Multipodal Phases in Graphs Lorenzo Sadun University of Texas at - - PowerPoint PPT Presentation

What’s the question? Graphons Graphs as graphons; large deviations The edge-triangle model A tour at e = 0.6 Star models Open questions

Multipodal Phases in Graphs

Lorenzo Sadun

University of Texas at Austin

ICERM; February 10, 2015

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

Outline

1

What’s the question?

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

Outline

1

What’s the question?

2

Graphons

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

Outline

1

What’s the question?

2

Graphons

3

Graphs as graphons; large deviations

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

Outline

1

What’s the question?

2

Graphons

3

Graphs as graphons; large deviations

4

The edge-triangle model

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

Outline

1

What’s the question?

2

Graphons

3

Graphs as graphons; large deviations

4

The edge-triangle model

5

A tour at e = 0.6

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

Outline

1

What’s the question?

2

Graphons

3

Graphs as graphons; large deviations

4

The edge-triangle model

5

A tour at e = 0.6

6

Star models

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

Outline

1

What’s the question?

2

Graphons

3

Graphs as graphons; large deviations

4

The edge-triangle model

5

A tour at e = 0.6

6

Star models

7

Open questions

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

1

What’s the question?

2

Graphons

3

Graphs as graphons; large deviations

4

The edge-triangle model

5

A tour at e = 0.6

6

Star models

7

Open questions

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 are random graphs?

Pick an appropriate ensemble 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

What are random graphs?

Pick an appropriate ensemble of large graphs Today, look at all graphs on N vertices with uniform weight.

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 are random graphs?

Pick an appropriate ensemble of large graphs Today, look at all graphs on N vertices with uniform weight. Within the ensemble, identify graphs with a particular property.

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 are random graphs?

Pick an appropriate ensemble of large graphs Today, look at all graphs on N vertices with uniform weight. Within the ensemble, identify graphs with a particular property. Today, look at edge and triangle densities.

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 are random graphs?

Pick an appropriate ensemble of large graphs Today, look at all graphs on N vertices with uniform weight. Within the ensemble, identify graphs with a particular property. Today, look at edge and triangle densities. How likely are certain values?

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 are random graphs?

Pick an appropriate ensemble of large graphs Today, look at all graphs on N vertices with uniform weight. Within the ensemble, identify graphs with a particular property. Today, look at edge and triangle densities. How likely are certain values? What other properties do graphs with those values have?

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 are random graphs?

Pick an appropriate ensemble of large graphs Today, look at all graphs on N vertices with uniform weight. Within the ensemble, identify graphs with a particular property. Today, look at edge and triangle densities. How likely are certain values? What other properties do graphs with those values have? How do the answers depend on the values?

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 graphs

Graphs = Incidence Matrices

• 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 graphs

Graphs = Incidence Matrices aij =

• 1

vertices i and j connected 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

Counting graphs

Graphs = Incidence Matrices aij =

• 1

vertices i and j connected Otherwise   1 1   =   1 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

Counting graphs

Graphs = Incidence Matrices aij =

• 1

vertices i and j connected Otherwise   1 1   =   1 1  

• 1

1 2 3 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

Counting Entropy

In edge-triangle model, pick N, δ, e0, t0.

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, e0, t0. e(G) = (number of edges)/ 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

In edge-triangle model, pick N, δ, e0, t0. e(G) = (number of edges)/ N 2

• .

t(G) = (number of triangles)/ N 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, e0, t0. e(G) = (number of edges)/ N 2

• .

t(G) = (number of triangles)/ N 3

• .

Z N,δ(e0, t0) = number of graphs G with N vertices, e(G) ∈ (e0 − δ, e0 + δ), t(G) ∈ (t0 − δ, t0 + 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, e0, t0. e(G) = (number of edges)/ N 2

• .

t(G) = (number of triangles)/ N 3

• .

Z N,δ(e0, t0) = number of graphs G with N vertices, e(G) ∈ (e0 − δ, e0 + δ), t(G) ∈ (t0 − δ, t0 + δ). Counting entropy is S(e0, t0) = lim

δ→0+ lim N→∞

ln(Z N,δ(e0, t0)) N2 .

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

edge density ǫ (0,0) density τ τ = ǫ3/2 triangle τ = ǫ(2ǫ − 1)

scallop

(1/2,0)

R

(1,1)

0.5 1 0.2 0.4 0.6 0.8 1 II I III

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

1

What’s the question?

2

Graphons

3

Graphs as graphons; large deviations

4

The edge-triangle model

5

A tour at e = 0.6

6

Star models

7

Open questions

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 x1, . . . , xN 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 x1, . . . , xN uniformly on [0, 1]. Once xj’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 x1, . . . , xN uniformly on [0, 1]. Once xj’s are chosen, edges are independent. Vertex i is connected to vertex j with probability g(xi, xj).

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 x1, . . . , xN uniformly on [0, 1]. Once xj’s are chosen, edges are independent. Vertex i is connected to vertex j with probability g(xi, xj). Proper graphs only. No loops, even if g(xi, xi) = 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).

Number of edges ≈ e(g) N 2

• with probability close to 1.

Number of triangles ≈ t(g) N 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).

Number of edges ≈ e(g) N 2

• with probability close to 1.

Number of triangles ≈ t(g) N 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 x1, . . . , xN are chosen Shannon entropy is −

• i<j

g(xi, xj) ln(g(xi, xj)) + (1 − g(xi, xj)) ln(1 − g(xi, xj)).

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 x1, . . . , xN are chosen Shannon entropy is −

• i<j

g(xi, xj) ln(g(xi, xj)) + (1 − g(xi, xj)) ln(1 − g(xi, xj)). Average over xi’s to get − N 2 g(x, y) ln(g(x, y)) + (1 − g) ln(1 − g)dx dy ≈ N2s(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 x1, . . . , xN are chosen Shannon entropy is −

• i<j

g(xi, xj) ln(g(xi, xj)) + (1 − g(xi, xj)) ln(1 − g(xi, xj)). Average over xi’s to get − N 2 g(x, y) ln(g(x, y)) + (1 − g) ln(1 − g)dx dy ≈ N2s(g), where s(g) := −1 2

• gln(g) + (1 − g) ln(1 − g)dx dy.

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

1

What’s the question?

2

Graphons

3

Graphs as graphons; large deviations

4

The edge-triangle model

5

A tour at e = 0.6

6

Star models

7

Open questions

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 gG(x, y) =

• 1

Vertices [Nx] + 1, [Ny + 1] connected by edge 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 gG(x, y) =

• 1

Vertices [Nx] + 1, [Ny + 1] connected by edge Otherwise

• 1

2 3

1 1 1 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

Large deviations

Oversimplification of a really important theorem: Theorem (Chatterjee-Varadhan) The number of graphs whose checkerboard graphon is close to g goes as exp

• N2s(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

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 Lp. 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 Lp. Convergence is in “cut metric” topology. Actual theorem involves open sets, closed sets, lim inf and lim sup of ln(#)/N2.

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(e0, t0) = sup s(g) = max s(g), where the sup is over all graphons with e(g) = e0 and t(g) = t0.

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(e0, t0) = sup s(g) = max s(g), where the sup is over all graphons with e(g) = e0 and t(g) = t0. Proof: Corollary of Chatterjee-Varadhan, using open and closed sets derived from rectangles in (e0, t0) 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(e0, t0) = sup s(g) = max s(g), where the sup is over all graphons with e(g) = e0 and t(g) = t0. Proof: Corollary of Chatterjee-Varadhan, using open and closed sets derived from rectangles in (e0, t0) 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

1

What’s the question?

2

Graphons

3

Graphs as graphons; large deviations

4

The edge-triangle model

5

A tour at e = 0.6

6

Star models

7

Open questions

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

edge density ǫ (0,0) density τ τ = ǫ3/2 triangle τ = ǫ(2ǫ − 1)

scallop

(1/2,0)

R

(1,1)

0.5 1 0.2 0.4 0.6 0.8 1 II I III

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¨

• s-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¨

• s-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¨

• s-R´

enyi (Radin-S)

(details tomorrow afternoon) As t → e3 from above, S(e, e3) − S(e, t) goes as (t − e3)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¨

• s-R´

enyi (Radin-S)

(details tomorrow afternoon) As t → e3 from above, S(e, e3) − S(e, t) goes as (t − e3)1. As t → e3 from below, S(e, e3) − S(e, t) goes as (e3 − 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) t e3 Not

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(β1, β2) = sup

e,t (β1e(g) + β2t(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(β1, β2) = sup

e,t (β1e(g) + β2t(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

edge density ǫ (0,0) density τ τ = ǫ3/2 triangle τ = ǫ(2ǫ − 1)

scallop

(1/2,0)

R

(1,1)

0.5 1 0.2 0.4 0.6 0.8 1 II I III

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)

g(x, y) =

• e + ν

x < 0.5 < y or y < 0.5 < x 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)

g(x, y) =

• e + ν

x < 0.5 < y or y < 0.5 < x e − ν x, y < 0.5 or x, y > 0.5.

e + ν e + ν e − ν e − ν 1/2 1/2 1 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

Properties of symmetric bipodal

e(g) = e; t(g) = e3 − ν3.

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) = e3 − ν3. 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) = e3 − ν3. 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) = e3 − ν3. 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 = e3. 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

edge density ǫ (0,0) density τ τ = ǫ3/2 triangle τ = ǫ(2ǫ − 1)

scallop

(1/2,0)

R

(1,1)

0.5 1 0.2 0.4 0.6 0.8 1 II I III

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 1 c c p p p p

22 11 12 12

Below ER curve, p11, p22 < p12.

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 1 c c p p p p

22 11 12 12

Below ER curve, p11, p22 < p12. 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 1 c c p p p p

22 11 12 12

Below ER curve, p11, p22 < p12. 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 1 c c p p p p

22 11 12 12

Below ER curve, p11, p22 < p12. 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 p11, p22 → 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

edge density ǫ (0,0) density τ τ = ǫ3/2 triangle τ = ǫ(2ǫ − 1)

scallop

(1/2,0)

R

(1,1)

0.5 1 0.2 0.4 0.6 0.8 1 II I III

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, p11 > p12 > p22.

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, p11 > p12 > p22. 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, p11 > p12 > p22. 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, p11 > p12 > p22. Highly social group mostly interacts with itself. Group with few social skills mostly uninvolved. As t → e3/2, p11 → 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:

pij

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

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 pij 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 pij 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 pij 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 pij 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 pij 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

1

What’s the question?

2

Graphons

3

Graphs as graphons; large deviations

4

The edge-triangle model

5

A tour at e = 0.6

6

Star models

7

Open questions

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

edge density ǫ (0,0) density τ τ = ǫ3/2 triangle τ = ǫ(2ǫ − 1)

scallop

(1/2,0)

R

(1,1)

0.5 1 0.2 0.4 0.6 0.8 1 II I III

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

Tripodal near the bottom

a b b p p d d c c

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

Tripodal near the bottom

a b b p p d d c c

At minimum (t ≈ 0.1415), a = b = 0, d = 1 c ≈ 0.4334, p ≈ 0.679.

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

Tripodal becomes (asymmetric) bipodal

a b b p p d d c c

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

Tripodal becomes (asymmetric) bipodal

a b b p p d d c c

When t = 0.1485, a = 0.0009, b = 0.2645, c = 0.4373, d = 0.999999999999, p = 0.4158.

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

Tripodal becomes (asymmetric) bipodal

a b b p p d d c c

When t = 0.1485, a = 0.0009, b = 0.2645, c = 0.4373, d = 0.999999999999, p = 0.4158. When t = 0.1486, a = 0.0014, b = 0.3397375, c = 0.4371, d = 0.99999999, p = 0.3397375

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

Tripodal becomes (asymmetric) bipodal

a b b p p d d c c

When t = 0.1485, a = 0.0009, b = 0.2645, c = 0.4373, d = 0.999999999999, p = 0.4158. When t = 0.1486, a = 0.0014, b = 0.3397375, c = 0.4371, d = 0.99999999, p = 0.3397375 At instant of transition ∂s/∂t and ∂s/∂e diverge.

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 becomes symmetric

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Graphon corresponding to t= 0.152 e= 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Graphon corresponding to t= 0.15328 e= 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t=0.152 and 0.1533

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

Long run of symmetric bipodal

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Graphon corresponding to t= 0.184 e= 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Graphon corresponding to t= 0.20576 e= 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t=0.184 and 0.20576

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 again

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Graphon corresponding to t= 0.2096 e= 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t=0.21

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

Erd¨

• s-R´

enyi

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Graphon corresponding to t= 0.216 e= 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t=0.216

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

Budding off a new cluster

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Graphon corresponding to t= 0.2201 e= 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t=0.22

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

Approaching 0-1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Graphon corresponding to t= 0.35033 e= 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Graphon corresponding to t= 0.44983 e= 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t=0.35 and 0.45

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

1

What’s the question?

2

Graphons

3

Graphs as graphons; large deviations

4

The edge-triangle model

5

A tour at e = 0.6

6

Star models

7

Open questions

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

k-stars

Instead of triangles, look at k − stars

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

k-stars

Instead of triangles, look at k − stars

2−star 3−star 4−star 1 1 1 2 2 3 3 4 2

t(g) =

• g(x0, x1)g(x0, x2)g(x0, x3) · · · g(x0, xk)dk+1x.

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

Calculating with k-stars

Degree d(x) := 1 g(x, y)dy.

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

Calculating with k-stars

Degree d(x) := 1 g(x, y)dy. Graph density t(g) = 1 d(x)kdx.

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

Euler-Lagrange equations (up to constants)

δs δg = α δe δg(x, y) + β δt δg(x, y)

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

Euler-Lagrange equations (up to constants)

δs δg = α δe δg(x, y) + β δt δg(x, y) ln

• g(x, y)

1 − g(x, y)

• = α + β
• d(x)k−1 + d(y)k−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

Euler-Lagrange equations (up to constants)

δs δg = α δe δg(x, y) + β δt δg(x, y) ln

• g(x, y)

1 − g(x, y)

• = α + β
• d(x)k−1 + d(y)k−1

g(x, y) = 1 1 + exp(α + β(d(x)k−1 + d(y)k−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

All entropy maximizers are multipodal

Theorem (Kenyon-Radin-Ren-S) In the k-star model, for all achievable values of (e, t), any entropy-maximizing graphon is multipodal. (Numerical evidence and perturbation theory suggest bipodal)

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

Proof of theorem, part 1

g(x, y) = 1 1 + exp(α + β(d(x)k−1 + d(y)k−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

Proof of theorem, part 1

g(x, y) = 1 1 + exp(α + β(d(x)k−1 + d(y)k−1)). d(x) = 1 dy 1 + exp(α + β(d(x)k−1 + d(y)k−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

Proof of theorem, part 1

g(x, y) = 1 1 + exp(α + β(d(x)k−1 + d(y)k−1)). d(x) = 1 dy 1 + exp(α + β(d(x)k−1 + d(y)k−1)). For fixed α, β and d(x), let F(z) = z − 1 dy 1 + exp(α + β(zk−1 + d(y)k−1)). All values

• f x must have F(d(x)) = 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

Proof, part 2

Claim: F(z) is analytic and not constant.

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

Proof, part 2

Claim: F(z) is analytic and not constant. Reason: Convolution of analytic function and bounded measure.

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

Proof, part 2

Claim: F(z) is analytic and not constant. Reason: Convolution of analytic function and bounded measure. Consequence: Only has finitely many roots 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

Proof, part 2

Claim: F(z) is analytic and not constant. Reason: Convolution of analytic function and bounded measure. Consequence: Only has finitely many roots on [0, 1]. Do measure-preserving transformation to make d(x) non-decreasing. Since d(x) is step function, so is g(x, y).

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

1

What’s the question?

2

Graphons

3

Graphs as graphons; large deviations

4

The edge-triangle model

5

A tour at e = 0.6

6

Star models

7

Open questions

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

Why multipodal?

Universal behavior near ER curve. (My other talk)

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

Why multipodal?

Universal behavior near ER curve. (My other talk) Is there something special about 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

Why multipodal?

Universal behavior near ER curve. (My other talk) Is there something special about edge-triangle? Can we prove multipodality for all of 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

Why multipodal?

Universal behavior near ER curve. (My other talk) Is there something special about edge-triangle? Can we prove multipodality for all of edge-triangle? Models with > 2 constraints. Are we missing something important by looking at edges + one other subgraph?

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

Why multipodal?

Universal behavior near ER curve. (My other talk) Is there something special about edge-triangle? Can we prove multipodality for all of edge-triangle? Models with > 2 constraints. Are we missing something important by looking at edges + one other subgraph? False conjecture: Any model with finitely many graph densities will yield multipodal graphons throughout. (Lov´ asz-Szegedy, finitely forcible graphons)

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

Why multipodal?

Universal behavior near ER curve. (My other talk) Is there something special about edge-triangle? Can we prove multipodality for all of edge-triangle? Models with > 2 constraints. Are we missing something important by looking at edges + one other subgraph? False conjecture: Any model with finitely many graph densities will yield multipodal graphons throughout. (Lov´ asz-Szegedy, finitely forcible graphons) Viable conjecture: Any model with finitely many graph densities will yield mutipodal graphons as entropy maximizers throughout interior of phase profile.

Lorenzo Sadun Multipodal Phases in Graphs