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On the degree distribution of random planar graphs Angelika Steger

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.

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On the degree distribution of random planar graphs Angelika Steger

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  1. On the degree distribution of random planar graphsAngelika Steger • (jointworkwithKonstantinos Panagiotou, SODA‘11) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

  2. Random Graphs fromClasseswithConstraints Motivation

  3. Classical Random Graph Theory Paul Erdős, Alfred Rényi On the evolution of random graphs Publ. Math. Inst. Int. Hungar. Acad. Sci., 1960 Given: a set of nvertices. Decideforeach potential edge randomly and independently whetheredgeispresent. edgeprobabilityp →randomgraphGn,p Key property: Independenceofedges.

  4. The Setup The obviousproblem: noindependence!  • Examples (classeswithconstraints): • Trees, Outerplanar Graphs, Planar Graphs, etc. • Generally: excluding a minor (or a fixedsubgraph) • Random Graph: • Thistalk: according to the uniform distribution • Typicalquestionsfor such a randomgraph: • Numberofedges? • DegreeSequence? [Numberofverticesofdegreeloglog(n)?] • Subgraphcount? • Evolution?

  5. Test Case: RandomPlanar Graphs Colin McDiarmid, AS, Dominic Welsh Random planar graphs Journal of Combinatorial Theory, Series B,  2005 Pn := setof all planargraphs on n (labelled) vertices Pn := graphdrawnrandomlyfromPn( → randomplanargraph)  c, C: 0 < c < Prob[Pnconnected] < C < 1

  6. Connectedness – ProofIdea # connectedplanargraphs on n vertices # planargraphs on n vertices Directapproach: Counting ... Prob[Pnconnected ] = We: rough, adhocmethods Giménez, Noy, 2009: |Pn| ≈p · n−7/2 · γn · n! wherep = 4.26094.. · 10−6 γ ≈ 27.2269.. |Cn| ≈c · n−7/2 · γn · n! wherec ≈ 4.10436.. · 10−6

  7. Techniques [in particular: Drmota, Giménez, Noy ...] • ”Classical” approach: • enumeration: count graphs with specific properties… • analytic combinatorics … • Lots of papers ... • This talk: • samplea graph • Boltzmann Sampling • analyze the construction during the execution of the algorithm • find andexploitindependencein theprobabilityspace

  8. Outline • 1) The Power of Independence • 2) Boltzmann Sampling • 3) Block Structure • 4) DegreeSequence

  9. Recall Azuma-Hoeffding : Ifforthereare s.t. then

  10. A General Decomposition Block Decomposition (of a connected graph)

  11. The Key Idea • Observation: cangenerate a graphwithgiven block structurebychoosingtheblocksindependently! • So weobtain a productprobabilityspace. • Condition on the block structure! Specify • Howmanyblocks of sizethereare, and • Howthey „touch“ eachother

  12. Outline • 1) The Power of Independence • 2) Boltzmann Sampling • 3) Block Structure • 4) DegreeSequence

  13. Generation of RandomObjects Duchon, Flajolet, Louchard, Schaeffer Boltzmann Samplers for the Random Generation of Combinatorial Structures Combinatorics, Probability and Computing, 2004

  14. BoltzmannSampler |G| = # of vertices of G G(x) := generatingfunctionforclassG An algorithmΓG (x) thatgenerates an elementG  Giscalled BoltzmannSampleriff • Observations: • Ifwe condition on |ΓG (x)| = n, thenΓG (x) is a uniform sampler. • Expectedsize of theoutputdepends on theparameter x:

  15. A SamplerforConnected Graphs forλCandμCappropriately (detailslater) ΓC (x): ... d ⟶ Po(λC) fori = 1,..., d: Bi ⟶ ΓB (μC) foru∈ Bi (excepttheroot) replaceuwithΓC(x) identifyroot of Biwithv (d1, d2 , …) (B1, B2, …) C ΓC (x) A BranchingProcess:

  16. WhyIsThisUseful ? ΓC (x) x=ρ (d1, d2 , …) (B1, B2, …) C Idea:properties of thesequences(d1, d2 , … , di , …) and (B1, B2 , … , Bi , …)that hold with „extremely high“ probability also hold for a randomobject The di‘sandtheBi‘saredrawnindependently! Underreasonableassumptions:

  17. WhyIsThisUseful (cont.) ? ΓC (ρ) (d1, d2 , …) (B1, B2, …) C • SupposethatthesamplerΓC (ρ)usedthevalues (d1, d2 , …, dn) and (B1, B2, …, Bm) to generateC • Byinspectingthesampler: • nisthe total number of vertices in C • msatisfies and (byChernoff)

  18. ΓC (ρ) (d1, d2 , …) (B1, B2, …) C Note: blocksare independent ... canapplye.g. Chernoffbounds E: ΓC (ρ) generates a graph on nvertices A: B: B1,..., BmsatisfypropertyP BΓ: blocks of ΓC (x) satisfypropertyP

  19. Summary ΓC (ρ) (d1, d2 , …) (B1, B2, …) C In order to bound itsuffices to bound where A: B: B1,..., BmsatisfypropertyP

  20. Outline • 1) The Power of Independence • 2) Boltzmann Sampling • 3) Block Structure • 4) DegreeSequence

  21. Nice Graph Classes [Norin, Seymour, Thomas, Wollan ‘06] • Letbetheset of biconnectedgraphs in • isniceif • Every lookslike • and are „small“: and • Examples: planar, outerplanar, minor-free, ...

  22. Block Structure Panagiotou, St. (SODA’09) Let C be a random graphfrom a ‘nice‘ class. Letbethesingularity of B(x). Thenthefollowingistruea.a.s. • If , then C has blocks of at most logarithmic size. • If , then • The largest block in Ccontainsvertices. • The second largest block containsvertices. • Thereare „many“ blocksthatcontainvertices.

  23. Complex Simple vs. Complex Simple „Plenty“ ofindependence A „lot“ ishidden in the large block e.g. outerplanargraphs, series-parallel graphs e.g. planargraphs

  24. Outline • 1) The Power of Independence • 2) Boltzmann Sampling • 3) Block Structure • 4) DegreeSequence

  25. SamplerConnected Graphs (λ1, λ2 , … ,λi , …) ΓC(x) C (B1, B2 , … ,Bi , …) List of parametersdistributedindep. according to Po(λC). List of vertexrootedbiconnectedgraphsdistr. indep. according to ΓB(μC).

  26. Subcriticalcase • Everyvertexisbornwith a certaindegree • Itthenreceives a certainnumber of newneighbors –indep. of itsbirthdegree pk-l = P [ receivek-lmoreneighborslater ] dl = P[ bornwithdegreel] Intuitively: • innervertexofbiconnectedcomponent • Poissonmanycopiesof arootof a biconnectedcomponent

  27. CriticalCase • Everyvertexisbornwith a certaindegree • Itthenreceives a certainnumber of newneighbors –indep. of itsbirthdegree Intuitively: large component „remainder“

  28. 2-connected ⟶ connected Panagiotou, St. (SODA’11) For‘nice‘ graphclasseswehave: if then wherek0‘(n)andc(.)depend on k0(n)respb(.)

  29. 3-connected ⟶ 2-connected Panagiotou, St. (SODA’11) For‘nice‘ graphclasseswehave: if then wherek0‘(n)andb(.)depend on k0(n)respt(.)

  30. Summary Bernasconi, Panagiotou, St (‘08): - degreesequenceofrandomdissections Bernasconi, Panagiotou, St (‘09): - degreesequenceofseries-parallel graphs Johannsen, Panagiotou (’10): - degreesequenceof3-connected planargraphs Panagiotou, St. (’11): - degreesequenceofplanargraphs Note: similarresultswereobtain (using different methods) byDrmota, Giménez, Noy ...

  31. Work in Progress Maximum degreeof a randomplanargraph: Reed, McDiarmid (`08): θ(log n) Boltzmann samplerapproach: ∃ a vertexofdegree ≧ (1- ε) c log n analyticcombinatoricsapproach: ∄ a vertexofdegree≦ (1+ ε) c log n

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