Split digraphs and their applications M. Drew LaMar The College of - - PowerPoint PPT Presentation

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Split digraphs and their applications M. Drew LaMar The College of William and Mary Quantitative Biology Laboratory Department of Biology National Institute of Standards and Technology Gaithersburg, MD Tuesday, June 19, 2012 Tuesday, July

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Split digraphs and their applications

  • M. Drew LaMar

The College of William and Mary Quantitative Biology Laboratory Department of Biology

National Institute of Standards and Technology Gaithersburg, MD Tuesday, June 19, 2012

1 Tuesday, July 3, 12

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Simple graphs and degree sequences

Undirected Directed

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Split graphs

A graph is split if it can be partitioned into a clique and an independent set:

  • Subset of perfect graphs
  • Superset of threshold graphs
  • Only chordal graphs whose complements are also

chordal

  • (2K2,C4,C5)-free
  • A graph is split if and only if its degree sequence

satisfies a particular Erdős-Gallai inequality with equality

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Graphic sequences

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Graphic sequences

Slack sequence

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Characterizations of split graphs

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Characterizations of split graphs

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Characterizations of split graphs

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Permutations:

Digraphic sequences

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Permutations: Lexicographic ordering (pref. to out-degree)

Digraphic sequences

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Permutations: Lexicographic ordering (pref. to out-degree) Lexicographic ordering (pref. to in-degree)

Digraphic sequences

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Permutations: Example: Lexicographic ordering (pref. to out-degree) Lexicographic ordering (pref. to in-degree)

Digraphic sequences

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Digraphic sequences

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Digraphic sequences

Slack sequences

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Structural characterization of split digraphs

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Structural characterization of split digraphs

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Structural characterization of split digraphs

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Structural characterization of split digraphs

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Structural characterization of split digraphs

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Structural characterization of split digraphs

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Structural characterization of split digraphs

X± X0 X+ X−

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Structural characterization of split digraphs

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Splittance of an undirected graph

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Splittance of an undirected graph

1 2 3 4 5

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1 2 3 4 5

Splittance of an undirected graph

1 2 3 4 5

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1 2 3 4 5

Splittance of an undirected graph

1 2 3 4 5

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1 2 3 4 5

Splittance of an undirected graph

1 2 3 4 5 1 2 3 4 5

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1 2 3 4 5

I I

Splittance of an undirected graph

1 2 3 4 5 1 2 3 4 5

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1 2 3 4 5 1 2 3 4 5

Splittance of an undirected graph

1 2 3 4 5

I C I C

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1 2 3 4 5

Splittance of an undirected graph

1 2 3 4 5 1 2 3 4 5

C I C I

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1 2 3 4 5

Splittance of an undirected graph

1 2 3 4 5 1 2 3 4 5

C I C I

NOT SPLIT!

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Splittance of a directed graph

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

X X

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

14 Tuesday, July 3, 12

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

14 Tuesday, July 3, 12

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

14 Tuesday, July 3, 12

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

16 Tuesday, July 3, 12

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

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Splittance of a directed graph

x1 x4 x2 x5 x3

1 2 3 4 5

SPLIT!

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Splittance of a directed graph - directed extensions

Undirected

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Splittance of a directed graph - directed extensions

Undirected Directed

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Splittance of a directed graph - directed extensions

Undirected Directed

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Splittance of a directed graph - directed extensions

Undirected Directed

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Splittance of a directed graph - directed extensions

Undirected Directed

NOT SPLIT! SPLIT!

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Canonical split decomposition

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Canonical split decomposition

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Canonical split decomposition of digraphs?

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Canonical split decomposition of digraphs?

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Realizing graphic and digraphic sequences

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Realizing graphic and digraphic sequences

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Uniform sampling algorithms Random walks

Random walk on set of realizations:

2-switch 3-cycle reorientation

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Uniform sampling algorithms Random walks

Example: Random walk on set of realizations:

2-switch 3-cycle reorientation

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Uniform sampling algorithms Random walks

Example: Random walk on set of realizations:

2-switch 3-cycle reorientation

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Uniform sampling algorithms Random walks

Example: Random walk on set of realizations:

2-switch 3-cycle reorientation

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Uniform sampling algorithms Random walks

Example: Random walk on set of realizations:

2-switch 3-cycle reorientation

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Uniform sampling algorithms Random walks

Example: Random walk on set of realizations:

2-switch 3-cycle reorientation

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Uniform sampling algorithms Random walks

Example: Random walk on set of realizations:

2-switch 3-cycle reorientation

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Uniform sampling algorithms Random walks

Example: Random walk on set of realizations:

2-switch 3-cycle reorientation

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Uniform sampling algorithms Random walks

Example: Random walk on set of realizations:

2-switch 3-cycle reorientation

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Uniform sampling algorithms Importance sampling

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Uniform sampling algorithms Importance sampling

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Uniform sampling algorithms Importance sampling

Algorithm

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Uniform sampling algorithms Importance sampling

Algorithm

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Uniform sampling algorithms Importance sampling

Algorithm

* Blitzstein and Diaconis, “Sequential Importance Sampling Algorithm for Generating Random Graphs with Prescribed Degrees.” Internet Mathematics. 2011 Mar. 9;6(4):489–522. (remained unpublished for 6 years) * del Genio et al, “Efficient and exact sampling of simple graphs with given arbitrary degree sequence.” PLoS ONE. 2010 Mar. 31;5(4):1–7.

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Uniform sampling algorithms Importance sampling

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Uniform sampling of realizations

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Uniform sampling of realizations

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Uniform sampling of realizations

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Uniform sampling of realizations

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Uniform sampling of realizations

x1 x2 x3 x7 x4 x5 x6 ✓ 5 5 5 2 1 1 1 1 1 1 5 4 4 4 ◆ (d)

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x4 x1 x2 x3 ✓ 2 1 1 1 2 1 1 1 ◆ x4 x1 x2 x3 ✓ 1 2 2 2 1 2 2 2 ◆ x4 x5 x1 x2 x3 ✓ 2 1 1 1 4 3 2 2 2 ◆ (a) (b) (c)

Examples:

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x4 x1 x2 x3 ✓ 2 1 1 1 2 1 1 1 ◆ x4 x1 x2 x3 ✓ 1 2 2 2 1 2 2 2 ◆ x4 x5 x1 x2 x3 ✓ 2 1 1 1 4 3 2 2 2 ◆ (a) (b) (c)

Examples:

U x1 x2 x3 U x1 x2 x3

~ F1 ~ F2

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U x1 x2 x3 U x1 x2 x3

~ F1 ~ F2

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X0 X± X+ X− F

      X± X+ X− X0 F X± 1 ∗ 1 ∗ 1 X+ 1 ∗ 1 ∗ 1 X− ∗ ∗ X0 ∗ ∗ F 1 1

    

U x1 x2 x3 U x1 x2 x3

~ F1 ~ F2

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X0 X± X+ X− F

      X± X+ X− X0 F X± 1 ∗ 1 ∗ 1 X+ 1 ∗ 1 ∗ 1 X− ∗ ∗ X0 ∗ ∗ F 1 1

    

U x1 x2 x3 U x1 x2 x3

~ F1 ~ F2

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Partial order of majorization and split partitions

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Partial order of majorization and split partitions

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Partial order of majorization and split partitions

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Partial order of majorization and split partitions

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Thank you!

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