on the degree distribution of random planar graphs angelika steger
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On the degree distribution of random planar graphs Angelika Steger. (j oint work with Konstantinos Panagiotou , SODA‘11 ) . TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A. Random Graphs from Classes with Constraints. Motivation. - PowerPoint PPT PresentationTRANSCRIPT
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 ≦