Universal Behavior near Erd os-R enyi Lorenzo Sadun University of - PowerPoint PPT Presentation

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Universal Behavior near Erd˝ os-R´ enyi Lorenzo Sadun University of Texas at Austin ICERM; February 11, 2015 Joint work with Rick Kenyon, Charles Radin and Kui Ren Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Outline Recap of graphs and graphons 1 Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Outline Recap of graphs and graphons 1 Results 2 Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Outline Recap of graphs and graphons 1 Results 2 Strategy 3 Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Outline Recap of graphs and graphons 1 Results 2 Strategy 3 First goal 4 Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Outline Recap of graphs and graphons 1 Results 2 Strategy 3 First goal 4 Second goal 5 Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Outline Recap of graphs and graphons 1 Results 2 Strategy 3 First goal 4 Second goal 5 Tradeoff between goals 6 Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Table of Contents Recap of graphs and graphons 1 Results 2 Strategy 3 First goal 4 Second goal 5 Tradeoff between goals 6 Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Graphons and Densities Graphon is a map g : [0 , 1] 2 → [0 , 1] with g ( x , y ) = g ( y , x ). Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Graphons and Densities Graphon is a map g : [0 , 1] 2 → [0 , 1] with g ( x , y ) = g ( y , x ). �� Edge density e ( g ) = g ( x , y ) dx dy . ��� Triangle density t ( g ) = g ( x , y ) g ( y , z ) g ( z , x ) dx dy dz . Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Graphons and Densities Graphon is a map g : [0 , 1] 2 → [0 , 1] with g ( x , y ) = g ( y , x ). �� Edge density e ( g ) = g ( x , y ) dx dy . ��� Triangle density t ( g ) = g ( x , y ) g ( y , z ) g ( z , x ) dx dy dz . �� Graphon entropy s ( g ) = − I 0 ( g ( x , y )) dx dy . I 0 ( u ) = 1 2 [ u ln( u ) + (1 − u ) ln(1 − u )]. Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Counting entropy S ( e 0 , t 0 ) measures how many graphs have edge density e 0 and triangle density t 0 Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Counting entropy S ( e 0 , t 0 ) measures how many graphs have edge density e 0 and triangle density t 0 S ( e 0 , t 0 ) = max { s ( g ) | e ( g ) = e 0 , t ( g ) = t 0 } . Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Counting entropy S ( e 0 , t 0 ) measures how many graphs have edge density e 0 and triangle density t 0 S ( e 0 , t 0 ) = max { s ( g ) | e ( g ) = e 0 , t ( g ) = t 0 } . For fixed e , S ( e , t ) maximized by Erd˝ os-R´ enyi graphon g ( x , y ) = e at t = e 3 . S ( e , e 3 ) = − I 0 ( e ). Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Counting entropy S ( e 0 , t 0 ) measures how many graphs have edge density e 0 and triangle density t 0 S ( e 0 , t 0 ) = max { s ( g ) | e ( g ) = e 0 , t ( g ) = t 0 } . For fixed e , S ( e , t ) maximized by Erd˝ os-R´ enyi graphon g ( x , y ) = e at t = e 3 . S ( e , e 3 ) = − I 0 ( e ). What happens when t is only close to e 3 ? Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Table of Contents Recap of graphs and graphons 1 Results 2 Strategy 3 First goal 4 Second goal 5 Tradeoff between goals 6 Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Phase portrait for edge-triangle model 1 (1,1) 0.8 τ = ǫ (2 ǫ − 1) 0.6 triangle density τ 0.4 τ = ǫ 3 / 2 I R 0.2 III scallop II (0,0) (1/2,0) 0 0 0.5 1 edge density ǫ Schematic Profile and Phase Portrait Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals What I’m going to show you: As t → e 3 from above, S ( e , e 3 ) − S ( e , t ) goes as ( t − e 3 ) 1 . Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals What I’m going to show you: As t → e 3 from above, S ( e , e 3 ) − S ( e , t ) goes as ( t − e 3 ) 1 . As t → e 3 from below, S ( e , e 3 ) − S ( e , t ) goes as ( e 3 − t ) 2 / 3 . Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals What I’m going to show you: As t → e 3 from above, S ( e , e 3 ) − S ( e , t ) goes as ( t − e 3 ) 1 . As t → e 3 from below, S ( e , e 3 ) − S ( e , t ) goes as ( e 3 − t ) 2 / 3 . The optimizing graphon takes a specific form just above the ER curve. Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals What I’m going to show you: As t → e 3 from above, S ( e , e 3 ) − S ( e , t ) goes as ( t − e 3 ) 1 . As t → e 3 from below, S ( e , e 3 ) − S ( e , t ) goes as ( e 3 − t ) 2 / 3 . The optimizing graphon takes a specific form just above the ER curve. The results above ER are universal , and apply to any model with edge density and one other graph density. Other graph can be triangle, k -star, K n , anything . Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals What I’m going to show you: As t → e 3 from above, S ( e , e 3 ) − S ( e , t ) goes as ( t − e 3 ) 1 . As t → e 3 from below, S ( e , e 3 ) − S ( e , t ) goes as ( e 3 − t ) 2 / 3 . The optimizing graphon takes a specific form just above the ER curve. The results above ER are universal , and apply to any model with edge density and one other graph density. Other graph can be triangle, k -star, K n , anything . Below ER, S ( e , e 3 ) − S ( e , t ) always goes as ( e 3 − t ) 2 / n for some n > 2. (Generically n = 3, but not always.) Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals What I’m going to show you: As t → e 3 from above, S ( e , e 3 ) − S ( e , t ) goes as ( t − e 3 ) 1 . As t → e 3 from below, S ( e , e 3 ) − S ( e , t ) goes as ( e 3 − t ) 2 / 3 . The optimizing graphon takes a specific form just above the ER curve. The results above ER are universal , and apply to any model with edge density and one other graph density. Other graph can be triangle, k -star, K n , anything . Below ER, S ( e , e 3 ) − S ( e , t ) always goes as ( e 3 − t ) 2 / n for some n > 2. (Generically n = 3, but not always.) Caveat: for some graphs, density is minimized at ER. In those models, results below ER are moot. Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals S ( e , t ) for fixed e S(e,t) Not t e 3 Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Graphon just above ER curve Graphon corresponding to t= 0.2201 e= 0.6 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi

Recap of graphs and graphons Results Strategy First goal Second goal Tradeoff between goals Table of Contents Recap of graphs and graphons 1 Results 2 Strategy 3 First goal 4 Second goal 5 Tradeoff between goals 6 Lorenzo Sadun Universal Behavior near Erd˝ os-R´ enyi