What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Counting Entropy In edge-triangle model, pick N , δ, e 0 , t 0 . � N � e ( G ) = (number of edges)/ . 2 Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Counting Entropy In edge-triangle model, pick N , δ, e 0 , t 0 . � N � e ( G ) = (number of edges)/ . 2 � N � t ( G ) = (number of triangles)/ . 3 Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Counting Entropy In edge-triangle model, pick N , δ, e 0 , t 0 . � N � e ( G ) = (number of edges)/ . 2 � N � t ( G ) = (number of triangles)/ . 3 Z N ,δ ( e 0 , t 0 ) = number of graphs G with N vertices, e ( G ) ∈ ( e 0 − δ, e 0 + δ ), t ( G ) ∈ ( t 0 − δ, t 0 + δ ). Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Counting Entropy In edge-triangle model, pick N , δ, e 0 , t 0 . � N � e ( G ) = (number of edges)/ . 2 � N � t ( G ) = (number of triangles)/ . 3 Z N ,δ ( e 0 , t 0 ) = number of graphs G with N vertices, e ( G ) ∈ ( e 0 − δ, e 0 + δ ), t ( G ) ∈ ( t 0 − δ, t 0 + δ ). ln( Z N ,δ ( e 0 , t 0 )) Counting entropy is S ( e 0 , t 0 ) = lim δ → 0 + lim . N 2 N →∞ Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Sneak peak at answers 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 (0,0) (1/2,0) 0 0.5 1 edge density ǫ Schematic Profile and Phase Portrait Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Vague description of phases Phases I, II, and III are bipodal . In typical graphs, vertices group into 2 clusters, each with consistent properties. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Vague description of phases Phases I, II, and III are bipodal . In typical graphs, vertices group into 2 clusters, each with consistent properties. Not the same thing as bipartite . In bipodal, can have edges within a cluster. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Vague description of phases Phases I, II, and III are bipodal . In typical graphs, vertices group into 2 clusters, each with consistent properties. Not the same thing as bipartite . In bipodal, can have edges within a cluster. Other observed phases are multipodal . Finitely many clusters. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Vague description of phases Phases I, II, and III are bipodal . In typical graphs, vertices group into 2 clusters, each with consistent properties. Not the same thing as bipartite . In bipodal, can have edges within a cluster. Other observed phases are multipodal . Finitely many clusters. What does all that mean? Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Vague description of phases Phases I, II, and III are bipodal . In typical graphs, vertices group into 2 clusters, each with consistent properties. Not the same thing as bipartite . In bipodal, can have edges within a cluster. Other observed phases are multipodal . Finitely many clusters. What does all that mean? Later. This slide is just a teaser. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Table of Contents What’s the question? 1 Graphons 2 Graphs as graphons; large deviations 3 The edge-triangle model 4 A tour at e = 0 . 6 5 Star models 6 Open questions 7 Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Definitions A graphon is a measurable function g : [0 , 1] 2 → [0 , 1] s.t. g ( x , y ) = g ( y , x ). Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Definitions A graphon is a measurable function g : [0 , 1] 2 → [0 , 1] s.t. g ( x , y ) = g ( y , x ). So what?! Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Definitions A graphon is a measurable function g : [0 , 1] 2 → [0 , 1] s.t. g ( x , y ) = g ( y , x ). So what?! Graphons are: Recipes for generating random graphs. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Definitions A graphon is a measurable function g : [0 , 1] 2 → [0 , 1] s.t. g ( x , y ) = g ( y , x ). So what?! Graphons are: Recipes for generating random graphs. Limits of large graphs. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Definitions A graphon is a measurable function g : [0 , 1] 2 → [0 , 1] s.t. g ( x , y ) = g ( y , x ). So what?! Graphons are: Recipes for generating random graphs. Limits of large graphs. Hat tip to Lov´ asz et al. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Graphons as recipes Given graphon g , to get a random graph with N vertices: Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Graphons as recipes Given graphon g , to get a random graph with N vertices: Pick N independent numbers x 1 , . . . , x N uniformly on [0 , 1]. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Graphons as recipes Given graphon g , to get a random graph with N vertices: Pick N independent numbers x 1 , . . . , x N uniformly on [0 , 1]. Once x j ’s are chosen, edges are independent. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Graphons as recipes Given graphon g , to get a random graph with N vertices: Pick N independent numbers x 1 , . . . , x N uniformly on [0 , 1]. Once x j ’s are chosen, edges are independent. Vertex i is connected to vertex j with probability g ( x i , x j ). Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Graphons as recipes Given graphon g , to get a random graph with N vertices: Pick N independent numbers x 1 , . . . , x N uniformly on [0 , 1]. Once x j ’s are chosen, edges are independent. Vertex i is connected to vertex j with probability g ( x i , x j ). Proper graphs only. No loops, even if g ( x i , x i ) � = 0. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Expected number of edges, triangles, etc. Probability that edge connects 2 vertices is �� g ( x , y ) dx dy := e ( g ). Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Expected number of edges, triangles, etc. Probability that edge connects 2 vertices is �� g ( x , y ) dx dy := e ( g ). Probability that 3 edges form triangle is ��� g ( x , y ) g ( y , z ) g ( z , x ) dx dy dz := t ( g ). Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Expected number of edges, triangles, etc. Probability that edge connects 2 vertices is �� g ( x , y ) dx dy := e ( g ). Probability that 3 edges form triangle is ��� g ( x , y ) g ( y , z ) g ( z , x ) dx dy dz := t ( g ). � N � Number of edges ≈ e ( g ) with probability close to 1. 2 � N � Number of triangles ≈ t ( g ) . 3 Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Expected number of edges, triangles, etc. Probability that edge connects 2 vertices is �� g ( x , y ) dx dy := e ( g ). Probability that 3 edges form triangle is ��� g ( x , y ) g ( y , z ) g ( z , x ) dx dy dz := t ( g ). � N � Number of edges ≈ e ( g ) with probability close to 1. 2 � N � Number of triangles ≈ t ( g ) . 3 Number of other embedded sub-graphs are easy integrals. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Expected Shannon entropy Shannon entropy of coin flip is − ( p ln( p ) + (1 − p ) ln(1 − p )). Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Expected Shannon entropy Shannon entropy of coin flip is − ( p ln( p ) + (1 − p ) ln(1 − p )). Once x 1 , . . . , x N are chosen Shannon entropy is � − g ( x i , x j ) ln( g ( x i , x j )) + (1 − g ( x i , x j )) ln(1 − g ( x i , x j )). i < j Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Expected Shannon entropy Shannon entropy of coin flip is − ( p ln( p ) + (1 − p ) ln(1 − p )). Once x 1 , . . . , x N are chosen Shannon entropy is � − g ( x i , x j ) ln( g ( x i , x j )) + (1 − g ( x i , x j )) ln(1 − g ( x i , x j )). i < j Average over x i ’s to get � N � �� g ( x , y ) ln( g ( x , y )) + (1 − g ) ln(1 − g ) dx dy ≈ − 2 N 2 s ( g ), where Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Expected Shannon entropy Shannon entropy of coin flip is − ( p ln( p ) + (1 − p ) ln(1 − p )). Once x 1 , . . . , x N are chosen Shannon entropy is � − g ( x i , x j ) ln( g ( x i , x j )) + (1 − g ( x i , x j )) ln(1 − g ( x i , x j )). i < j Average over x i ’s to get � N � �� g ( x , y ) ln( g ( x , y )) + (1 − g ) ln(1 − g ) dx dy ≈ − 2 N 2 s ( g ), where s ( g ) := − 1 �� gln ( g ) + (1 − g ) ln(1 − g ) dx dy . 2 is the graphon entropy Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Table of Contents What’s the question? 1 Graphons 2 Graphs as graphons; large deviations 3 The edge-triangle model 4 A tour at e = 0 . 6 5 Star models 6 Open questions 7 Lorenzo Sadun Multipodal Phases in Graphs
�� �� �� �� �� �� �� �� �� �� �� �� What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Checkerboard graphons Given graph G on N vertices, define Lorenzo Sadun Multipodal Phases in Graphs
�� �� �� �� �� �� �� �� �� �� �� �� What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Checkerboard graphons Given graph G on N vertices, define � 1 Vertices [ Nx ] + 1 , [ Ny + 1] connected by edge g G ( x , y ) = 0 Otherwise Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Checkerboard graphons Given graph G on N vertices, define � 1 Vertices [ Nx ] + 1 , [ Ny + 1] connected by edge g G ( x , y ) = 0 Otherwise 2 0 1 0 �� �� �� �� 1 0 1 0 1 0 �� �� �� �� �� �� �� �� 1 3 Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Large deviations Oversimplification of a really important theorem: Theorem (Chatterjee-Varadhan) The number of graphs whose checkerboard graphon is close to g N 2 s ( g ) � � goes as exp . Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Caveats Renumbering vertices doesn’t change statistical properties of graph. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Caveats Renumbering vertices doesn’t change statistical properties of graph. Applying measure-preserving transformation to [0 , 1] doesn’t change statistical properties of graphon. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Caveats Renumbering vertices doesn’t change statistical properties of graph. Applying measure-preserving transformation to [0 , 1] doesn’t change statistical properties of graphon. All statements should be read as “up to measure-preserving transformation”. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Caveats Renumbering vertices doesn’t change statistical properties of graph. Applying measure-preserving transformation to [0 , 1] doesn’t change statistical properties of graphon. All statements should be read as “up to measure-preserving transformation”. “Close to” is not pointwise or in L p . Convergence is in “cut metric” topology. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Caveats Renumbering vertices doesn’t change statistical properties of graph. Applying measure-preserving transformation to [0 , 1] doesn’t change statistical properties of graphon. All statements should be read as “up to measure-preserving transformation”. “Close to” is not pointwise or in L p . Convergence is in “cut metric” topology. Actual theorem involves open sets, closed sets, lim inf and lim sup of ln(#) / N 2 . Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Counting entropy is graphon entropy Theorem (Radin-S) S ( e 0 , t 0 ) = sup s ( g ) = max s ( g ) , where the sup is over all graphons with e ( g ) = e 0 and t ( g ) = t 0 . Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Counting entropy is graphon entropy Theorem (Radin-S) S ( e 0 , t 0 ) = sup s ( g ) = max s ( g ) , where the sup is over all graphons with e ( g ) = e 0 and t ( g ) = t 0 . Proof: Corollary of Chatterjee-Varadhan, using open and closed sets derived from rectangles in ( e 0 , t 0 ) space. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Counting entropy is graphon entropy Theorem (Radin-S) S ( e 0 , t 0 ) = sup s ( g ) = max s ( g ) , where the sup is over all graphons with e ( g ) = e 0 and t ( g ) = t 0 . Proof: Corollary of Chatterjee-Varadhan, using open and closed sets derived from rectangles in ( e 0 , t 0 ) space. Further corollary: We can stop doing combinatorics and do functional analysis instead. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Table of Contents What’s the question? 1 Graphons 2 Graphs as graphons; large deviations 3 The edge-triangle model 4 A tour at e = 0 . 6 5 Star models 6 Open questions 7 Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions The phase portrait 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 (0,0) (1/2,0) 0 0.5 1 edge density ǫ Schematic Profile and Phase Portrait Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Provable phase transition at Erd¨ os-R´ enyi (Radin-S) Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Provable phase transition at Erd¨ os-R´ enyi (Radin-S) (details tomorrow afternoon) Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Provable phase transition at Erd¨ os-R´ enyi (Radin-S) (details tomorrow afternoon) As t → e 3 from above, S ( e , e 3 ) − S ( e , t ) goes as ( t − e 3 ) 1 . Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Provable phase transition at Erd¨ os-R´ enyi (Radin-S) (details tomorrow afternoon) 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 Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Same thing in pictures S(e,t) Not t e 3 Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions ERGMs don’t work! In ERGM, apply Legendre transform to S ( e , t ) to get F ( β 1 , β 2 ) = sup e , t ( β 1 e ( g ) + β 2 t ( g ) + S ( e , t )). Analogue of grand canonical ensemble vs. microcanonical ensemble. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions ERGMs don’t work! In ERGM, apply Legendre transform to S ( e , t ) to get F ( β 1 , β 2 ) = sup e , t ( β 1 e ( g ) + β 2 t ( g ) + S ( e , t )). Analogue of grand canonical ensemble vs. microcanonical ensemble. S ( e , t ) has wrong concavity in much of phase profile. Transform is not invertible. Behavior near ER is universal. So is failure of ERGMs. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions The phase portrait 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 (0,0) (1/2,0) 0 0.5 1 edge density ǫ Schematic Profile and Phase Portrait Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions The symmetric bipodal phase (aka Phase II) � e + ν x < 0 . 5 < y or y < 0 . 5 < x g ( x , y ) = e − ν x , y < 0 . 5 or x , y > 0 . 5 . Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions The symmetric bipodal phase (aka Phase II) � e + ν x < 0 . 5 < y or y < 0 . 5 < x g ( x , y ) = e − ν x , y < 0 . 5 or x , y > 0 . 5 . 1 e + ν e − ν 1/2 e − ν e + ν 0 1 1/2 Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Properties of symmetric bipodal e ( g ) = e ; t ( g ) = e 3 − ν 3 . Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Properties of symmetric bipodal e ( g ) = e ; t ( g ) = e 3 − ν 3 . Reaches lower boundary when e ≤ 1 2, not when e > 1 2. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Properties of symmetric bipodal e ( g ) = e ; t ( g ) = e 3 − ν 3 . Reaches lower boundary when e ≤ 1 2, not when e > 1 2. Vertices form two clusters of equal size, with edge probability e + ν between clusters and e − ν within cluster. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Properties of symmetric bipodal e ( g ) = e ; t ( g ) = e 3 − ν 3 . Reaches lower boundary when e ≤ 1 2, not when e > 1 2. Vertices form two clusters of equal size, with edge probability e + ν between clusters and e − ν within cluster. Structure is only proven for e = 1 2, t = 0, and t = e 3 . Elsewhere supported by numerics and perturbation theory. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions The phase portrait 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 (0,0) (1/2,0) 0 0.5 1 edge density ǫ Schematic Profile and Phase Portrait Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Asymmetric bipodal phases: III = Dating Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Asymmetric bipodal phases: III = Dating 1 p p 12 22 c p p 11 12 c 0 1 Below ER curve, p 11 , p 22 < p 12 . Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Asymmetric bipodal phases: III = Dating 1 p p 12 22 c p p 11 12 c 0 1 Below ER curve, p 11 , p 22 < p 12 . Members of each group like to interact with other group, not themselves. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Asymmetric bipodal phases: III = Dating 1 p p 12 22 c p p 11 12 c 0 1 Below ER curve, p 11 , p 22 < p 12 . Members of each group like to interact with other group, not themselves. Can determine exactly where symmetric phase becomes unstable w.r.t. changing relative size of clusters. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Asymmetric bipodal phases: III = Dating 1 p p 12 22 c p p 11 12 c 0 1 Below ER curve, p 11 , p 22 < p 12 . Members of each group like to interact with other group, not themselves. Can determine exactly where symmetric phase becomes unstable w.r.t. changing relative size of clusters. Limit as p 11 , p 22 → 0 is bipartite. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions The phase portrait 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 (0,0) (1/2,0) 0 0.5 1 edge density ǫ Schematic Profile and Phase Portrait Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Asymmetric bipodal phases: I = Socialites Above ER curve, p 11 > p 12 > p 22 . Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Asymmetric bipodal phases: I = Socialites Above ER curve, p 11 > p 12 > p 22 . Highly social group mostly interacts with itself. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Asymmetric bipodal phases: I = Socialites Above ER curve, p 11 > p 12 > p 22 . Highly social group mostly interacts with itself. Group with few social skills mostly uninvolved. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Asymmetric bipodal phases: I = Socialites Above ER curve, p 11 > p 12 > p 22 . Highly social group mostly interacts with itself. Group with few social skills mostly uninvolved. As t → e 3 / 2 , p 11 → 1, others → 0. Complete graph on some vertices, plus spectators. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Multipodality All observed phases are multipodal: Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Multipodality All observed phases are multipodal: p ij j i Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions What does that mean? m different clusters, of possibly different sizes. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions What does that mean? m different clusters, of possibly different sizes. Probability p ij of edge between vertices in clusters i and j . Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions What does that mean? m different clusters, of possibly different sizes. Probability p ij of edge between vertices in clusters i and j . Self-organization. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions What does that mean? m different clusters, of possibly different sizes. Probability p ij of edge between vertices in clusters i and j . Self-organization. Phase transitions as clusters split/merge, bud, or shrink to size 0. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions What does that mean? m different clusters, of possibly different sizes. Probability p ij of edge between vertices in clusters i and j . Self-organization. Phase transitions as clusters split/merge, bud, or shrink to size 0. For each m , finite-dimensional space of m -podal graphons. Functional analysis becomes ordinary calculus. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions What does that mean? m different clusters, of possibly different sizes. Probability p ij of edge between vertices in clusters i and j . Self-organization. Phase transitions as clusters split/merge, bud, or shrink to size 0. For each m , finite-dimensional space of m -podal graphons. Functional analysis becomes ordinary calculus. Multipodal structure is observed , not proven for edge-triangle. Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions Table of Contents What’s the question? 1 Graphons 2 Graphs as graphons; large deviations 3 The edge-triangle model 4 A tour at e = 0 . 6 5 Star models 6 Open questions 7 Lorenzo Sadun Multipodal Phases in Graphs
What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0 . 6 Star models Open questions The phase portrait 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 (0,0) (1/2,0) 0 0.5 1 edge density ǫ Schematic Profile and Phase Portrait Lorenzo Sadun Multipodal Phases in Graphs
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