THE DEGREE DISTRIBUTION OF RANDOM PLANAR GRAPHS

Transcription

1 THE DEGREE DISTRIBUTION OF RANDOM PLANAR GRAPHS Michael Drmota joint work with Omer Giménez and Marc Noy Institut für Diskrete Mathematik und Geometrie TU Wien supported by the Austrian Science Foundation FWF, grant S th LL-Seminar on Graph Theory, Leoben, April 25, 2009

2 Contents Short history of random planar graphs Generating function for random planar graphs Asymptotic enumeration of planar graphs The degree distribution of random planar graphs Outerplanar graphs Series-parallel graphs

3 Random Planar Graphs History R n... labelled planar graphs with n vertices with uniform distribution X n... number of edges is a random planar graph with n vertices Denise, Vasconcellos, Welsh 1996) P{X n > 3 2 n} 1, P{X n < 5 2 n} 1. Note that 0 e 3n for all planar graphs.)

4 Random Planar Graphs History McDiarmid, Steger, Welsh 2005) P{H appears in R n at least αn times} 1 H... any fixed planar graph, α > 0 sufficiently small. H H H H...

5 Random Planar Graphs Consequences: P{There are αn vertices of degree k} 1 k > 0 a given integer, α > 0 sufficiently small. for some C > 1. P{There are C n automorphisms} 1

6 Random Planar Graphs Further Results: P{R n is connected} γ > 0 [McDiarmid+Reed] n... maximum degree in R n E n = Θlog n)

7 Random Planar Graphs The number of planar graphs [Bender, Gao, Wormald 2002)] b n... number of 2-connected labelled planar graphs b n c n 7 2 γ n 2 n!, γ 2 = [Gimenez+Noy 2005)] g n... number of all labelled planar graphs g n c n 7 2 γ n n!, γ =

8 Random Planar Graphs Precise distributional results [Gimenez+Noy 2005)] X n satisfies a central limit theorem: E X n n, V X n c n. P{ X n n > εn} e αε) n Connectedness: P{R n is connected} e ν = number of components of R n =: C n 1 + P oν).

9 Random Planar Graphs Degree Distribution Theorem [D.+Giménez+Noy] Let p n,k be the probability that a random node in a random planar graph R n has degree k. Then the limit p k := lim n p n,k exists. The probability generating function pw) = p k w k k 1 can be explicitly computed; p k c k 1 2q k for some c > 0 and 0 < q < 1. p 1 p 2 p 3 p 4 p 5 p

10 Random Planar Graphs Implicit equation for D 0 y, w): SD0 t 1) + t) 1 + D 0 = 1 + y w ) exp 43t + 1)D 0 + 1) ) D2 0 t4 12t t 9) + D 0 2t 4 + 6t 3 6t t 12) + t 4 + 6t 3 + 9t 2, 4t + 3)D 0 + 1)3t + 1) 1 + 2t where t = ty) satisfies y+1 = 1 + 3t)1 t) exp 1 ) t 2 1 t) t + 5t 2 ) t)1 + 2t)1 + 3t) 2 and S = D 0 t 1) + t)d 0 t 1) 3 + tt + 3) 2 ). Explicit expressions in terms of D 0 y, w) SEVERAL PAGES!!!!): B 0 y, w), B 2 y, w), B 3 y, w) Explict expression for pw): pw) = e B 01,w) B 0 1,1) B 2 1, w) + e B 01,w) B 0 1,1) 1 + B 21, 1) B 3 1, w) B 3 1, 1)

11 Random Planar Graphs Conjecture for maximum degree n n log n 1 log1/q) in probability and E n log n log1/q) where q = appear in the asymptotics of p k c k 1 2q k ; 1/ log1/q) =

12 Degree Distribution X k) n... number of vertices of degree k in a random labelled planar graph of size n p n,k... probability that a random vertex in a random labelled planar graph of size n has degree k ˆp n,k... probability that the root vertex in a random labelled vertex rooted planar graph of size n has degree k p n,k = ˆp n,k E X k) n = n p n,k

13 Degree Distribution Generating functions for counting planar graphs b n,m... number of 2-connected labelled planar graphs with n vertices and m edges, b n = m b n,m Bx, y) = x n b n,m n,m n! ym c n,m... number of connected labelled planar graphs with n vertices and m edges, c n = m c n,m Cx, y) = x n c n,m n,m n! ym g n,m... number of all labelled planar graphs with n vertices and m edges, g n = m g n,m Gx, y) = x n g n,m n,m n! ym

14 Degree Distribution Generating functions for counting planar graphs Gx, y) = exp Cx, y)), Cx, y) B Cx, y) = exp x, y x x x Bx, y) y Mx, D) 2x 2 D )) = x2 1 + Dx, y), y ) 1 + D = log xd2 1 + y 1 + xd, Mx, y) = x 2 y xy y U)2 1 + V ) U + V ) 3 U = xy1 + V ) 2, V = y1 + U) 2., ),

15 Degree Distribution Asymptotic enumeration of planar graphs b n = b ρ n 1 n 7 2n! 1 + O c n = c ρ n 2 n 7 2n! g n = g ρ n 2 n 7 2n! 1 + O 1 + O )) 1 n )) 1 n )) 1 n,, ρ 1 = , ρ 2 = , b = , c = , g =

16 Degree Distribution Generating functions for the degree distribution of planar graphs C = C x... GF for graphs with a root vertex which is not counted w... additional variable that counts the degree of the root vertex Generating functions: G x, y, w) C x, y, w) B x, y, w) T x, y, w) all rooted planar graphs connected rooted planar graphs 2-connected rooted planar graphs 3-connected rooted planar graphs Note that G x, y, 1) = G x, y) etc. x

17 Degree Distribution more precisely... g n,m,k... number of vertex rooted labelled planar graphs with n + 1 vertices, m edges, where the uncounted and unlabelled) root vertex has degree k. G x, y, w) = g x n n,m,k n,m,k n! ym w k k g n 1,m,k = n g n,m, p n,k = g n 1,m,k n g n,m k 1 p n,k w k = 1 n g n,m k 1 g n 1,m,k wk = [xn 1 ]G x, 1, w) [x n 1 ]G x, 1, 1) = pw) = k 1 p n w k = lim n [x n 1 ]G x, 1, w) [x n 1 ]G x, 1, 1)

18 Degree Distribution G x, y, w) = exp Cx, y, 1)) C x, y, w), C x, y, w) = exp B xc x, y, 1), y, w )), w B x, y, w) w = xyw exp Sx, y, w) + 1 x 2 Dx, y, w) T 1 Dx, y, w) = 1 + yw) exp Sx, y, w) + x 2 Dx, y, w) T Dx, y, w) x, Dx, y, 1), Dx, y, 1) Sx, y, w) = xdx, y, 1) Dx, y, w) Sx, y, w)), T x, y, w) = x2 y 2 w wy xy 1 u + 1) 2 w 1 u, v, w) + u w + 1) x, Dx, y, 1), )) 1 2wvw + u 2 + 2u + 1)1 + u + v) 3 ux, y) = xy1 + vx, y)) 2, vx, y) = y1 + ux, y)) 2. w 2 u, v, w) Dx, y, w) Dx, y, 1) ), ))

19 Degree Distribution with polynomials w 1 = w 1 u, v, w) and w 2 = w 2 u, v, w) given by w 1 = uvw 2 + w1 + 4v + 3uv 2 + 5v 2 + u 2 + 2u + 2v 3 + 3u 2 v + 7uv) + u + 1) 2 u + 2v v 2 ), w 2 =u 2 v 2 w 2 2wuv2u 2 v + 6uv + 2v 3 + 3uv 2 + 5v 2 + u 2 + 2u + 4v + 1) + u + 1) 2 u + 2v v 2 ) 2.

20 Planar Maps vs. Planar Graphs Whitney s Theorem Every 3-connected planar graph has a unique embedding into the plane. = The counting problem of rooted 3-connected planar maps is equivalent to the counting problem of rooted labelled) 3-connected planar graphs despite of a factor n 1)!) Furthermore, the counting problem of rooted 3-connected planar maps is equivalent to the counting problem of rooted simple quadrangulations.

21 3-Connected Maps 3-connected map simple quadrangulations

22 3-Connected Maps 3-connected map simple quadrangulations

23 3-Connected Maps 3-connected map simple quadrangulations

24 3-Connected Maps 3-connected map simple quadrangulations

25 3-Connected Maps 3-connected map simple quadrangulations

26 3-Connected Maps 3-connected map simple quadrangulations

27 3-Connected Maps q ijk... number of simple quadrangulations with i + 1 vertices of type 1 ), j + 1 vertices of type 2 ), and with root vertex of degree k + 1 Qx, y, w) = i,j,k q i,j,k x i y j w k Theorem [Mullin+Schellenberg, D+Gimenez+Noy] Qx, y, w) = xyw with wy x 1 ) UV W R, S, w) 1 + U + V ) 3

28 3-Connected Maps with algebraic function U = Ux, y), V = V x, y) given by U = xv + 1) 2, V = yu + 1) 2 and W U, V, w) = w 1U, V, w) + U w + 1) w 2 U, V, w) 2V + 1) 2 V w + U 2 + 2U + 1) with polynomials w 1 = w 1 U, V, w) and w 2 = w 2 U, V, w) given by w 1 = UV w 2 + w1 + 4V + 3UV 2 + 5V 2 + U 2 + 2U + 2V 3 + 3U 2 V + 7UV ) + U + 1) 2 U + 2V V 2 ), w 2 =U 2 V 2 w 2 2wUV 2U 2 V + 6UV + 2V 3 + 3UV 2 + 5V 2 + U 2 + 2U + 4V + 1) + U + 1) 2 U + 2V V 2 ) 2.

29 3-Connected Planar Graphs Corollary By Whitney s theorem: T x, y, w) = xw 2 Qxy, y, w).

30 2-Connected Planar Graphs Planar networks A network N is a multi-)graph with two distinguished vertices, called its poles usually labelled 0 and ) such that the multi-)graph ˆN obtained from N by adding an edge between the poles of N is 2- connected. Let M be a network and X = N e, e EM)) a system of networks indexed by the edge-set EM) of M. Then N = MX) is called the superposition with core M and components N e and is obtained by replacing all edges e EM) by the corresponding network N e and, of course, by identifying the poles of N e with the end vertices of e accordingly). A network N is called an h-network if it can be represented by N = MX), where the core M has the property that the graph ˆM obtained by adding an edge joining the poles is 3-connected and has at least 4 vertices. Similarly N = MX) is called a p-network if M consists of 2 or more edges that connect the poles, and it is called an s-network if M consists of 2 or more edges that connect the poles in series.

31 2-Connected Planar Graphs Planar networks Trakhtenbrot s canonical network decomposition theorem: any network with at least 2 edges belongs to exactly one of the 3 classes of h-, p- or s-networks. Furthermore, any h-network has a unique decomposition of the form N = MX), and a p-network or any s- network) can be uniquely decomposed into components which are not themselves p-networks or s-networks).

32 2-Connected Planar Graphs Planar networks Lwt Dx, y, w) and Sx, y, w), respectively, the GFs of planar) networks and series networks, with the same meaning for the variables x, y and w: Then by a variant of [Walsh 1982)] Dx, y, w) = 1 + yw) exp Sx, y, w) + Sx, y, w) = xdx, y, 1) Dx, y, w) Sx, y, w)), 1 x 2 Dx, y, w) T x, Ex, y), Dx, y, w) Dx, y, 1) )

33 2-Connected Planar Graphs A planar network with non-adjacent poles is obtained by distinguishing, orienting and then deleting any edge of an arbitrary 2-connected planar graph: w B x, y, w) w = xyw exp Sx, y, w) + 1 x 2 Dx, y, w) T x, Dx, y, 1), Dx, y, w) Dx, y, 1) ))

34 Connected Planar Graphs C x, y, w) = e B xc x,y,1),x,w) xc B B B xc xc xc xc xc xc

35 All Planar Graphs G x, y, w) = exp Cx, y, 1)) C x, y, w) C C C C C G

36 Asymptotics for Random Planar Graphs Functional equations Suppose that Ax, u) = Φx, u, Ax, u)), where Φx, u, a) has a power series expansion at 0, 0, 0) with non-negative coefficients and Φ aa x, u, a) 0. Let x 0 > 0, a 0 > 0 inside the region of convergence) satisfy the system of equations: a 0 = Φx 0, 1, a 0 ), 1 = Φ a x 0, 1, a 0 ). Then there exists analytic function gx, u), hx, u), and ρu) such that locally Ax, u) = gx, u) hx, u) 1 x ρu).

37 Asymptotics for Random Planar Graphs Asymptotics for coefficients Ax) = gx) hx) 1 x ρ + some technical conditions) = [x n ] Ax) = hρ) 2 π ρ n n O )) 1 n. Similarly: Ax, u) = gx, u) hx, u) 1 x ρu) + some technical conditions) = [x n ] Ax, u) = hρu), u) 2 π ρu) n n O )) 1 n.

38 Asymptotics for Random Planar Graphs Asymptotics for coefficients and Ax) = gx) + hx) 1 x ρ) α + some technical conditions) = [x n ] Ax) = hρ) Γ α) ρ n n α O 1 n )).

39 Asymptotics for Random Planar Graphs Singular expansion Ax) = gx) hx) 1 x ρ = g 0 + g 1 x ρ) + g 2 x ρ) 2 + ) + h 0 + h 1 x ρ) + h 2 x ρ) 2 + ) 1 x ρ = a 0 + a 1 1 x )1 2 + a2 1 x )2 2 + a3 1 x 2 + ρ ρ ρ)3 = a 0 + a 1 X + a 2 X 2 + a 3 X 3 + with X = 1 x ρ.

40 Asymptotics for Random Planar Graphs Ux, y) = xy1 + V x, y)) 2, V x, y) = y1 + Ux, y)) 2 = Ux, y) = xy1 + y1 + Ux, y)) 2 ) 2 = Ux, y) = gx, y) hx, y) = V x, y) = g 2 x, y) h 2 x, y) 1 y τx) 1 y τx) Mx, y) = x 2 y xy y U)2 1 + V ) 2 ) 1 + U + V ) 3!!! = Mx, y) = g 3 x, y) + h 3 x, y) 1 y τx) due to cancellation of the 1 y/τx)-term )3 2

41 Asymptotics for Random Planar Graphs Mx, D) 2x 2 D = log 1 + D 1 + y!!! = Dx, y) = g 4 x, y) + h 4 x, y) due to interaction of the singularities!!! ) xd2 1 + xd 1 x Ry) )3 2 Bx, y) y = x Dx, y), 1 + y!!! = Bx, y) = g 5 x, y) + h 5 x, y) = b n b R1) n n 7 2n! 1 x Ry) )5 2

42 Asymptotics for Random Planar Graphs B x, y) = g 6 x, y) + h 6 x, y) C x, y) = e B xc x,y),y),!!! = C x, y) = g 7 x, y) + h 7 x, y) due to interaction of the singularities!!! 1 x Ry) 1 x ry) )3 2, )3 2 = Cx, y) = g 8 x, y) + h 8 x, y) 1 x ry) )5 2 = c n c r1) n n 7 2n!

43 Asymptotics for Random Planar Graphs Cx, y) = g 8 x, y) + h 8 x, y) 1 x ry) = Gx, y) = e Cx,y) = g 9 x, y) + h 9 x, y) )5 2 1 x ry) )5 2. = g n g r1) n n 7 2n!

44 Asymptotic Degree Distribution 3-connected planar graphs T x, y, w) = x2 y 2 w wy xy 1 U + 1) 2 w 1 U, V, w) + U w + 1) 2wV w + U 2 + 2U + 1)1 + U + V ) 3 w 2 U, V, w) ), ũ 0 y) = y, ry) = ũ 0 y) y1 + y1 + ũ 0 y)) 2 ) 2, X = 1 x ry) = T x, y, w) = T 0 y, w) + T 2 y, w) X 2 + T 3 y, w) X 3 + O X 4 ) due to cancellation of the 1 x/rz)-term.

45 Asymptotic Degree Distribution Planar networks Dx, y, w) = 1 + yw) exp Sx, y, w) + T x, Dx, y, 1), 1 x 2 Dx, y, w) Sx, y, w) = xdx, y, 1) Dx, y, w) Sx, y, w)) Dx, y, w) Dx, y, 1) )) 1 τx)... inverse function of ry) DRy), y, 1) = τry)) X = 1 x Ry) = Dx, y, w) = D 0 y, w) + D 2 y, w)x 2 + D 3 y, w)x 3 + OX 4 ),

46 Asymptotic Degree Distribution 2-connected planar graphs w B x, y, w) w = xyw exp Sx, y, w) + 1 x 2 Dx, y, w) T x, Dx, y, 1), Dx, y, w) Dx, y, 1) )) = B x, y, w) = B 0 y, w) + B 2 y, w)x 2 + B 3 y, w)x 3 + OX 4 ) Remark. All these functions B j y, w) can be explicitly computed.

47 Asymptotic Degree Distribution connected planar graphs C x, 1, w) = exp B xc x), 1, w ))???

48 Asymptotic Degree Distribution Lemma fx) = n 0 a n x n n! = f 0 + f 2 X 2 + f 3 X 3 + OX 4 ), X = 1 x ρ, Hx, z, w) = h 0 x, w) + h 2 x, w)z 2 + h 3 x, w) Z 3 + OZ 4 ), Z = 1 z fρ), f H x) = Hx, fx), w) = n 0 b n w) xn n! = lim n b n w) a n = h 2ρ, w) f 0 + h 3ρ, w) f 3 f 2 f 0 ) 3/2.

49 Asymptotic Degree Distribution connected planar graphs C x, 1, w) = exp B xc x), 1, w )) Application of the lemma with and = pw) = lim n b n w) a n fx) = xc x) Hx, z, w) = xe B z,1,w). = e B 01,w) B 0 1,1) B 2 1, w) + e B 01,w) B 0 1,1) 1 + B 21, 1) B 3 1, w) B 3 1, 1)

50 Random Planar Graphs Classes of planar graphs Outerplanar graphs: all vertices are on the infinite face equivalently no K 4 and no K 2,3 as a minor). Series-parallel graphs: series-parellel extension of a tree or forest equivalently no K 4 as a minor). Planar graphs. no K 5 and no K 3,3 as a minor) Remark. outerplanar series-parallel planar

51 Outerplanar Graphs All vertices are on the infinite face.

52 Outerplanar Graphs Theorem 1 Let p n,k be the probability that a random node in a random 2-connected, connected or unrestricted outerplanar graph with n vertices has degree k. Then the limit p k := lim n p n,k exists. The probability generating function pw) = p k w k k 1 can be explicitly computed.

53 Outerplanar Graphs pw) = p k w k k 1 2-connected pw) = )w )w) 2 connected or unrestricted: pw) = c 1w 2 1 c 2 w) 2 exp with certain constants c 1, c 2, c 3, c 4 > 0). c 3 w + c 4w 2 1 c 2 w) )

54 Outerplanar Graphs Theorem 2 X k) n... number of vertices of degree k in random 2-connected, connected or unrestricted labelled outerplanar graphs with n vertices. = X k) n satisfies a central limit theorem with E X k) n µ k n and V X k) n σ 2 k n. Remark. µ k = p k.

55 Outerplanar Graphs Theorem 3 n... maximum degree in random 2-connected, connected or unrestricted labelled outerplanar graphs with n vertices. = n log n c in probability E n c log n, where c = 1/ log1/q) and 1/q in radius of convergence of pw).

56 Series-Parallel Graphs Series-parallel extension of a tree or forest Series-extension: Parallel-extension:

57 Series-Parallel Graphs Theorem 1 Let p n,k be the probability that a random node in a random 2-connected, connected or unrestricted series-parallel graph with n vertices has degree k. Then the limit p k := lim n p n,k exists. The probability generating function pw) = p k w k k 1 can be explicitly computed.

58 Series-Parallel Graphs We just mention the case of 2-connected series-parallel graphs pw) = k 1 p k w k : pw) = B 11, w) B 1 1, 1), where B 1 y, w) is given by the following procedure...

59 Series-Parallel Graphs E 0 y) 3 E 0 y) 1 = log 1 + E ) 2 0y) 1 + Ry) E 0y), 1 1/E0 y) 1 Ry) =, E 0 y) ) 2Ry)E 0 y) Ry)E 0 y)) E 1 y) =, 2Ry)E 0 y) + Ry) 2 E 0 y) 2 ) 2 + 2Ry)1 + Ry)E 0 y)) Ry)E 0y) 1+Ry)E D 0 y, w) = 1 + yw)e D0y,w) 0 y) 1, D 1 y, w) = 1 + D 0y, w)) Ry)E 1y)D 0 y,w) 1+Ry)E 0 y), D 0 y, w)) Ry)E 0y)D 0 y,w) 1+Ry)E 0 y) B 0 y, w) = Ry)D 0y, w) 1 + Ry)E 0 y) Ry)2 E 0 y)d 0 y, w) Ry)E 0 y)), B 1 y, w) = Ry)D 1y, w) 1 + Ry)E 0 y) Ry)2 E 0 y)d 0 y, w)d 1 y, w) 1 + Ry)E 0 y) Ry)2 E 1 y)d 0 y, w)1 + D 0 y, w)/2) 1 + Ry)E 0 y)) 2.

60 Series-Parallel Graphs Theorem 2 X k) n... number of vertices of degree k in random 2-connected, connected or unrestricted labelled series-parallel graphs with n vertices. = X k) n satisfies a central limit theorem with E X k) n µ k n and V X k) n σ 2 k n. Remark. µ k = p k.

61 Series-Parallel Graphs Theorem 3 n... maximum degree in random 2-connected, connected or unrestricted labelled series-parallel graphs with n vertices. = n log n c in probability E n c log n, where c = 1/ log1/q) and 1/q in radius of convergence of pw).

62 References Articles O. Giménez and M. Noy, Asymptotic enumeration and limit laws of planar graphs, J. Amer. Math. Soc. to appear). M. Drmota, O. Giménez, and M. Noy, The number of vertices of given degree in series-parallel graphs Random Structures and Algorithms to appear). M. Drmota, O. Giménez, and M. Noy, Degree distribution in random planar graphs, manuscript. M. Drmota, O. Giménez, and M. Noy, The maximum degree of planar graphs, draft.

63 General References Books Michael Drmota, Random Trees, Springer, Wien-New York, Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge University Press,

64 Thank You!