On the degree distribution of random planar graphs

Angelika Steger

(joint work with Konstantinos Panagiotou, SODA‘11)

Random Graphs from

Classes with Constraints

Motivation

Classical Random Graph Theory

Paul Erdős, Alfred RényiOn the evolution of random graphsPubl. Math. Inst. Int. Hungar. Acad. Sci., 1960

Given: a set of n vertices.Decide for each potential edge randomly and independently whether edge is present.

edge probability p → random graph Gn,p

Key property: Independence of edges.

The Setup

• Examples (classes with constraints):– Trees, Outerplanar Graphs, Planar Graphs, etc.– Generally: excluding a minor (or a fixed subgraph)

• Random Graph: – This talk: according to the uniform distribution

• Typical questions for such a random graph:– Number of edges?– Degree Sequence? [Number of vertices of degree loglog(n)?]– Subgraph count?– Evolution?

The obvious problem: no independence!

Test Case: Random Planar Graphs

Colin McDiarmid, AS, Dominic WelshRandom planar graphsJournal of Combinatorial Theory, Series B, 2005

c, C: 0 < c < Prob[Pn connected ] < C < 1

Pn := set of all planar graphs on n (labelled) vertices

Pn := graph drawn randomly from Pn ( → random planar graph)

Connectedness – Proof Idea

Direct approach: Counting ...

Prob[Pn connected ] =

We: rough, adhoc methods

Giménez, Noy, 2009:

|Pn| ≈ p · n−7/2 · γn · n! where p = 4.26094.. · 10−6

γ ≈ 27.2269.. |Cn| ≈ c · n−7/2 · γn · n! where c ≈ 4.10436.. · 10−6

# connected planar graphs on n vertices

# planar graphs on n vertices

Techniques

• ”Classical” approach:– enumeration: count graphs with specific properties…– analytic combinatorics …– Lots of papers ...

• This talk:– sample a graph

• Boltzmann Sampling• analyze the construction during the execution of the

algorithm– find and exploit independence in the probability

space

[in particular: Drmota, Giménez, Noy ...]

Outline

• 1) The Power of Independence

• 2) Boltzmann Sampling

• 3) Block Structure

• 4) Degree Sequence

Recall

Azuma-Hoeffding :

If for there are s.t.

then

A General Decomposition

Block Decomposition (of a connected graph)

The Key Idea

• Condition on the block structure! Specify– How many blocks of size there are, and– How they „touch“ each other

• Observation: can generate a graph with given block structure by choosing the blocks independently!

• So we obtain a product probability space.

Outline

• 1) The Power of Independence

• 2) Boltzmann Sampling

• 3) Block Structure

• 4) Degree Sequence

Generation of Random Objects

Duchon, Flajolet, Louchard, SchaefferBoltzmann Samplers for the Random Generation of Combinatorial StructuresCombinatorics, Probability and Computing, 2004

Boltzmann Sampler

Observations:

• If we condition on |ΓG (x)| = n, then ΓG (x) is a uniform sampler.

• Expected size of the output depends on the parameter x:

An algorithm ΓG (x) that generates an element G G is called Boltzmann Sampler iff

|G| = # of vertices of G

G(x) := generating function for class G

A Sampler for Connected Graphs

A Branching Process:

...

C(d1, d2 , …)(B1, B2, … )

ΓC (x):

ΓC (x)

d Po(λ⟶ C)for i = 1,..., d: Bi ⟶ ΓB (μC)

for u ∈Bi (except the root) replace u with ΓC (x)

identify root of Bi with v

for λC and μC

appropriately(details later)

Why Is This Useful ?

• The di‘s and the Bi‘s are drawn independently!• Under reasonable assumptions:

C(d1, d2 , …)(B1, B2, … )

ΓC (x)

Idea: properties of the sequences (d1, d2 , … , di , …) and (B1, B2 , … , Bi , …) that hold with „extremely high“ probability also hold for a random object

x=ρ

Why Is This Useful (cont.) ?

• Suppose that the sampler ΓC (ρ) used the values (d1, d2 , …, dn) and (B1, B2, …, Bm) to generate C• By inspecting the sampler:

- n is the total number of vertices in C- m satisfies and (by Chernoff)

C(d1, d2 , …)(B1, B2, … )

ΓC (ρ)

E: ΓC (ρ) generates a graph on n verticesA:B: B1,..., Bm satisfy property PBΓ: blocks of ΓC (x) satisfy property P

C(d1, d2 , …)(B1, B2, … )

ΓC (ρ)

Note: blocks are independent ... can apply e.g. Chernoff bounds

Summary

In order to bound

it suffices to bound

where A:B: B1,..., Bm satisfy property P

C(d1, d2 , …)(B1, B2, … )

ΓC (ρ)

Outline

• 1) The Power of Independence

• 2) Boltzmann Sampling

• 3) Block Structure

• 4) Degree Sequence

• Let be the set of biconnected graphs in • is nice if– Every looks like

– and are „small“:

and

• Examples: planar, outerplanar, minor-free, ...

Nice Graph Classes

[Norin, Seymour, Thomas, Wollan ‘06]

Panagiotou, St. (SODA’09)Let C be a random graph from a ‘nice‘ class. Let be the singularity of B(x). Then the following is true a.a.s.– If , then

C has blocks of at most logarithmic size.– If , then• The largest block in C contains

vertices.• The second largest block contains vertices.• There are „many“ blocks that contain vertices.

Block Structure

Simple Complex

Simple vs. Complex

„Plenty“ of independence

A „lot“ is hidden in the large block

e.g. outerplanar graphs, series-parallel graphs e.g. planar graphs

Outline

• 1) The Power of Independence

• 2) Boltzmann Sampling

• 3) Block Structure

• 4) Degree Sequence

Sampler Connected Graphs

ΓC (x) C(λ1, λ2 , … , λi , …)(B1, B2 , … , Bi , …)

List of parameters distributed indep. according to Po(λC).

List of vertex rooted biconnected graphs distr. indep. according to ΓB(μC).

Subcritical case

Intuitively:

Every vertex is born with a certain degree It then receives a certain number of new

neighbors – indep. of its birth degree

dl = P[ born with degree l]pk-l = P [ receive k-l more neighbors later ]

inner vertex of biconnected component Poisson many copies of a root of a biconnected component

Critical Case

Intuitively:

Every vertex is born with a certain degree It then receives a certain number of new

neighbors – indep. of its birth degree

large component „remainder“

2-connected connected⟶

Panagiotou, St. (SODA’11)For ‘nice‘ graph classes we have: if

then

where k0‘(n) and c(.) depend on k0(n) resp b(.)

3-connected 2-connected⟶

Panagiotou, St. (SODA’11)For ‘nice‘ graph classes we have: if

then

where k0‘(n) and b(.) depend on k0(n) resp t(.)

Summary

Bernasconi, Panagiotou, St (‘08): - degree sequence of random dissectionsBernasconi, Panagiotou, St (‘09): - degree sequence of series-parallel graphsJohannsen, Panagiotou (’10): - degree sequence of 3-connected planar graphsPanagiotou, St. (’11): - degree sequence of planar graphs

Note: similar results were obtain (using different methods) by Drmota, Giménez, Noy ...

Work in Progress

Maximum degree of a random planar graph:

Reed, McDiarmid (`08): θ(log n)

Boltzmann sampler approach: ∃ a vertex of degree (1- ε) c log n≧

analytic combinatorics approach: ∄ a vertex of degree (1+ ε) c log n ≦