Deciding on the type of a graph from a BFS Wang Xiaomin Joint work - PowerPoint PPT Presentation

Deciding on the type of a graph from a BFS Reporter: Deciding on the type of a graph from a BFS Wang Xiaomin Joint work with Matthieu Latapy and Mich` ele Soria Reporter: Wang Xiaomin Outline Joint work with Matthieu Latapy and Mich` ele Soria Introduction Rebuilding Validation 6 mars 2011 Deciding without m Perspectives Al´ ea 2011

Deciding on the type of a Outline graph from a BFS Reporter: Wang Xiaomin Joint work with Matthieu 1 Introduction Latapy and Mich` ele Soria Outline 2 Rebuilding Introduction Rebuilding Validation 3 Validation Deciding without m Perspectives 4 Deciding without m 5 Perspectives

Deciding on the type of a Outline graph from a BFS Reporter: Wang Xiaomin Joint work with Matthieu 1 Introduction Latapy and Mich` ele Soria Outline 2 Rebuilding Introduction Rebuilding Validation 3 Validation Deciding without m Perspectives 4 Deciding without m 5 Perspectives

Deciding on the type of a The difficulty of the measurement graph from a BFS of the Internet Reporter: Wang Xiaomin Joint work with Matthieu Latapy and Mich` ele Soria Outline 1 Internet: Complex Introduction Rebuilding 2 Measurement: Sampling Validation 3 Problem: Partial and Biased Deciding without m Perspectives Definition Degree distribution � the fraction P k of nodes with k links. Type of distribution: Poisson, Power-law, Regular...

Deciding on the type of a Traceroute graph from a BFS Reporter: Wang Xiaomin Traceroute → route from a monitor to a destination. Joint work with Matthieu Latapy and e Mich` ele Soria Outline f Introduction c k Rebuilding a j Validation Deciding i d without m h b Perspectives g route ∼ shortest path

Deciding on the type of a Traceroute graph from a BFS Reporter: Wang Xiaomin Traceroute → route from a monitor to a destination. Joint work with Matthieu Latapy and e Mich` ele Soria Outline f Introduction c k Rebuilding a j Validation Deciding i d without m h h b b Perspectives g g route ∼ shortest path

Deciding on the type of a BFS Tree graph from a BFS Reporter: Wang Xiaomin 1 monitor, many destinations Joint work with Matthieu → BFS: Breadth First Search Tree Latapy and Mich` ele Soria Outline e Introduction f Rebuilding c k Validation Deciding a j without m Perspectives i d h b g

Deciding on the type of a BFS Tree graph from a BFS Reporter: Wang Xiaomin 1 monitor, many destinations Joint work with Matthieu → BFS: Breadth First Search Tree Latapy and Mich` ele Soria Outline e e Introduction f f Rebuilding c c k k Validation Deciding a j j without m Perspectives i i d d h h b b g g

Deciding on the type of a BFS Tree: Power-law degree graph from a BFS distribution Reporter: Wang Xiaomin Joint work with Matthieu Latapy and [ Achliotas , Clauset , Kempe , Moore , JACM , 2005] Mich` ele Soria Outline �� 1 � i − 1 � � Introduction p vis ( t ) m (1 − p vis ( t )) i − 1 − m dt � a obs it i − 1 m +1 = a i Rebuilding m 0 i Validation (1) Deciding � k without m �� j ja j t j 1 � ka k t k p vis ( t ) = (2) Perspectives j ja j t j δ t 2 � k � 1 � g ′ ( t ) � t − (1 − z ) �� g obs ( z ) = z g ′ g ′ (1) g ′ (3) dt g ′ (1) 0

Deciding on the type of a BFS Tree: Power-law degree graph from a BFS distribution Reporter: Wang Xiaomin Joint work with Matthieu Degree distribution of the BFS is always Power-law. Latapy and Mich` ele Soria 1 Poisson: a m = λ m e − λ → a obs m +1 ∼ m − 1 Outline m ! 2 Regular: a r = 1 → a obs 1 Introduction m +1 ∼ rm Rebuilding 3 Power-law: a m ∼ m − α → underestimate α Validation Problem: How to get information on the type of the graph Deciding without m from a BFS tree (always Power-law)? Perspectives 1 Current approach : collect samples as large as possible → still biased? 2 Our approach : infer the properties of the graph from a BFS.

Deciding on the type of a Outline graph from a BFS Reporter: Wang Xiaomin Joint work with Matthieu 1 Introduction Latapy and Mich` ele Soria Outline 2 Rebuilding Introduction Rebuilding Validation 3 Validation Deciding without m Perspectives 4 Deciding without m 5 Perspectives

Deciding on the type of a Methodology graph from a BFS Reporter: Distance: Wang Xiaomin 1 Distribution 1: from the reconstructed graph (G1 or G2). Joint work with Matthieu Latapy and 2 Distribution 2: calculated with (n,m,type). Mich` ele Soria Distance Outline G1 theo.Poisson Introduction Poisson Strategy Rebuilding Validation measure infer G BFS,n,m min Type of G Deciding without m Perspectives PL Strategy Distance G2 theo.PL Step 1: decide on the type of a graph from (n,m,BFS) Step 2: decide on the type of a graph from (n,BFS)

Deciding on the type of a Rebuilding: Methodology graph from a BFS Reporter: Wang Xiaomin Joint work with Matthieu Latapy and Mich` ele Soria Unknown connected G → BFS tree T , n , m → G ′ Outline Method: add m − n + 1 links to T Introduction How to rebuild: Rebuilding Validation 1 Forbidden positions Deciding without m 2 RR, PP and RP strategies Perspectives

Deciding on the type of a Rebuilding: Forbidden positions graph from a BFS Reporter: Wang Xiaomin Joint work a with Matthieu Latapy and Mich` ele Soria Outline c b Introduction Rebuilding Validation g e d f Deciding without m Perspectives h Any link of G is necessarily between two nodes in consecutive levels of T , or in the same level of T .

Deciding on the type of a Rebuilding: Forbidden positions graph from a BFS Reporter: Wang Xiaomin Joint work a with Matthieu Latapy and Mich` ele Soria Outline c b Introduction Rebuilding Validation g e d f Deciding without m Perspectives h Any link of G is necessarily between two nodes in consecutive levels of T , or in the same level of T .

Deciding on the type of a Rebuilding: Forbidden positions graph from a BFS Reporter: Wang Xiaomin Joint work a with Matthieu Latapy and Mich` ele Soria Outline c b Introduction Rebuilding Validation g e d f Deciding without m Perspectives h Any link of G is necessarily between two nodes in consecutive levels of T , or in the same level of T .

Deciding on the type of a Rebuilding strategies: RR and PP graph from a BFS Reporter: Among all allowed positions: Wang Xiaomin 1 RR: Inspired from Erd¨ os-R´ enyi Model, the two extremities Joint work with Matthieu are chosen with uniform probability. Latapy and Mich` ele Soria l Outline E ( d G ′ ( v ) = l ) = 1 � � n jk P ( k → l , j ) (4) Introduction n j > 0 k =1 Rebuilding Validation 2 PP: Inspired from Barab´ asi-Albert Model, the two Deciding without m extremities are chosen with probability proportional to Perspectives their degree. l E ( d G ′ ( v ) = l ) = 1 � � n jk P ( k → l , j , m ′ ) (5) n j > 0 k =1 3 Other strategies

Deciding on the type of a Outline graph from a BFS Reporter: Wang Xiaomin Joint work with Matthieu 1 Introduction Latapy and Mich` ele Soria Outline 2 Rebuilding Introduction Rebuilding Validation 3 Validation Deciding without m Perspectives 4 Deciding without m 5 Perspectives

Deciding on the type of a Validation: process graph from a BFS Reporter: Wang Xiaomin Joint work Distance with Matthieu G1 Latapy and theo.Poisson Mich` ele Soria RR Outline Introduction measure infer Rebuilding G BFS,n,m min Type of G Validation Deciding without m PP Perspectives Distance G2 theo.PL

Deciding on the type of a Validation: process graph from a BFS Reporter: Wang Xiaomin Joint work Distance with Matthieu G1 Latapy and theo.Poisson Mich` ele Soria RR Outline Introduction measure infer Rebuilding G BFS,n,m min Type of G Validation Deciding without m PP Perspectives Distance G2 theo.PL Validate

Deciding on the type of a Validation: datasets graph from a BFS Reporter: Wang Xiaomin Joint work with Matthieu Latapy and Mich` ele Soria Model graphs Outline 1 Simple, connected Introduction [ F . Viger , M . Latapy , 11 thICCC , 2005] Rebuilding 2 Poisson: 3 to 10 Validation Power-law: 2.1 to 2.5 Deciding without m 3 Size: 1000 to 100000 nodes Perspectives 4 Sample: 10 each

Deciding on the type of a Graphic Validation on model graph from a BFS graphs Reporter: Wang Xiaomin Joint work Poisson 10 Power-law 2.2 with Matthieu Latapy and Mich` ele Soria Outline Introduction Rebuilding Validation Deciding without m Perspectives Poisson 10: RR is best. Power-law 2.2: PP is best. Our strategies work on model graphs

Deciding on the type of a Outline graph from a BFS Reporter: Wang Xiaomin Joint work with Matthieu 1 Introduction Latapy and Mich` ele Soria Outline 2 Rebuilding Introduction Rebuilding Validation 3 Validation Deciding without m Perspectives 4 Deciding without m 5 Perspectives