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

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 3, 12 1

Simple graphs and degree sequences Undirected Directed Tuesday, July 3, 12 2

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 • (2K 2 ,C 4 ,C 5 )-free • A graph is split if and only if its degree sequence satisfies a particular Erd ő s-Gallai inequality with equality Tuesday, July 3, 12 3

Graphic sequences Tuesday, July 3, 12 4

Graphic sequences Slack sequence Tuesday, July 3, 12 4

Characterizations of split graphs Tuesday, July 3, 12 5

Characterizations of split graphs Tuesday, July 3, 12 5

Characterizations of split graphs Tuesday, July 3, 12 5

Digraphic sequences Permutations: Tuesday, July 3, 12 6

Digraphic sequences Permutations: Lexicographic ordering (pref. to out-degree) Tuesday, July 3, 12 6

Digraphic sequences Permutations: Lexicographic ordering Lexicographic ordering (pref. to out-degree) (pref. to in-degree) Tuesday, July 3, 12 6

Digraphic sequences Permutations: Lexicographic ordering Lexicographic ordering (pref. to out-degree) (pref. to in-degree) Example: Tuesday, July 3, 12 6

Digraphic sequences Tuesday, July 3, 12 7

Digraphic sequences Slack sequences Tuesday, July 3, 12 7

Structural characterization of split digraphs Tuesday, July 3, 12 8

Structural characterization of split digraphs Tuesday, July 3, 12 8

Structural characterization of split digraphs Tuesday, July 3, 12 8

Structural characterization of split digraphs Tuesday, July 3, 12 8

Structural characterization of split digraphs Tuesday, July 3, 12 8

Structural characterization of split digraphs Tuesday, July 3, 12 8

Structural characterization of split digraphs X ± X 0 X + X − Tuesday, July 3, 12 8

Structural characterization of split digraphs Tuesday, July 3, 12 8

Splittance of an undirected graph Tuesday, July 3, 12 9

Splittance of an undirected graph 1 2 3 4 5 Tuesday, July 3, 12 9

Splittance of an undirected graph 1 2 3 4 5 0 1 2 3 4 5 Tuesday, July 3, 12 9

Splittance of an undirected graph 1 2 3 4 5 0 1 2 3 4 5 Tuesday, July 3, 12 9

Splittance of an undirected graph 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 Tuesday, July 3, 12 9

Splittance of an undirected graph 1 2 3 4 5 0 1 2 3 4 5 I I 0 1 2 3 4 5 Tuesday, July 3, 12 9

Splittance of an undirected graph 1 2 3 4 5 C I 0 1 2 3 4 5 C I 0 1 2 3 4 5 Tuesday, July 3, 12 10

Splittance of an undirected graph 1 2 3 4 5 I C 0 1 2 3 4 5 C I 0 1 2 3 4 5 Tuesday, July 3, 12 11

Splittance of an undirected graph 1 2 3 4 5 I C 0 1 2 3 4 5 C I NOT SPLIT! 0 1 2 3 4 5 Tuesday, July 3, 12 12

Splittance of a directed graph Tuesday, July 3, 12 13

Splittance of a directed graph 1 2 3 4 5 x 4 x 2 x 5 x 3 x 1 Tuesday, July 3, 12 13

Splittance of a directed graph 1 2 3 4 5 x 4 x 2 x 5 x 3 x 1 Tuesday, July 3, 12 13

Splittance of a directed graph 1 2 3 4 5 X x 4 x 2 x 5 x 3 X x 1 Tuesday, July 3, 12 13

Splittance of a directed graph 1 2 3 4 5 x 4 x 2 x 5 x 3 x 1 Tuesday, July 3, 12 14

Splittance of a directed graph 1 2 3 4 5 x 4 x 2 x 5 x 3 x 1 Tuesday, July 3, 12 14

Splittance of a directed graph 1 2 3 4 5 x 4 x 2 x 5 x 3 x 1 Tuesday, July 3, 12 14

Splittance of a directed graph 1 2 3 4 5 x 4 x 2 x 5 x 3 x 1 Tuesday, July 3, 12 14

Splittance of a directed graph 1 2 3 4 5 x 4 x 2 x 5 x 3 x 1 Tuesday, July 3, 12 15

Splittance of a directed graph 1 2 3 4 5 x 4 x 2 x 5 x 3 x 1 Tuesday, July 3, 12 16

Splittance of a directed graph 1 2 3 4 5 x 4 x 2 x 5 x 3 x 1 Tuesday, July 3, 12 16

Splittance of a directed graph 1 2 3 4 5 x 4 x 2 x 5 x 3 x 1 Tuesday, July 3, 12 16

Splittance of a directed graph 1 2 3 4 5 x 4 x 2 x 5 x 3 x 1 Tuesday, July 3, 12 17

Splittance of a directed graph 1 2 3 4 5 x 4 x 2 x 5 x 3 x 1 Tuesday, July 3, 12 17

Splittance of a directed graph 1 2 3 4 5 x 4 x 2 x 5 x 3 SPLIT! x 1 Tuesday, July 3, 12 17

Splittance of a directed graph - directed extensions Undirected Tuesday, July 3, 12 18

Splittance of a directed graph - directed extensions Undirected Directed Tuesday, July 3, 12 18

Splittance of a directed graph - directed extensions Undirected Directed Tuesday, July 3, 12 18

Splittance of a directed graph - directed extensions Undirected Directed Tuesday, July 3, 12 18

Splittance of a directed graph - directed extensions Undirected Directed NOT SPLIT! SPLIT! Tuesday, July 3, 12 18

Canonical split decomposition Tuesday, July 3, 12 19

Canonical split decomposition Tuesday, July 3, 12 19

Canonical split decomposition of digraphs? Tuesday, July 3, 12 20

Canonical split decomposition of digraphs? Tuesday, July 3, 12 20

Realizing graphic and digraphic sequences Tuesday, July 3, 12 21

Realizing graphic and digraphic sequences Tuesday, July 3, 12 21

Uniform sampling algorithms Random walks Random walk on set of realizations: 2-switch 3-cycle reorientation Tuesday, July 3, 12 22

Uniform sampling algorithms Random walks Random walk on set of realizations: 2-switch 3-cycle reorientation Example: Tuesday, July 3, 12 22

Uniform sampling algorithms Random walks Random walk on set of realizations: 2-switch 3-cycle reorientation Example: Tuesday, July 3, 12 22

Uniform sampling algorithms Random walks Random walk on set of realizations: 2-switch 3-cycle reorientation Example: Tuesday, July 3, 12 22

Uniform sampling algorithms Random walks Random walk on set of realizations: 2-switch 3-cycle reorientation Example: Tuesday, July 3, 12 22

Uniform sampling algorithms Random walks Random walk on set of realizations: 2-switch 3-cycle reorientation Example: Tuesday, July 3, 12 22

Uniform sampling algorithms Random walks Random walk on set of realizations: 2-switch 3-cycle reorientation Example: Tuesday, July 3, 12 22

Uniform sampling algorithms Random walks Random walk on set of realizations: 2-switch 3-cycle reorientation Example: Tuesday, July 3, 12 22

Uniform sampling algorithms Random walks Random walk on set of realizations: 2-switch 3-cycle reorientation Example: Tuesday, July 3, 12 22

Uniform sampling algorithms Importance sampling Tuesday, July 3, 12 23

Uniform sampling algorithms Importance sampling Tuesday, July 3, 12 23

Uniform sampling algorithms Importance sampling Algorithm Tuesday, July 3, 12 23

Uniform sampling algorithms Importance sampling Algorithm Tuesday, July 3, 12 23

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. Tuesday, July 3, 12 23

Uniform sampling algorithms Importance sampling Tuesday, July 3, 12 23

Uniform sampling of realizations Tuesday, July 3, 12 24

Uniform sampling of realizations Tuesday, July 3, 12 24

Uniform sampling of realizations Tuesday, July 3, 12 24

Uniform sampling of realizations Tuesday, July 3, 12 24

Uniform sampling of realizations ( d ) x 7 x 1 x 2 x 4 x 5 x 3 x 6 ✓ 5 ◆ 5 5 2 1 1 1 1 1 1 5 4 4 4 Tuesday, July 3, 12 24

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Examples: ( a ) ( b ) ( c ) x 4 x 4 x 4 x 5 x 1 x 2 x 1 x 2 x 1 x 2 x 3 x 3 x 3 ✓ 2 ✓ 1 ✓ 2 ◆ ◆ ◆ 1 1 1 2 2 2 1 1 1 4 2 1 1 1 1 2 2 2 3 2 2 2 0 Tuesday, July 3, 12 26

Examples: ( a ) ( b ) ( c ) x 4 x 4 x 4 x 5 x 1 x 2 x 1 x 2 x 1 x 2 x 3 x 3 x 3 ✓ 2 ✓ 1 ✓ 2 ◆ ◆ ◆ 1 1 1 2 2 2 1 1 1 4 2 1 1 1 1 2 2 2 3 2 2 2 0 ~ ~ F 1 F 2 U U x 1 x 2 x 1 x 2 x 3 x 3 Tuesday, July 3, 12 26

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~ ~ F 1 F 2 U U x 1 x 2 x 1 x 2 x 3 x 3 Tuesday, July 3, 12 27