, using [2, Theorem 8, p. 133] we may pass from the model §(n,p) to the model §(n, M). Denote by LT{G) the rth largest order of a component of a graph G.

THEOREM 8. Let 0 < 7 < |, a = ?nl~~< and t = n/2 + s. Then for every m G N and w(n) —+00 a.e. Gt is such that

Li(G) = 4a + 0{u>{n)n/sx/2 + (logn)a2/n)

and

fco

where fco and fc2 are as in Theorem 3. D

Before extending the range of t in the theorem above, we investigate the distri-bution of the small components of Gp in the case when p is not as close to 1/n as has been required so far.

5. Components of order less than n2//3. First we consider the small com-ponents of Gp with p < 1/n.

THEOREM 9. Let 0 < e = e(n) = o(l) be such that ￡nn —* c x> for every fixed n > 0 and set p = (1 — ￡)/n. Given A G R+, choose l\ = l\(n) in such a way that p(n,￡,lx) = (2/7r)1/2nZA-5/2((l -e)e')h￡-7 - A

and denote by Z = Z(GP) the number of components of Gp having at least l\ vertices. Then the distribution of Z tends to P\, the Poisson distribution with mean A.

PROOF. The assumptions imply easily that l\ —? 00, l\ = o(nn) for every ? > 0 and the expected number of vertices on components containing cycles is bounded. Hence we may assume that Z is the number of tree-components of order at least l\. Since also l\￡2 —> 00, a trite calculation shows that

e(z) = y Ewk> -1)) ~ 4= ? fc-5'2?1 - ￡y)k

?si ^27r f etr,

i;5/2((l - ￡)Ofc/(l - (1 - ￡)e￡) ~ p(n, ￡, /A) ~ A.

270BéLA BOLLOBáS

Furthermore, by making use of l\ = o(nn) it is easily shown that for every r € N the rth factorial moment

Er{Z) = E(Z{Z-l)---(Z-r + l))

converges to Ar. Hence Z —> P\, as claimed. D

COROLLARY 10. Let ￡, p and l\ be as in Theorem 9, and let ui(n) —? oo. Then for every m G N a.e. Gp satisfies

h - w(n)/e < Lm{Gp) < ??< Li(Gp)

PROOF. It is easily seen that for every fixed A we have

lx ~ (logn)/(e-log(l/(l-e))) ~2(logn)/￡ and /i-w(n)/e < l\ < li+u(n)/e. Hence the assertion follows from Theorem 9. D

If p = c/n for some constant c < 1 then we need not be able to find numbers l\ = l\{n) ensuring that p(n, 1 —c, l\) —> A. Nevertheless, with some simple changes the results above carry over to this case without any difficulty. In fact, our task is easier since the tree-components we have to consider have only O(logn) vertices. Thus one arrives at the following result of Erd?s and Rényi [6, p. 49].

If 0 < c < 1 is a constant, p = c/n, a — c — 1 — l oge,

fco =- < l og n - - log log n - l0 > G N and l0 = 0(1)

then the number of components of Gp with at least fco v ertices has asymptotically Poisson distribution with mean

1 ?5/

2 in

cy/2^ 1 - e-°

Though the simple method of means is sufficient to prove this assertion, we would like to point out that recently Barbour [1] applied a more sophisticated and powerful method to prove that the number of certain components has asymptotically Poisson distribution.

Let us turn to the case p = (1 + ￡)/n > 1/n. The proof of Theorem 6 is easily adapted to show that if ￡ = o(l) then the distribution of the number of components of order less than n2/3 in Gp is almost the same as the distribution in GP' with p' = (l-￡)/n. Furthermore, it is easily seen that ￡ can be rather small for a slightly weaker version of Theorem 6 to remain valid. As in Theorem 8, we state the result for Gt rather than Gp.

THEOREM 11. Lett = n/2 + a, a = o(n), an"2/3(logn)2 -* oo, ￡ = 2a/n and for A > 0 choose l\ = l\{n) in such a way that

(2/7r)1/2n/;5/2((l - ￡)e￡)^￡-2 -? A.

Then for w(n) —? oo and m G N a.e. Gt is such that

Li(Gt) = (4 + o(l))a,

h - w(n)/e < Lm(Gt) < < L2(Gt) < h + w(n)/e,

Li(G) = (l + o(l))/i, t = 2,...,m.

If En11 -> oo for every n > 0 then Z* -i Pa, where Z* = max{m- 1: Lm > l\}. D

THE EVOLUTION OF RANDOM GRAPHS271

6. The small components after time n/2. As t increases, our task of locating the orders of the components of G? becomes easier. Indeed, if c > 1 is a constant and t — cn/2 = o(n) then the expected number of vertices on components containing cycles and having fewer than n2/3 vertices is bounded and so is the expected number of vertices on components whose order is between ci logn and n2//3. Hence if u>(n) —? oo then a.e. Gt is such that with the exception of uj(n) vertices all vertices belong to the giant component or else to trees of order at most ci logn. Now the expectation of the number of vertices on these trees is about C2n and by a slight variant of Theorem 2 the variance is 0(n(logn)2). Therefore we are led to the following result.

THEOREM 12. Let c > 1 be a constant and let t = [cn/2j, w(n) —? oo. Then a.e. Gt is such that, with the exception of at most w(n) vertices, all vertices of Gt belong to the giant component or to components which are trees. Furthermore,

Li(Gt)-n 1-TV(C^

< ?jj(n)nx'2 logn

c^ fc!

fc=i

and if ko = ?{logn- § loglogn-/o} G N, where a = c- 1 -loge and Iq = 0(1), then Z* = max{m— 1: Lm(Gt) > fco} bas asymptotically Poisson distribution with mean

A~——-el°. D

cs/2?l-e~a

In the range of t covered by Theorem 11 the giant component increases about four times as fast as t. Another way of proving Theorem 12 would be to establish this fact first and then use it to deduce the assertion about the size of the giant component. What is the expectation of the increase of Li(Gt) as í changes to t + 1? The probability that the (t + l)st edge will join the giant component to a component of order Lj is about LiLj/Q). Hence the expectation is about

n2/3 / x /. x fc-1 / , , x fcn-fc2/2

n2/3

~(2L1/n2)^-^=fc1/2exp{-￡2fc/2}

fc=i v

~(2V/a(Ll/n) ?°° xxl2e~^l2dx

= (^y (Li/n)r Q) {￡2/2)-x'2 = 2Li/{￡n) = LJs.

From this one can deduce that if t is o(n) but not too small then Li(i) =?i(n/2 + a) ~ Cia. By considering the crude order of Li(Gt) as t ceases to be o(n), one can show easily that the constant ci is 4.

Using Theorem 2, for t > (l + ￡)n/2 it is easy to obtain fairly precise information about the distribution of the orders of the components of Gt- We shall do a little more than that: we shall prove some results about all graphs G( of a graph process after time (1 + ￡)n/2. Let us start with a rather crude result.

272BêLA BOLLOBáS

THEOREM 13. Let ￡ > 0 be fixed and for c0 = c0(n) > I + ￡ set ci = 3/(co — 1 — l og Co). Then a.e. graph process G = (Gt)o ?S such that for t > con/2 the graph Gt does not contain a component whose order is between ci logn and n2'3.

PROOF. Let co < c < 3 logn and set p = c/n. By inequality (3) the expected number of components of Gp whose order fc satisfies fci = [ci log nj < fc < k2 = [n2/3\ is at most

-oix)t(t)^&-'(i-T'k)

k=ki v '

= o(n)cex~c ' = o(n~2).

Since the property of containing a component of order fc is convex and a.e. graph process becomes connected by time n log n, this implies our assertion. D THEOREM 14. (i) Let c0 > 1 be fixed. Then a.e. G is such that for t > c0n/2 the graph Gt does not contain a (fc, d)-component with d > 1 ond fc < n2?3'.

(ii) For w(n) -?oo a.e. G is such that for t > w(n)n every component of Gt, with the exception of its giant component is a tree.

PROOF. Since for t > n log n a.e. Gt is connected, it suffices to restrict our attention to the range con/2 < t < n logn. Furthermore, by Theorem 11 it suffices to consider components of order at most fci = [ci lognj, where

ci =3/(c0 - 1 - logc0).

Note that the expected number of (fc, d)-components with 4 < fc < fci and d > 1 a graph process contans between times to = [con/2J and ?i = [2n lognj is at most

t=t0k=4 v/d>lv/ N/ N '

On the other hand, it is easily seen (cf. Theorem 9c of [6]) that for to < t < ti = [n lognj and 1 < fc < fci the life-time of a component of order fc in Gt has approximately exponential distribution with mean n/(2fc). In particular, a component of order fc in Gt will be a component of Gt+i,Gt+2,... ,Gt+i with probability at least \, where I = [n/(3fci)J. Consequently the probability that G is such that there is a time t with (0 < t < tx for which Gt has a (fc, d)-component with 4 < fc < fci and d > 1 is at most

0(l)(2//) = 0((logn)/n).

This proves (i). Assertion (ii) is proved analogously. D

Let fc = O(logn), A = {(fc,d): 1 < fc < fc, d = —1 o r 0}, and define the Xt as in Theorem 2. Then for every fixed i there is a constant d, such that for

THE EVOLUTION OF RANDOM GRAPHS 273

n/2

?? - *?<*?> s?1 (:y-2+- (§r (? - s )""~" s *■"-*and, using a slight variant of Theorem 2,

4i a2(Xi) < p2x + —?2+i/n < d,p2l.

m Consequently by Chebyshev's inequality

(14) Pt(\Xi(Gt)-K\>u)<^.

This relation enables us to deduce uniform bounds for the X?. Here we state only a rather simple result about wt = w(Gt), the number of components of Gt.

THEOREM 15. Suppose 0 < ￡, w(n) —> oo and c(n) = logn-u;(n)

—? oo. Then

a.e. G is such that for every t satisfying n <2t < c(n)n we have (15)

\wt - n?{2t/n)\ < ￡n?{2t/n),

where i ~ kk-2

PROOF. Let n >0 and set tj = L(l+jn)n/2j, j = 0,1,... ,m = [c(n)/nj. Put fc = [(3 logn)/(n - log(l + n))] and let A and Xi be as above. Then a.e. G is such that for tj < í < tm we have wt = w(Gt) = Xo(Gt) +1. This implies that it suffices to estimate Ao(Gt) instead of w(Gt).

It is easily seen that if n > 0 is sufficiently small then for n/2 < t' < t < c(n)n/2

(16)

\Et,{Xo)-n?(2t/n)\<￡-n?2t/n

and (17) \?(2t/n)-?(2t'/n)\<-t?(2t/n).

25d0 . 50d0 _, ■< —=— n eJ '.E2Note that ?(c) > ex~c. Hence by (14) and (16)

Ptj(|X0(GtJ-^(Xo)|>^tj(Xo))<^^

Since X!7Li e3T} — °(l)i a-e- G is such that

(18) \X0(Gtj) - Etj(X0)\ < 6-Et](X0)

for every j = l,...,m. A.e. G is such that u;t+i(G) < wt(G) if t > ii so a.e. G is such that Xo = Xo(Gt) is a monotone decreasing function of t for t > t?. Therefore relations (16), (17) and (18) imply that a.e. G satisfies (15) if ti < t < c(n)n/2.

Finally, if n is sufficiently small then a.e. G is such that \wt - n/2| < ￡n/5 and \?(2t/n) - §| < e/5 whenever n/2 < t < ij, so (15) holds in this range as well.

274BêLA BOLLOBáS

In Theorem 15 the approximation of Wt becomes more precise as t grows. From inequality (14) one can also obtain approximations to the same degree for every value of t. For example, it is easy to show that if w(n) —? oo then G is such that

fci

MG)-n4t"

k=\k\

2_/2t

\n

I -e-2t/"

for every t > n/2.

References

< u;(n)n2/3

10. ii.

12.

13.A. D. Barbour, Poisson convergence and random gmphs, Math. Proc. Cambridge Philos. Soc.

92 (1982), 349-359.

B. Bollobas, Graph theory—An introductory course, Graduate Texts in Math., Springer-Verlag, New York, Heidelberg and Berlin, 1979.

_, Random graphs, Combinatorics (H. N. V. Temperley, ed.), London Math. Soc. Lecture Notes Series, vol. 52, Cambridge Univ. Press, New York, 1981, pp. 80-102.

_, The evolution of sparse gmphs, Graph Theory and Combinatorics (B. Bollobas, ed.), Academic Press, London, 1984.

P. Erd?s and A. Rényi, On random g raphs. I, Publ. Math. Debrecen 6 (1959), 290-297.

_, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sei. 5 (1960), 17-61. _, Onthe evolution o f random graphs, Bull. Inst. Internat. Statist. Tokyo 38 (1961), 343-347.

C. W. Ford and G. E. Uhlenbeck, Combinatorial problems in the theory of graphs. Proc. Nat. Acad. Sei. U.S.A. 43 (1957), 163-167.

L. Katz, Probability of maecomposabikty of a random mapping function, Ann. Math. Statist. 26 (1955), 512-517.

A. Rényi, On connected g mphs. I, Publ. Math. Inst. Hungar. Acad. Sei. 4 (1959), 385-387.

E. M. Wright, Asymptotic enumeration of connected graphs, Proc. Roy. Soc. Edinburgh Sect. A

68 (1968/69), 298-308.

_, The number of connected sparsely-edged graphs. II: Smooth graphs and blocks, J. Graph Theo.-y 2 (1978), 299-305.

_, The number of connected sparsely-edged graphs. Ill: Asymptotic results, J. Graph Theory 4 (1980), 393-407.

Department of Mathematics, Louisiana State University, Baton Rouge, louisiana 70803

实验5 JAVA常用类

山西大学计算机与信息技术学院 实验报告 姓名学号专业班级 课程名称 Java实验实验日期成绩指导教师批改日期 实验5 JAVA常用类实验名称 一．实验目的： （1）掌握常用的String，StringBuffer(StringBuilder)类的构造方法的使用；（2）掌握字符串的比较方法，尤其equals方法和==比较的区别； （3）掌握String类常用方法的使用； （4）掌握字符串与字符数组和byte数组之间的转换方法； （5）Date，Math，PrintWriter，Scanner类的常用方法。 二．实验内容 1.二进制数转换为十六进制数（此程序参考例题249页9. 2.13） 程序源代码 import java.util.*; public class BinToHexConversion{ //二进制转化为十六进制的方法 public static String binToHex(String bin){ int temp; //二进制转化为十六进制的位数 if(bin.length()%4==0) temp = bin.length()/4; else temp = bin.length()/4 + 1; char []hex = new char[temp]; //十六进制数的字符形式 int []hexDec = new int[temp];//十六进制数的十进制数形式 int j = 0; for(int i=0;i=0&&dec<10) return (char)('0'+dec-0); else if(dec>=10&&dec<=15) return (char)('A'+dec-10); else return '@'; }

尊重的素材

尊重的素材（为人处世） 思路 人与人之间只有互相尊重才能友好相处 要让别人尊重自己，首先自己得尊重自己 尊重能减少人与人之间的摩擦 尊重需要理解和宽容 尊重也应坚持原则 尊重能促进社会成员之间的沟通 尊重别人的劳动成果 尊重能巩固友谊 尊重会使合作更愉快 和谐的社会需要彼此间的尊重 名言 施与人，但不要使对方有受施的感觉。帮助人，但给予对方最高的尊重。这是助人的艺术，也是仁爱的情操。—刘墉 卑己而尊人是不好的，尊己而卑人也是不好的。———徐特立 知道他自己尊严的人，他就完全不能尊重别人的尊严。———席勒 真正伟大的人是不压制人也不受人压制的。———纪伯伦 草木是靠着上天的雨露滋长的，但是它们也敢仰望穹苍。———莎士比亚 尊重别人，才能让人尊敬。———笛卡尔 谁自尊，谁就会得到尊重。———巴尔扎克 人应尊敬他自己，并应自视能配得上最高尚的东西。———黑格尔 对人不尊敬，首先就是对自己的不尊敬。———惠特曼

每当人们不尊重我们时，我们总被深深激怒。然而在内心深处，没有一个人十分尊重自己。———马克·吐温 忍辱偷生的人，绝不会受人尊重。———高乃依 敬人者，人恒敬之。———《孟子》 人必自敬，然后人敬之；人必自侮，然后人侮之。———扬雄 不知自爱反是自害。———郑善夫 仁者必敬人。———《荀子》 君子贵人而贱己，先人而后己。———《礼记》 尊严是人类灵魂中不可糟蹋的东西。———古斯曼 对一个人的尊重要达到他所希望的程度，那是困难的。———沃夫格纳 经典素材 1元和200元 （尊重劳动成果） 香港大富豪李嘉诚在下车时不慎将一元钱掉入车下，随即屈身去拾，旁边一服务生看到了，上前帮他拾起了一元钱。李嘉诚收起一元钱后，给了服务生200元酬金。 这里面其实包含了钱以外的价值观念。李嘉诚虽然巨富，但生活俭朴，从不挥霍浪费。他深知亿万资产，都是一元一元挣来的。钱币在他眼中已抽象为一种劳动，而劳动已成为他最重要的生存方式，他的所有财富，都是靠每天20小时以上的劳动堆积起来的。200元酬金，实际上是对劳动的尊重和报答，是不能用金钱衡量的。 富兰克林借书解怨 （尊重别人赢得朋友）

三角函数,反三角函数公式大全

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那一刻我感受到了幸福_初中作文

那一刻我感受到了幸福 本文是关于初中作文的那一刻我感受到了幸福，感谢您的阅读！ 每个人民的心中都有一粒幸福的种子，当它拥有了雨水的滋润和阳光的沐浴，它就会绽放出最美丽的姿态。那一刻，我们都能够闻到幸福的芬芳，我们都能够感受到幸福的存在。 在寒假期间，我偶然在授索电视频道，发现(百家讲坛)栏目中大学教授正在解密幸福，顿然引起我的好奇心，我放下了手中的遥控器，静静地坐在电视前，注视着频道上的每一个字，甚至用笔急速记在了笔记本上。我还记得，那位大学教授讲到了一个故事：一位母亲被公司升职到外国工作，这位母亲虽然十分高兴，但却又十分无奈，因为她的儿子马上要面临中考了，她不能撇下儿子迎接中考的挑战，于是她决定拒绝这了份高薪的工作，当有人问她为什么放弃这么好的机会时，她却毫无遗憾地说，纵然我能给予儿子最贵的礼物，优异的生活环境，但我却无当给予他关键时刻的那份呵护与关爱，或许以后的一切会证明我的选择是正确的。听完这样一段故事，我心中有种说不出的感觉，刹那间，我仿拂感觉那身边正在包饺子的妈妈，屋里正在睡觉的爸爸，桌前正在看小说的妹妹给我带来了一种温馨，幸福感觉。正如教授所说的那种解密幸福。就要选择一个明确的目标，确定自已追求的是什么，或许那时我还不能完全诠释幸福。 当幸福悄悄向我走来时，我已慢慢明白，懂得珍惜了。 那一天的那一刻对我来说太重要了，原本以为出差在外的父母早已忘了我的生日，只有妹妹整日算着日子。我在耳边唠叨个不停，没想到当日我失落地回到家中时，以为心中并不在乎生日，可是眼前的一切，让我心中涌现的喜悦，脸上露出的微笑证明我是在乎的。

爸爸唱的英文生日快乐歌虽然不是很动听，但爸爸对我的那份爱我听得很清楚，妈妈为我做的长寿面，我细细的品尝，吃出了爱的味道。妹妹急忙让我许下三个愿望，嘴里不停的唠叨：我知道你的三个愿望是什么?我问：为什么呀!我们是一家人，心连心呀!她高兴的说。 那一刻我才真正解开幸福的密码，感受到了真正的幸福，以前我无法理解幸福，即使身边有够多的幸福也不懂得欣赏，不懂得珍惜，只想拥有更好更贵的，其实幸福比物质更珍贵。 那一刻的幸福就是爱的升华，许多时候能让我们感悟幸福不是名利，物质。而是在血管里涌动着的，漫过心底的爱。 也许每一个人生的那一刻，就是我们幸运的降临在一个温馨的家庭中，而不是降临在孤独的角落里。 家的感觉就是幸福的感觉，幸福一直都存在于我们的身边!

JAVA中常用类的常用方法

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反正割 反余割 运算公式 余角关系 2 arccos sin arc π = +x x 2 cot tan arc π =+x arc x 2 csc ec a π = +x arc x rcs 负数关系 x x sin arc )sin(arc -=- x x rc arccos )cos(a -=-π x x tan arc )tan(arc -=- x rc x c cot a )(ot arc -=-π

x rc x sec a )(arcsec -=-π x arc x c sec )(sc arc -=- 倒数关系 x arc x csc )1 arcsin(= x arc x sec )1 arccos(= x arc x arc x cot 2cot )1arctan(-==π x x x arc arctan 23arctan )1cot(-=+=ππ x x arc arccos )1 sec(= x x arc arcsin )1 csc(= 三角函数关系

加减法公式 1. ) 10,0()11arcsin(arcsin arcsin ) 10,0()11arcsin(arcsin arcsin ) 10()11arcsin(arcsin arcsin 22222 2 222222>+<<-+---=+>+>>-+--=+≤+≤-+-=+y x y x x y y x y x y x y x x y y x y x y x xy x y y x y x ，，或ππ 2. ) 10,0()11arcsin(arcsin arcsin ) 10,0()11arcsin(arcsin arcsin ) 10()11arcsin(arcsin arcsin 22222 2 222222>+><-----=->+<>----=-≤+≥---=-y x y x x y y x y x y x y x x y y x y x y x xy x y y x y x ，，或ππ 3. ) 0() 11arccos(2arccos arccos ) 0() 11arccos(arccos arccos 2 2 22<+----=+≥+---=+y x x y xy y x y x x y xy y x π 4. ) () 11arccos(arccos arccos ) () 11arccos(arccos arccos 2 2 22y x x y xy y x y x x y xy y x <--+=-≥--+-=- 5. ) 1,0(1arctan arctan arctan ) 1,0(1arctan arctan arctan ) 1(1arctan arctan arctan ><-++-=+>>-++=+<-+=+xy x xy y x y x xy x xy y x y x xy xy y x y x ππ

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小学生作文《感悟幸福》范文五篇汇总

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常用反三角函数公式表

反三角函数公式

反三角函数图像与特征 1 ：

反三角函数的定义域与主值范围 式中n为任意整数.

反三角函数的相互关系 sin x = x-x3/3!+x5/5!-...(-1)k-1*x2k-1/(2k-1)!+... (-∞= -1 And x < -0.5 Then ArcSin = -Atn(Sqr(1 - x * x) / x) - 2 * Atn(1) If x >= -0.5 And x <= 0.5 Then ArcSin = Atn(x / Sqr(1 - x * x))

If x > 0.5 And x <= 1 Then ArcSin = -Atn(Sqr(1 - x * x) / x) + 2 * Atn(1) End Function ArcCos(x) 函数 功能：返回一个指定数的反余弦值，以弧度表示，返回类型为Double。 语法：ArcCos(x)。 说明：其中，x的取值范围为[-1，1]，x的数据类型为Double。 程序代码： Function ArcCos(x As Double) As Double If x >= -1 And x < -0.5 Then ArcCos = Atn(Sqr(1 - x *x) / x) + 4 * Atn(1) If x >= -0.5 And x <= 0.5 Then ArcCos = -Atn(x/ Sqr(1 - x * x)) + 2 * Atn(1) If x> 0.5 And x <= 1 Then ArcCos = Atn(Sqr(1 - x*x) / x) End Function

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大学高数 函数与反三角函数图像

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sin cos), a x b x x? +=+其中tan b a ?=，?所在的象限与点(,) a b所在的象限一 致。

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