Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Pim van der Hoorn

Workshop on Random Geometric Graphs Toronto

June 20, 2017

Joint work with: Gabor Lippner and Dmitri Krioukov

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Complex Network Random graph ensemble

• Number of edges, triangles • Degree distribution Typical graph with the given • Clustering, Communities topology • High order topological properties

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

The challenge

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Complex Network Random graph ensemble

• Number of edges, triangles • Degree distribution Typical graph with the given • Clustering, Communities topology • High order topological properties

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

The challenge

Given: a degree distribution P (D = k)

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Complex Network Random graph ensemble

• Number of edges, triangles • Degree distribution Typical graph with the given • Clustering, Communities topology • High order topological properties

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

The challenge

Given: a degree distribution P (D = k)

Goal: deﬁne random graph model that generates typical graphs with this distribution

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Random graph ensemble

• Number of edges, triangles • Degree distribution Typical graph with the given • Clustering, Communities topology • High order topological properties

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

The challenge Complex Network

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Random graph ensemble

Typical graph with the given topology

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

The challenge Complex Network

• Number of edges, triangles • Degree distribution • Clustering, Communities • High order topological properties

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Typical graph with the given topology

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

The challenge Complex Network Random graph ensemble

• Number of edges, triangles • Degree distribution • Clustering, Communities • High order topological properties

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

The challenge Complex Network Random graph ensemble

• Number of edges, triangles • Degree distribution Typical graph with the given • Clustering, Communities topology • High order topological properties

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

The challenge Complex Network Random graph ensemble

• Number of edges, triangles • Degree distribution Typical graph with the given • Clustering, Communities topology • High order topological properties

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Create a typical network with 8 nodes and 5 edges.

A typical network with n nodes and m edges is Gn,m.

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Typical networks

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn A typical network with n nodes and m edges is Gn,m.

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Typical networks

Create a typical network with 8 nodes and 5 edges.

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn A typical network with n nodes and m edges is Gn,m.

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Typical networks

Create a typical network with 8 nodes and 5 edges.

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn A typical network with n nodes and m edges is Gn,m.

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Typical networks

Create a typical network with 8 nodes and 5 edges.

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn A typical network with n nodes and m edges is Gn,m.

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Typical networks

Create a typical network with 8 nodes and 5 edges.

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn A typical network with n nodes and m edges is Gn,m.

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Typical networks

Create a typical network with 8 nodes and 5 edges.

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn A typical network with n nodes and m edges is Gn,m.

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Typical networks

Create a typical network with 8 nodes and 5 edges.

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn A typical network with n nodes and m edges is Gn,m.

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Typical networks

Create a typical network with 8 nodes and 5 edges.

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn A typical network with n nodes and m edges is Gn,m.

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Typical networks

Create a typical network with 8 nodes and 5 edges.

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn A typical network with n nodes and m edges is Gn,m.

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Typical networks

Create a typical network with 8 nodes and 5 edges.

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Typical networks

Create a typical network with 8 nodes and 5 edges.

A typical network with n nodes and m edges is Gn,m.

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn ⇒ model

Gn,m Hard constraints

2) m edges on average Gn,p p = m/(n − 1) Soft constraints

Let Gn be the set of all simple graphs of size n. Consider a probability distribution P on Gn X S[P] = − P(G) log(P(G)) Gibbs entropy

G∈Gn Given a set of constraints C (number of edges, triangles ect.) Typical networks with given constraints are sampled from

P∗ = arg max S[P] P∈C

1) exactly m edges

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn ⇒ model

Gn,m Hard constraints

2) m edges on average Gn,p p = m/(n − 1) Soft constraints

X S[P] = − P(G) log(P(G)) Gibbs entropy

G∈Gn Given a set of constraints C (number of edges, triangles ect.) Typical networks with given constraints are sampled from

P∗ = arg max S[P] P∈C

1) exactly m edges

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy

Let Gn be the set of all simple graphs of size n. Consider a probability distribution P on Gn

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn ⇒ model

Gn,m Hard constraints

2) m edges on average Gn,p p = m/(n − 1) Soft constraints

Given a set of constraints C (number of edges, triangles ect.) Typical networks with given constraints are sampled from

P∗ = arg max S[P] P∈C

1) exactly m edges

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy

Let Gn be the set of all simple graphs of size n. Consider a probability distribution P on Gn X S[P] = − P(G) log(P(G)) Gibbs entropy

G∈Gn

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Gn,m Hard constraints

2) m edges on average Gn,p p = m/(n − 1) Soft constraints

⇒ model

1) exactly m edges

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy

Let Gn be the set of all simple graphs of size n. Consider a probability distribution P on Gn X S[P] = − P(G) log(P(G)) Gibbs entropy

G∈Gn Given a set of constraints C (number of edges, triangles ect.) Typical networks with given constraints are sampled from

P∗ = arg max S[P] P∈C

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Gn,m Hard constraints

2) m edges on average Gn,p p = m/(n − 1) Soft constraints

1) exactly m edges

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy

Let Gn be the set of all simple graphs of size n. Consider a probability distribution P on Gn X S[P] = − P(G) log(P(G)) Gibbs entropy

G∈Gn Given a set of constraints C (number of edges, triangles ect.) Typical networks with given constraints are sampled from

P∗ = arg max S[P] ⇒ model P∈C

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Gn,m Hard constraints

2) m edges on average Gn,p p = m/(n − 1) Soft constraints

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy

Let Gn be the set of all simple graphs of size n. Consider a probability distribution P on Gn X S[P] = − P(G) log(P(G)) Gibbs entropy

G∈Gn Given a set of constraints C (number of edges, triangles ect.) Typical networks with given constraints are sampled from

P∗ = arg max S[P] ⇒ model P∈C

1) exactly m edges

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Hard constraints

2) m edges on average Gn,p p = m/(n − 1) Soft constraints

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy

Let Gn be the set of all simple graphs of size n. Consider a probability distribution P on Gn X S[P] = − P(G) log(P(G)) Gibbs entropy

G∈Gn Given a set of constraints C (number of edges, triangles ect.) Typical networks with given constraints are sampled from

P∗ = arg max S[P] ⇒ model P∈C

1) exactly m edges Gn,m

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Hard constraints

Gn,p p = m/(n − 1) Soft constraints

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy

Let Gn be the set of all simple graphs of size n. Consider a probability distribution P on Gn X S[P] = − P(G) log(P(G)) Gibbs entropy

G∈Gn Given a set of constraints C (number of edges, triangles ect.) Typical networks with given constraints are sampled from

P∗ = arg max S[P] ⇒ model P∈C

1) exactly m edges Gn,m 2) m edges on average

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Hard constraints Soft constraints

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy

Let Gn be the set of all simple graphs of size n. Consider a probability distribution P on Gn X S[P] = − P(G) log(P(G)) Gibbs entropy

G∈Gn Given a set of constraints C (number of edges, triangles ect.) Typical networks with given constraints are sampled from

P∗ = arg max S[P] ⇒ model P∈C

1) exactly m edges Gn,m

2) m edges on average Gn,p p = m/(n − 1)

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Soft constraints

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy

Let Gn be the set of all simple graphs of size n. Consider a probability distribution P on Gn X S[P] = − P(G) log(P(G)) Gibbs entropy

G∈Gn Given a set of constraints C (number of edges, triangles ect.) Typical networks with given constraints are sampled from

P∗ = arg max S[P] ⇒ model P∈C

1) exactly m edges Gn,m Hard constraints

2) m edges on average Gn,p p = m/(n − 1)

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy

Let Gn be the set of all simple graphs of size n. Consider a probability distribution P on Gn X S[P] = − P(G) log(P(G)) Gibbs entropy

G∈Gn Given a set of constraints C (number of edges, triangles ect.) Typical networks with given constraints are sampled from

P∗ = arg max S[P] ⇒ model P∈C

1) exactly m edges Gn,m Hard constraints

2) m edges on average Gn,p p = m/(n − 1) Soft constraints

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Soft conﬁguration model

Soft constraint E [Di ] = di

Consider a network of size n Measure (compute) degree sequence dn = d1,..., dn

Hard constraint Di = di Conﬁguration model

1 Connect each pair (i, j) with probability wij = λ +λ e i j +1 Where λi solve the system of equations: X 1 di = eλi +λj + 1 j6=i

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given (expected) degrees

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Soft conﬁguration model

Soft constraint E [Di ] = di

Measure (compute) degree sequence dn = d1,..., dn

Hard constraint Di = di Conﬁguration model

1 Connect each pair (i, j) with probability wij = λ +λ e i j +1 Where λi solve the system of equations: X 1 di = eλi +λj + 1 j6=i

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given (expected) degrees

Consider a network of size n

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Soft conﬁguration model

Soft constraint E [Di ] = di

Hard constraint Di = di Conﬁguration model

1 Connect each pair (i, j) with probability wij = λ +λ e i j +1 Where λi solve the system of equations: X 1 di = eλi +λj + 1 j6=i

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given (expected) degrees

Consider a network of size n Measure (compute) degree sequence dn = d1,..., dn

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Soft conﬁguration model

Soft constraint E [Di ] = di

1 Connect each pair (i, j) with probability wij = λ +λ e i j +1 Where λi solve the system of equations: X 1 di = eλi +λj + 1 j6=i

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given (expected) degrees

Consider a network of size n Measure (compute) degree sequence dn = d1,..., dn

Hard constraint Di = di Conﬁguration model

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Soft conﬁguration model

1 Connect each pair (i, j) with probability wij = λ +λ e i j +1 Where λi solve the system of equations: X 1 di = eλi +λj + 1 j6=i

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given (expected) degrees

Consider a network of size n Measure (compute) degree sequence dn = d1,..., dn

Hard constraint Di = di Conﬁguration model Soft constraint E [Di ] = di

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn 1 Connect each pair (i, j) with probability wij = λ +λ e i j +1 Where λi solve the system of equations: X 1 di = eλi +λj + 1 j6=i

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given (expected) degrees

Consider a network of size n Measure (compute) degree sequence dn = d1,..., dn

Hard constraint Di = di Conﬁguration model Soft constraint E [Di ] = di Soft conﬁguration model

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given (expected) degrees

Consider a network of size n Measure (compute) degree sequence dn = d1,..., dn

Hard constraint Di = di Conﬁguration model Soft constraint E [Di ] = di Soft conﬁguration model

1 Connect each pair (i, j) with probability wij = λ +λ e i j +1 Where λi solve the system of equations: X 1 di = eλi +λj + 1 j6=i

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given (expected) degrees

Consider a network of size n Measure (compute) degree sequence dn = d1,..., dn

Hard constraint Di = di Conﬁguration model Soft constraint E [Di ] = di Soft conﬁguration model

1 Connect each pair (i, j) with probability wij = λ +λ e i j +1 Where λi solve the system of equations: X 1 di = eλi +λj + 1 j6=i

X X E [Di ] = P (i − j) = wij = di . j6=i j6=i

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn lim (Dn = t) = (D = t) lim [Dn] = [D] := ν n→∞ P P n→∞ E E

Hypersoft Constraint: Sequences of graphs (Gn)n≥1

Given Constraint:

Hard/Soft Constraint: Gn for ﬁxed n

What is the correct optimization problem?

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given degree distribution

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn lim (Dn = t) = (D = t) lim [Dn] = [D] := ν n→∞ P P n→∞ E E

Hypersoft Constraint: Sequences of graphs (Gn)n≥1

Constraint:

Hard/Soft Constraint: Gn for ﬁxed n

What is the correct optimization problem?

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given degree distribution

Given (expected) degree sequence dn = d1,..., dn

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn lim (Dn = t) = (D = t) lim [Dn] = [D] := ν n→∞ P P n→∞ E E

Hypersoft Constraint: Sequences of graphs (Gn)n≥1

Constraint:

Hard/Soft Constraint: Gn for ﬁxed n

What is the correct optimization problem?

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given degree distribution

Given distribution P (D = t)

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Hypersoft Constraint: Sequences of graphs (Gn)n≥1

lim ( = t) = (D = t) lim [Dn] = [D] := ν n→∞ P P n→∞ E E

Hard/Soft Constraint: Gn for ﬁxed n

What is the correct optimization problem?

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given degree distribution

Given distribution P (D = t) Constraint:

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Hypersoft Constraint: Sequences of graphs (Gn)n≥1

lim [Dn] = [D] := ν n→∞ E E

Hard/Soft Constraint: Gn for ﬁxed n

What is the correct optimization problem?

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given degree distribution

Given distribution P (D = t) Constraint:

lim (Dn = t) = (D = t) n→∞ P P

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Hypersoft Constraint: Sequences of graphs (Gn)n≥1

Hard/Soft Constraint: Gn for ﬁxed n

What is the correct optimization problem?

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given degree distribution

Given distribution P (D = t) Constraint:

lim (Dn = t) = (D = t) lim [Dn] = [D] := ν n→∞ P P n→∞ E E

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Hypersoft Constraint: Sequences of graphs (Gn)n≥1

Hard/Soft Constraint: Gn for ﬁxed n

What is the correct optimization problem?

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given degree distribution

Given distribution P (D = t) Hypersoft Constraint:

lim (Dn = t) = (D = t) lim [Dn] = [D] := ν n→∞ P P n→∞ E E

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Hypersoft Constraint: Sequences of graphs (Gn)n≥1

What is the correct optimization problem?

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given degree distribution

Given distribution P (D = t) Hypersoft Constraint:

lim (Dn = t) = (D = t) lim [Dn] = [D] := ν n→∞ P P n→∞ E E

Hard/Soft Constraint: Gn for ﬁxed n

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn What is the correct optimization problem?

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given degree distribution

Given distribution P (D = t) Hypersoft Constraint:

lim (Dn = t) = (D = t) lim [Dn] = [D] := ν n→∞ P P n→∞ E E

Hard/Soft Constraint: Gn for ﬁxed n

Hypersoft Constraint: Sequences of graphs (Gn)n≥1

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphs with given degree distribution

Given distribution P (D = t) Hypersoft Constraint:

lim (Dn = t) = (D = t) lim [Dn] = [D] := ν n→∞ P P n→∞ E E

Hard/Soft Constraint: Gn for ﬁxed n

Hypersoft Constraint: Sequences of graphs (Gn)n≥1

What is the correct optimization problem?

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn i)( An)n≥1 a sequence of growing subsets An ⊆ An+1 ⊆ R ii)( µn)n≥1 probability measures on An iii) W : R → [0, 1], µn-integrable Graphon

A graphon ensemble (W , (An)n≥1, (µn)n≥1) consists of:

To sample a graph Gn of size n:

1) Take an i.i.d. sample xn = x1,..., xn from µn

2) Connect each pair of nodes i, j with probability W (xi , xj )

Classic setting of dense graphs:

An = I = [0, 1] and µn = µ uniform measure on I

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon ensembles

To sample a graph Gn of size n:

1) Take an i.i.d. sample xn = x1,..., xn from µn

2) Connect each pair of nodes i, j with probability W (xi , xj )

Classic setting of dense graphs:

An = I = [0, 1] and µn = µ uniform measure on I

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon ensembles

A graphon ensemble (W , (An)n≥1, (µn)n≥1) consists of:

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn ii)( µn)n≥1 probability measures on An iii) W : R → [0, 1], µn-integrable Graphon

To sample a graph Gn of size n:

1) Take an i.i.d. sample xn = x1,..., xn from µn

2) Connect each pair of nodes i, j with probability W (xi , xj )

Classic setting of dense graphs:

An = I = [0, 1] and µn = µ uniform measure on I

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon ensembles

A graphon ensemble (W , (An)n≥1, (µn)n≥1) consists of:

i)( An)n≥1 a sequence of growing subsets An ⊆ An+1 ⊆ R

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn iii) W : R → [0, 1], µn-integrable Graphon

To sample a graph Gn of size n:

1) Take an i.i.d. sample xn = x1,..., xn from µn

2) Connect each pair of nodes i, j with probability W (xi , xj )

Classic setting of dense graphs:

An = I = [0, 1] and µn = µ uniform measure on I

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon ensembles

A graphon ensemble (W , (An)n≥1, (µn)n≥1) consists of:

i)( An)n≥1 a sequence of growing subsets An ⊆ An+1 ⊆ R ii)( µn)n≥1 probability measures on An

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn To sample a graph Gn of size n:

1) Take an i.i.d. sample xn = x1,..., xn from µn

2) Connect each pair of nodes i, j with probability W (xi , xj )

Classic setting of dense graphs:

An = I = [0, 1] and µn = µ uniform measure on I

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon ensembles

A graphon ensemble (W , (An)n≥1, (µn)n≥1) consists of:

i)( An)n≥1 a sequence of growing subsets An ⊆ An+1 ⊆ R ii)( µn)n≥1 probability measures on An iii) W : R → [0, 1], µn-integrable Graphon

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Classic setting of dense graphs:

An = I = [0, 1] and µn = µ uniform measure on I

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon ensembles

A graphon ensemble (W , (An)n≥1, (µn)n≥1) consists of:

i)( An)n≥1 a sequence of growing subsets An ⊆ An+1 ⊆ R ii)( µn)n≥1 probability measures on An iii) W : R → [0, 1], µn-integrable Graphon

To sample a graph Gn of size n:

1) Take an i.i.d. sample xn = x1,..., xn from µn

2) Connect each pair of nodes i, j with probability W (xi , xj )

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon ensembles

A graphon ensemble (W , (An)n≥1, (µn)n≥1) consists of:

i)( An)n≥1 a sequence of growing subsets An ⊆ An+1 ⊆ R ii)( µn)n≥1 probability measures on An iii) W : R → [0, 1], µn-integrable Graphon

To sample a graph Gn of size n:

1) Take an i.i.d. sample xn = x1,..., xn from µn

2) Connect each pair of nodes i, j with probability W (xi , xj )

Classic setting of dense graphs:

An = I = [0, 1] and µn = µ uniform measure on I

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn AnAn+1 An+2 W : R × R → [0, 1]

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon model: Illustration

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn AnAn+1 An+2

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon model: Illustration

W : R × R → [0, 1]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn An+1 An+2

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon model: Illustration

An W : R × R → [0, 1]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn An+1 An+2

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon model: Illustration

An W : R × R → [0, 1]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn An An+2

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon model: Illustration

An+1 W : R × R → [0, 1]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn An An+2

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon model: Illustration

An+1 W : R × R → [0, 1]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn AnAn+1

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon model: Illustration

An+2 W : R × R → [0, 1]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn AnAn+1

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon model: Illustration

An+2 W : R × R → [0, 1]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn H(p) = −p log(p) − (1 − p) log(1 − p) Bernoulli entropy

ZZ σ[W , µ, A] = H(W (x, y)) dµ(x) dµ(y) Graphon Entropy A2

Let A ⊆ R, W : A → [0, 1] a graphon and µ a measure on A.

Graphon entropy maximization [vdH, Lippner and Krioukov 2017] Let w : A → [0, 1], µ-integrable. Then the solution Z W ∗ = arg max σ[W , µ, A]: w(x) = W (x, y) dµ(y) W is given by A 1 W ∗(x, y) = eλ(x)+λ(y) + 1

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon entropy maximization

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn H(p) = −p log(p) − (1 − p) log(1 − p) Bernoulli entropy

ZZ σ[W , µ, A] = H(W (x, y)) dµ(x) dµ(y) Graphon Entropy A2

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon entropy maximization

Let A ⊆ R, W : A → [0, 1] a graphon and µ a measure on A.

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn H(p) = −p log(p) − (1 − p) log(1 − p) Bernoulli entropy

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon entropy maximization

Let A ⊆ R, W : A → [0, 1] a graphon and µ a measure on A. ZZ σ[W , µ, A] = H(W (x, y)) dµ(x) dµ(y) Graphon Entropy A2

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Graphon entropy maximization [vdH, Lippner and Krioukov 2017] Let w : A → [0, 1], µ-integrable. Then the solution Z W ∗ = arg max σ[W , µ, A]: w(x) = W (x, y) dµ(y) W is given by A 1 W ∗(x, y) = eλ(x)+λ(y) + 1

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon entropy maximization

Let A ⊆ R, W : A → [0, 1] a graphon and µ a measure on A. ZZ σ[W , µ, A] = H(W (x, y)) dµ(x) dµ(y) Graphon Entropy A2 H(p) = −p log(p) − (1 − p) log(1 − p) Bernoulli entropy

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon entropy maximization

Let A ⊆ R, W : A → [0, 1] a graphon and µ a measure on A. ZZ σ[W , µ, A] = H(W (x, y)) dµ(x) dµ(y) Graphon Entropy A2 H(p) = −p log(p) − (1 − p) log(1 − p) Bernoulli entropy

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Graphon entropy maximization

Let A ⊆ R, W : A → [0, 1] a graphon and µ a measure on A. ZZ σ[W , µ, A] = H(W (x, y)) dµ(x) dµ(y) Graphon Entropy A2 H(p) = −p log(p) − (1 − p) log(1 − p) Bernoulli entropy

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Z 1 f (x) = W ∗(x, y) dy 0

Random graphs with a given degree sequence [Chatterjee et al. 2011]

Consider a sequence of degree sequences (dn)n≥1, dn = d1,..., dn Suppose there exists a function f : [0, 1] → [0, 1] Dn lim P ≤ t = P (f (U) ≤ t) n→∞ n

Convergence Theorem [Chatterjee et al. 2011]

Let (Gn)n≥1 be a sequence of graphs where Gn u.a.r. from all graphs with degree sequence dn. Then the limit of (Gn)n≥1 is given by the graphon 1 W ∗(x, y) = , eλ(x)+λ(y) + 1

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Dense graphs with given degrees

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Z 1 f (x) = W ∗(x, y) dy 0

Consider a sequence of degree sequences (dn)n≥1, dn = d1,..., dn Suppose there exists a function f : [0, 1] → [0, 1] Dn lim P ≤ t = P (f (U) ≤ t) n→∞ n

Convergence Theorem [Chatterjee et al. 2011]

Let (Gn)n≥1 be a sequence of graphs where Gn u.a.r. from all graphs with degree sequence dn. Then the limit of (Gn)n≥1 is given by the graphon 1 W ∗(x, y) = , eλ(x)+λ(y) + 1

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Dense graphs with given degrees

Random graphs with a given degree sequence [Chatterjee et al. 2011]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Z 1 f (x) = W ∗(x, y) dy 0

Suppose there exists a function f : [0, 1] → [0, 1] Dn lim P ≤ t = P (f (U) ≤ t) n→∞ n

Convergence Theorem [Chatterjee et al. 2011]

Let (Gn)n≥1 be a sequence of graphs where Gn u.a.r. from all graphs with degree sequence dn. Then the limit of (Gn)n≥1 is given by the graphon 1 W ∗(x, y) = , eλ(x)+λ(y) + 1

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Dense graphs with given degrees

Random graphs with a given degree sequence [Chatterjee et al. 2011]

Consider a sequence of degree sequences (dn)n≥1, dn = d1,..., dn

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Z 1 f (x) = W ∗(x, y) dy 0

Convergence Theorem [Chatterjee et al. 2011]

Let (Gn)n≥1 be a sequence of graphs where Gn u.a.r. from all graphs with degree sequence dn. Then the limit of (Gn)n≥1 is given by the graphon 1 W ∗(x, y) = , eλ(x)+λ(y) + 1

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Dense graphs with given degrees

Random graphs with a given degree sequence [Chatterjee et al. 2011]

Consider a sequence of degree sequences (dn)n≥1, dn = d1,..., dn Suppose there exists a function f : [0, 1] → [0, 1] Dn lim P ≤ t = P (f (U) ≤ t) n→∞ n

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Z 1 f (x) = W ∗(x, y) dy 0

Convergence Theorem [Chatterjee et al. 2011]

Let (Gn)n≥1 be a sequence of graphs where Gn u.a.r. from all graphs with degree sequence dn. Then the limit of (Gn)n≥1 is given by the graphon 1 W ∗(x, y) = , eλ(x)+λ(y) + 1

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Dense graphs with given degrees

Random graphs with a given degree sequence [Chatterjee et al. 2011]

Consider a sequence of degree sequences (dn)n≥1, dn = d1,..., dn Suppose there exists a function f : [0, 1] → [0, 1] Dn lim P ≤ t = P (f (U) ≤ t) n→∞ n

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Dense graphs with given degrees

Random graphs with a given degree sequence [Chatterjee et al. 2011]

Consider a sequence of degree sequences (dn)n≥1, dn = d1,..., dn Suppose there exists a function f : [0, 1] → [0, 1] Dn lim P ≤ t = P (f (U) ≤ t) n→∞ n

Convergence Theorem [Chatterjee et al. 2011]

Let (Gn)n≥1 be a sequence of graphs where Gn u.a.r. from all graphs with degree sequence dn. Then the limit of (Gn)n≥1 is given by the graphon 1 Z 1 W ∗(x, y) = , f (x) = W ∗(x, y) dy λ(x)+λ(y) e + 1 0

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Let (Gn)n≥1 be a sequence of dense graphs where Gn u.a.r. from all graphs with degree sequence dn. Then the limit of (Gn)n≥1 is given by the graphon

Convergence Theorem [Chatterjee et al. 2011]

Z 1 W ∗ f (x) = W ∗(x, y) dy 0

∗ What happens when (Gn)n≥1 is sampled from (W , A, µ)?

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy graphons with given degrees

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn ∗ What happens when (Gn)n≥1 is sampled from (W , A, µ)?

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy graphons with given degrees

Convergence Theorem [Chatterjee et al. 2011]

Let (Gn)n≥1 be a sequence of dense graphs where Gn u.a.r. from all graphs with degree sequence dn. Then the limit of (Gn)n≥1 is given by the graphon 1 Z 1 W ∗(x, y) = , f (x) = W ∗(x, y) dy λ(x)+λ(y) e + 1 0

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn ∗ What happens when (Gn)n≥1 is sampled from (W , A, µ)?

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy graphons with given degrees

Convergence Theorem [Chatterjee et al. 2011]

Let (Gn)n≥1 be a sequence of dense graphs where Gn u.a.r. from all graphs with degree sequence dn. Then the limit of (Gn)n≥1 is given by the graphon Z 1 W ∗ = arg max σ[W , µ, A]: f (x) = W ∗(x, y) dy W 0

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy graphons with given degrees

Convergence Theorem [Chatterjee et al. 2011]

Z 1 W ∗ = arg max σ[W , µ, A]: f (x) = W ∗(x, y) dy W 0

∗ What happens when (Gn)n≥1 is sampled from (W , A, µ)?

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy graphons with given degrees

Convergence Theorem [Chatterjee et al. 2011]

Z 1 W ∗ = arg max σ[W , µ, A]: f (x) = W ∗(x, y) dy W 0

∗ What happens when (Gn)n≥1 is sampled from (W , A, µ)? Dn lim P ≤ t = P (f (U) ≤ t) n→∞ n

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy graphons with given degrees

Convergence Theorem [Chatterjee et al. 2011]

Z 1 W ∗ = arg max σ[W , µ, A]: f (x) = W ∗(x, y) dy W 0

∗ What happens when (Gn)n≥1 is sampled from (W , A, µ)?

S[Gn] lim − σ[W ∗, µ, A] = 0 [Janson 2005] n→∞ n 2

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Maximum entropy graphons with given degrees

Convergence Theorem [Chatterjee et al. 2011]

Z 1 W ∗ = arg max σ[W , µ, A]: f (x) = W ∗(x, y) dy W 0

∗ What happens when (Gn)n≥1 is sampled from (W , A, µ)?

S[Gn] lim − 1 = 0 [Janson 2005] n→∞ n ∗ 2 σ[W , µ, A]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Z := W ∗(x, y) dµ(y) A

∗ ⇒ lim σ[W , An, µn] = 0 n→∞

Find (An)n≥1, An ⊆ R, µn measure on An and λ(x) such that if 1 W ∗(x, y) = eλ(x)+λ(y) + 1 ∗ then for (Gn)n≥1 sampled from (W , (An)≥1, (µn)≥1):

H.1 lim (Dn ≤ k) = (D ≤ k) n→∞ P P H.2 lim [Dn] = ν n→∞ E

S[G ] H.3 lim n − 1 = 0 n→∞ n ∗ 2 σ[W , µn, An]

Given: distribution P (D = k) with E [D] = ν < ∞

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Hypersoft entropy maximization problem

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Z := W ∗(x, y) dµ(y) A

∗ ⇒ lim σ[W , An, µn] = 0 n→∞

Find (An)n≥1, An ⊆ R, µn measure on An and λ(x) such that if 1 W ∗(x, y) = eλ(x)+λ(y) + 1 ∗ then for (Gn)n≥1 sampled from (W , (An)≥1, (µn)≥1):

H.1 lim (Dn ≤ k) = (D ≤ k) n→∞ P P H.2 lim [Dn] = ν n→∞ E

S[G ] H.3 lim n − 1 = 0 n→∞ n ∗ 2 σ[W , µn, An]

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Hypersoft entropy maximization problem

Given: distribution P (D = k) with E [D] = ν < ∞

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn ∗ ⇒ lim σ[W , An, µn] = 0 n→∞

Z := W ∗(x, y) dµ(y) A

H.1 lim (Dn ≤ k) = (D ≤ k) n→∞ P P H.2 lim [Dn] = ν n→∞ E

S[G ] H.3 lim n − 1 = 0 n→∞ n ∗ 2 σ[W , µn, An]

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Hypersoft entropy maximization problem

Given: distribution P (D = k) with E [D] = ν < ∞ Find (An)n≥1, An ⊆ R, µn measure on An and λ(x) such that if 1 W ∗(x, y) = eλ(x)+λ(y) + 1 ∗ then for (Gn)n≥1 sampled from (W , (An)≥1, (µn)≥1):

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Z := W ∗(x, y) dµ(y) A

∗ ⇒ lim σ[W , An, µn] = 0 n→∞

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Hypersoft entropy maximization problem

H.1 lim (Dn ≤ k) = (D ≤ k) n→∞ P P H.2 lim [Dn] = ν n→∞ E

S[G ] H.3 lim n − 1 = 0 n→∞ n ∗ 2 σ[W , µn, An]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn ∗ ⇒ lim σ[W , An, µn] = 0 n→∞

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Hypersoft entropy maximization problem

Given: distribution P (D = k) with E [D] = ν < ∞ Find (An)n≥1, An ⊆ R, µn measure on An and λ(x) such that if 1 Z W ∗(x, y) = f (x) := W ∗(x, y) dµ(y) λ(x)+λ(y) e + 1 A ∗ then for (Gn)n≥1 sampled from (W , (An)≥1, (µn)≥1):

H.1 lim (Dn ≤ k) = (D ≤ k) n→∞ P P H.2 lim [Dn] = ν n→∞ E

S[G ] H.3 lim n − 1 = 0 n→∞ n ∗ 2 σ[W , µn, An]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn ∗ ⇒ lim σ[W , An, µn] = 0 n→∞

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Hypersoft entropy maximization problem

Given: distribution P (D = k) with E [D] = ν < ∞ Find (An)n≥1, An ⊆ R, µn measure on An and λ(x) such that if 1 Z W ∗(x, y) = w (x) := W ∗(x, y) dµ (y) λ(x)+λ(y) n n e + 1 An ∗ then for (Gn)n≥1 sampled from (W , (An)≥1, (µn)≥1):

H.1 lim (Dn ≤ k) = (D ≤ k) n→∞ P P H.2 lim [Dn] = ν n→∞ E

S[G ] H.3 lim n − 1 = 0 n→∞ n ∗ 2 σ[W , µn, An]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn ∗ ⇒ lim σ[W , An, µn] = 0 n→∞

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Hypersoft entropy maximization problem

H.1 lim (Dn ≤ k) = (D ≤ k) n→∞ P P H.2 lim [Dn] = ν n→∞ E

S[G ] H.3 lim n − 1 = 0 n→∞ n ∗ 2 σ[W , µn, An]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Hypersoft entropy maximization problem

H.1 lim (Dn ≤ k) = (D ≤ k) n→∞ P P ∗ H.2 lim [Dn] = ν ⇒ lim σ[W , An, µn] = 0 n→∞ E n→∞

S[G ] H.3 lim n − 1 = 0 n→∞ n ∗ 2 σ[W , µn, An]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn ∗ ⇒ lim σ[W , An, µn] = 0 n→∞

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Hypersoft entropy maximization problem

Given: distribution P (D = k) with E [D] = ν< ∞ Find (An)n≥1, An ⊆ R, µn measure on An and λ(x) such that if 1 Z W ∗(x, y) = w (x) := W ∗(x, y) dµ (y) λ(x)+λ(y) n n e + 1 An ∗ then for (Gn)n≥1 sampled from (W , (An)≥1, (µn)≥1):

H.1 lim (Dn ≤ k) = (D ≤ k) n→∞ P P H.2 lim [Dn] = ν n→∞ E

S[G ] H.3 lim n − 1 = 0 n→∞ n ∗ 2 σ[W , µn, An]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn ∗ ⇒ lim σ[W , An, µn] = 0 n→∞

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Hypersoft entropy maximization problem

Given: distribution P (D = k) with E [D] = ν< ∞ Find( An)n≥1, An ⊆ R, µn measure on An and λ(x) such that if 1 Z W ∗(x, y) = w (x) := W ∗(x, y) dµ (y) λ(x)+λ(y) n n e + 1 An ∗ then for (Gn)n≥1 sampled from (W , (An)≥1, (µn)≥1):

H.1 lim (Dn ≤ k) = (D ≤ k) n→∞ P P H.2 lim [Dn] = ν n→∞ E

S[G ] H.3 lim n − 1 = 0 n→∞ n ∗ 2 σ[W , µn, An]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Y k (D = k) = e−Y P E k!

γx 1 n Take µ(x) = e and deﬁne Rn = 2 log( νβ2 ). Then the solution to the hypersoft entropy maximization problem is given by:

µ|An (−∞, Rn] µ(An)

Let γ > 1, ν > 0 and β = (γ − 1)/γ. ( (νβ)γt−γ t ≥ νβ P (Y > t) = 1 else

Main result [vdH, Lippner, Krioukov 2017]

1 W (x, y) = A = µ = e+ + 1 n n

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scale-free hypersoft entropy maximization problem

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn γx 1 n Take µ(x) = e and deﬁne Rn = 2 log( νβ2 ). Then the solution to the hypersoft entropy maximization problem is given by:

µ|An (−∞, Rn] µ(An)

Y k (D = k) = e−Y P E k!

Main result [vdH, Lippner, Krioukov 2017]

1 W (x, y) = A = µ = e+ + 1 n n

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scale-free hypersoft entropy maximization problem Let γ > 1, ν > 0 and β = (γ − 1)/γ. ( (νβ)γt−γ t ≥ νβ P (Y > t) = 1 else

µ|An (−∞, Rn] µ(An)

Main result [vdH, Lippner, Krioukov 2017]

1 W (x, y) = A = µ = e+ + 1 n n

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scale-free hypersoft entropy maximization problem Let γ > 1, ν > 0 and β = (γ − 1)/γ.

( γ −γ k (νβ) t t ≥ νβ Y −Y P (Y > t) = P (D = k) = E e 1 else k!

µ|An (−∞, Rn] µ(An)

Main result [vdH, Lippner, Krioukov 2017]

1 W (x, y) = A = µ = e+ + 1 n n

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scale-free hypersoft entropy maximization problem Let γ > 1, ν > 0 and β = (γ − 1)/γ.

( γ −γ k (νβ) t t ≥ νβ Y −Y P (Y > t) = P (D = k) = E e 1 else k!

µ|An (−∞, Rn] µ(An)

γ −γ P (D > t) ∼ γ(νβ) k as k → ∞

Main result [vdH, Lippner, Krioukov 2017]

1 W (x, y) = A = µ = e+ + 1 n n

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scale-free hypersoft entropy maximization problem Let γ > 1, ν > 0 and β = (γ − 1)/γ.

( γ −γ k (νβ) t t ≥ νβ Y −Y P (Y > t) = P (D = k) = E e 1 else k!

E [D] = E [Y ] = ν

µ|An (−∞, Rn] µ(An)

Main result [vdH, Lippner, Krioukov 2017]

1 W (x, y) = A = µ = e+ + 1 n n

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scale-free hypersoft entropy maximization problem Let γ > 1, ν > 0 and β = (γ − 1)/γ.

( γ −γ k (νβ) t t ≥ νβ Y −Y P (Y > t) = P (D = k) = E e 1 else k!

γ −γ E [D] = E [Y ] = ν P (D > t) ∼ γ(νβ) k as k → ∞

µ|An (−∞, Rn] µ(An)

Main result [vdH, Lippner, Krioukov 2017]

1 W (x, y) = A = µ = e+ + 1 n n

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scale-free hypersoft entropy maximization problem Let γ > 1, ν> 0 and β = (γ − 1)/γ.

( γ −γ k (νβ) t t ≥ νβ Y −Y P (Y > t) = P (D = k)= E e 1 else k!

µ|An (−∞, Rn] µ(An)

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scale-free hypersoft entropy maximization problem Let γ > 1, ν> 0 and β = (γ − 1)/γ.

( γ −γ k (νβ) t t ≥ νβ Y −Y P (Y > t) = P (D = k)= E e 1 else k!

Main result [vdH, Lippner, Krioukov 2017]

1 W (x, y) = A = µ = eλ(x)+λ(y) + 1 n n

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn µ|An (−∞, Rn] µ(An)

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scale-free hypersoft entropy maximization problem Let γ > 1, ν> 0 and β = (γ − 1)/γ.

( γ −γ k (νβ) t t ≥ νβ Y −Y P (Y > t) = P (D = k)= E e 1 else k!

Main result [vdH, Lippner, Krioukov 2017] γx 1 n Take µ(x) = e and deﬁne Rn = 2 log( νβ2 ). Then the solution to the hypersoft entropy maximization problem is given by: 1 W (x, y) = A = µ = eλ(x)+λ(y) + 1 n n

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scale-free hypersoft entropy maximization problem Let γ > 1, ν> 0 and β = (γ − 1)/γ.

( γ −γ k (νβ) t t ≥ νβ Y −Y P (Y > t) = P (D = k)= E e 1 else k!

Main result [vdH, Lippner, Krioukov 2017] γx 1 n Take µ(x) = e and deﬁne Rn = 2 log( νβ2 ). Then the solution to the hypersoft entropy maximization problem is given by:

1 µ|An W (x, y) = x+y An = (−∞, Rn] µn = e + 1 µ(An)

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scale-free hypersoft entropy maximization problem Let γ > 1, ν > 0 and β = (γ − 1)/γ.

( γ −γ k (νβ) t t ≥ νβ Y −Y P (Y > t) = P (D = k) = E e 1 else k!

Main result [vdH, Lippner, Krioukov 2017] γx 1 n Take µ(x) = e and deﬁne Rn = 2 log( νβ2 ). Then the solution to the hypersoft entropy maximization problem is given by:

1 µ|An W (x, y) = x+y An = (−∞, Rn] µn = e + 1 µ(An)

( γ(x−Rn) γe if x ≤ Rn µn(x) = 0 else

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scale-free hypersoft entropy maximization problem Let γ > 1, ν > 0 and β = (γ − 1)/γ.

( γ −γ k (νβ) t t ≥ νβ Y −Y P (Y > t) = P (D = k) = E e 1 else k!

Main result [vdH, Lippner, Krioukov 2017] γx 1 n Take µ(x) = e and deﬁne Rn = 2 log( νβ2 ). Then the solution to the hypersoft entropy maximization problem is given by:

1 µ|An W (x, y) = x+y An = (−∞, Rn] µn = e + 1 µ(An)

Hypersoft Conﬁguration Model (HSCM)

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Some simulations

0 10 100 Theory Theory 4 HSCM n=10 HSCM n=104 5 HSCM n=10 HSCM n=105 6 HSCM n=10 HSCM n=106 -5 10 Internet 10-5

-10 10 10-10

-15 10 10-15 0 2 4 6 10 10 10 10 100 101 102 103 104 105

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Hyperbolic random graphs [Krioukov et al. 2010]

Parameters: β ζ α Rn ≈ log(n)

Sample n points xi = (θi , ri ) independently sinh(αr) θ u.a.r. ∈ [0, 2π] ρ(r) = α cosh(αRn) − 1

1 W (xi , xj ) = βζ (d(x ,x )−R ) e 2 i j n + 1

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

So, where is the geometry?

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Parameters: β ζ α Rn ≈ log(n)

Sample n points xi = (θi , ri ) independently sinh(αr) θ u.a.r. ∈ [0, 2π] ρ(r) = α cosh(αRn) − 1

1 W (xi , xj ) = βζ (d(x ,x )−R ) e 2 i j n + 1

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

So, where is the geometry? Hyperbolic random graphs [Krioukov et al. 2010]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Sample n points xi = (θi , ri ) independently sinh(αr) θ u.a.r. ∈ [0, 2π] ρ(r) = α cosh(αRn) − 1

1 W (xi , xj ) = βζ (d(x ,x )−R ) e 2 i j n + 1

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

So, where is the geometry? Hyperbolic random graphs [Krioukov et al. 2010]

Parameters: β ζ α Rn ≈ log(n)

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn 1 W (xi , xj ) = βζ (d(x ,x )−R ) e 2 i j n + 1

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

So, where is the geometry? Hyperbolic random graphs [Krioukov et al. 2010]

Parameters: β ζ α Rn ≈ log(n)

Sample n points xi = (θi , ri ) independently sinh(αr) θ u.a.r. ∈ [0, 2π] ρ(r) = α cosh(αRn) − 1

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

So, where is the geometry? Hyperbolic random graphs [Krioukov et al. 2010]

Parameters: β ζ α Rn ≈ log(n)

Sample n points xi = (θi , ri ) independently sinh(αr) θ u.a.r. ∈ [0, 2π] ρ(r) = α cosh(αRn) − 1

1 W (xi , xj ) = βζ (d(x ,x )−R ) e 2 i j n + 1

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

So, where is the geometry? Hyperbolic random graphs [Krioukov et al. 2010]

Parameters: β ζ α Rn ≈ log(n)

Sample n points xi = (θi , ri ) independently sinh(αr) θ u.a.r. ∈ [0, 2π] ρ(r) = α cosh(αRn) − 1

1 W (xi , xj ) = βζ (d(x ,x )−R ) e 2 i j n + 1 1 d(x , x ) = arccos (cosh(ζr ) cosh(ζr ) − sinh(ζr ) sinh(ζr ) cos(∆θ )) i j ζ i j i j ij

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn 1 = ezi +zj + 1

η R r − n → z 2 i 2 i

Sample n points xi = (θi , ri ) independently (Rn ≈ log(n))

θ u.a.r. ∈ [0, 2π] ρ() ≈ αeα()

1 W (xi , xj ) = βζ (d(x ,x )−R ) e 2 i j n + 1

ζ → ∞ β → 0 η := βζ constant

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Really, where is the geometry?

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn η R r − n → z 2 i 2 i

1 = ezi +zj + 1

ζ → ∞ β → 0 η := βζ constant

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Really, where is the geometry?

Sample n points xi = (θi , ri ) independently (Rn ≈ log(n))

θ u.a.r. ∈ [0, 2π] ρ(r) ≈ αeα(r−Rn)

1 W (xi , xj ) = βζ (d(x ,x )−R ) e 2 i j n + 1

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn 1 = ezi +zj + 1

η R r − n → z 2 i 2 i

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Really, where is the geometry?

Sample n points xi = (θi , ri ) independently (Rn ≈ log(n))

θ u.a.r. ∈ [0, 2π] ρ(r) ≈ αeα(r−Rn)

1 W (xi , xj ) = βζ (d(x ,x )−R ) e 2 i j n + 1

ζ → ∞ β → 0 η := βζ constant

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn 1 = ezi +zj + 1

η R r − n → z 2 i 2 i

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Really, where is the geometry?

Sample n points xi = (θi , ri ) independently (Rn ≈ log(n))

θ u.a.r. ∈ [0, 2π] ρ(r) ≈ αeα(r−Rn)

1 W (xi , xj ) = βζ (d(x ,x )−R ) e 2 i j n + 1

ζ → ∞ β → 0 η := βζ constant

2 ∆θ For large ζ : d(x , x ) ≈ r + r + log ij i j i j ζ 2

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn 1 = ezi +zj + 1

η R r − n → z 2 i 2 i

2 ∆θ + log ij ζ 2

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Really, where is the geometry?

Sample n points xi = (θi , ri ) independently (Rn ≈ log(n))

θ u.a.r. ∈ [0, 2π] ρ(r) ≈ αeα(r−Rn)

1 W (xi , xj ) = βζ (d(x ,x )−R ) e 2 i j n + 1

ζ → ∞ β → 0 η := βζ constant

For large ζ : d(xi , xj ) ≈ ri + rj

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn 1 = ezi +zj + 1

2 ∆θ + log ij ζ 2

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Really, where is the geometry?

Sample n points xi = (θi , ri ) independently (Rn ≈ log(n))

θ u.a.r. ∈ [0, 2π] ρ(r) ≈ αeα(r−Rn)

1 W (xi , xj ) = βζ (d(x ,x )−R ) e 2 i j n + 1 η R ζ → ∞ β → 0 η := βζ constant r − n → z 2 i 2 i

For large ζ : d(xi , xj ) ≈ ri + rj

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn 1 = ezi +zj + 1

2 ∆θ + log ij ζ 2

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Really, where is the geometry?

Sample n points xi = (θi , ri ) independently (Rn ≈ log(n))

0 α0(z− Rn ) θ u.a.r. ∈ [0, 2π] ρ(z) ≈ α e 2

1 W (xi , xj ) = βζ (d(x ,x )−R ) e 2 i j n + 1 η R ζ → ∞ β → 0 η := βζ constant r − n → z 2 i 2 i

For large ζ : d(xi , xj ) ≈ ri + rj

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn 2 ∆θ + log ij ζ 2

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Really, where is the geometry?

Sample n points xi = (θi , ri ) independently (Rn ≈ log(n))

0 α0(z− Rn ) θ u.a.r. ∈ [0, 2π] ρ(z) ≈ α e 2

1 1 W (xi , xj ) = = βζ (d(x ,x )−R ) zi +zj e 2 i j n + 1 e + 1 η R ζ → ∞ β → 0 η := βζ constant r − n → z 2 i 2 i

For large ζ : d(xi , xj ) ≈ ri + rj

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Really, where is the geometry?

Sample n points xi = (θi , ri ) independently (Rn ≈ log(n))

0 α0(z− Rn ) θ u.a.r. ∈ [0, 2π] ρ(z) ≈ α e 2

1 1 W (xi , xj ) = = βζ (d(x ,x )−R ) zi +zj e 2 i j n + 1 e + 1 η R ζ → ∞ β → 0 η := βζ constant r − n → z 2 i 2 i

HSCM is zero clustering limit of Hyperbolic Random Geometric Graphs

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn H.2 lim [] = ν n→∞ E n X D1 = I1j E [Iij | xi , xj ] = W (xi , xj ) j=2 E [D1| x1] =

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Average degree

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn n X D1 = I1j E [Iij | xi , xj ] = W (xi , xj ) j=2 E [D1| x1] =

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Average degree

H.2 lim [Dn] = ν n→∞ E

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn n X D1 = I1j E [Iij | xi , xj ] = W (xi , xj ) j=2 E [D1| x1] =

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Average degree

H.2 lim [D1] = ν n→∞ E

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn E [D1| x1] =

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Average degree

H.2 lim [D1] = ν n→∞ E n X D1 = I1j E [Iij | xi , xj ] = W (xi , xj ) j=2

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Average degree

H.2 lim [D1] = ν n→∞ E n X D1 = I1j E [Iij | xi , xj ] = W (xi , xj ) j=2

E [D1| x1] =

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Average degree

H.2 lim [D1] = ν n→∞ E n X D1 = I1j E [Iij | xi , xj ] = W (xi , xj ) j=2   n X E [D1| x1] = E  W (x1, xj ) x1

j=2

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Average degree

H.2 lim [D1] = ν n→∞ E n X D1 = I1j E [Iij | xi , xj ] = W (xi , xj ) j=2 Z E [D1| x1] = (n − 1) W (xi , y) dµn(y) An

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Average degree

H.2 lim [D1] = ν n→∞ E n X D1 = I1j E [Iij | xi , xj ] = W (xi , xj ) j=2

E [D1| x1] = (n − 1)wn(x1)

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Average degree

H.2 lim [D1] = ν n→∞ E n X D1 = I1j E [Iij | xi , xj ] = W (xi , xj ) j=2

E [D1| x1] = (n − 1)wn(x1)

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Average degree

H.2 lim [D1] = ν n→∞ E n X D1 = I1j E [Iij | xi , xj ] = W (xi , xj ) j=2

E [D1| x1] = (n − 1)wn(x1)

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0

2 Z Rn −x nE [wn(x)] = n e dµn(x) + o(1) 0

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Average degree

H.2 lim [D1] = ν n→∞ E n X D1 = I1j E [Iij | xi , xj ] = W (xi , xj ) j=2

E [D1| x1] = (n − 1)wn(x1)

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0

n n [w (x)] = e−2Rn + o(1) = ν + o(1) E n β2

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn d d Poi ((n − 1)wn(X )) = Poi (Y ) = D

H.1 lim ( ≤ k) = (D ≤ k) n→∞ P P

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0 n X D1 = I1j E [D1| x1] = (n − 1)wn(x1) j=2

d D1 = X ∼ µn

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Degree distribution

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn d d Poi ((n − 1)wn(X )) = Poi (Y ) = D

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0 n X D1 = I1j E [D1| x1] = (n − 1)wn(x1) j=2

d D1 = X ∼ µn

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Degree distribution

H.1 lim (Dn ≤ k) = (D ≤ k) n→∞ P P

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn d d Poi ((n − 1)wn(X )) = Poi (Y ) = D

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0 n X D1 = I1j E [D1| x1] = (n − 1)wn(x1) j=2

d D1 = X ∼ µn

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Degree distribution

H.1 lim (D1 ≤ k) = (D ≤ k) n→∞ P P

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn d d Poi ((n − 1)wn(X )) = Poi (Y ) = D

n X D1 = I1j E [D1| x1] = (n − 1)wn(x1) j=2

d D1 = X ∼ µn

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Degree distribution

H.1 lim (D1 ≤ k) = (D ≤ k) n→∞ P P

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn d d Poi ((n − 1)wn(X )) = Poi (Y ) = D

d D1 = X ∼ µn

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Degree distribution

H.1 lim (D1 ≤ k) = (D ≤ k) n→∞ P P

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0 n X D1 = I1j E [D1| x1] = (n − 1)wn(x1) j=2

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn d d Poi ((n − 1)wn(X )) = Poi (Y ) = D

d D1 = X ∼ µn

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Degree distribution

H.1 lim (D1 ≤ k) = (D ≤ k) n→∞ P P

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0 n X D1 = I1j E [D1| x1] = (n − 1)wn(x1) j=2

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn =d Poi (Y ) =d D

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Degree distribution

H.1 lim (D1 ≤ k) = (D ≤ k) n→∞ P P

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0 n X D1 = I1j E [D1| x1] = (n − 1)wn(x1) j=2

d D1 = Poi ((n − 1)wn(X )) X ∼ µn

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn =d Poi (Y ) =d D

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Degree distribution

H.1 lim (D1 ≤ k) = (D ≤ k) n→∞ P P

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0 n X D1 = I1j E [D1| x1] = (n − 1)wn(x1) j=2

d D1 = Poi ((n − 1)wn(X )) X ∼ µn

γ −γ P ((n − 1)wn(X ) > t) ≈ (νβ) t (1+o(1)) = P (Y > t) (1+o(1))

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn =d Poi (Y ) =d D

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Degree distribution

H.1 lim (D1 ≤ k) = (D ≤ k) n→∞ P P

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0 n X D1 = I1j E [D1| x1] = (n − 1)wn(x1) j=2

d D1 = Poi ((n − 1)wn(X )) X ∼ µn

Z d wn(x) = W (x, y) dµn(y) if X ∼ µn wn(X ) → Y An

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Degree distribution

H.1 lim (D1 ≤ k) = (D ≤ k) n→∞ P P

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0 n X D1 = I1j E [D1| x1] = (n − 1)wn(x1) j=2

d d d D1 = Poi ((n − 1)wn(X )) = Poi (Y ) = D X ∼ µn

Z d wn(x) = W (x, y) dµn(y) if X ∼ µn wn(X ) → Y An

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

HSCM Degree distribution

H.1 lim (D1 ≤ k) = (D ≤ k) n→∞ P P

Z Rn −x−y −x −y W (x, y) ≈ e x, y ≥ 0 wn(x) ≈ e e dµn(y) 0 n X D1 = I1j E [D1| x1] = (n − 1)wn(x1) j=2

d d d D1 = Poi ((n − 1)wn(X )) = Poi (Y ) = D X ∼ µn

Z d wn(x) = W (x, y) dµn(y) if X ∼ µn wn(X ) → Y An

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn W (Rn, Rn) → 0

lim σ[W , µn, An] = n→∞

H(W (x, y)) = (x + y)W (x, y) − (1 − W (x, y)) log(1 − W (x, y))

( γ(x−Rn) γe if x ≤ Rn µn(x) = 0 else

σ[W , µn, An] ≈

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scaling of the entropy

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn W (Rn, Rn) → 0

H(W (x, y)) = (x + y)W (x, y) − (1 − W (x, y)) log(1 − W (x, y))

( γ(x−Rn) γe if x ≤ Rn µn(x) = 0 else

σ[W , µn, An] ≈

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scaling of the entropy

lim anσ[W , µn, An] = a n→∞

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn W (Rn, Rn) → 0

H(W (x, y)) = (x + y)W (x, y) − (1 − W (x, y)) log(1 − W (x, y))

( γ(x−Rn) γe if x ≤ Rn µn(x) = 0 else

σ[W , µn, An] ≈

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scaling of the entropy

n lim σ[W , µn, An] = ν n→∞ log(n)

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn W (Rn, Rn) → 0

( γ(x−Rn) γe if x ≤ Rn µn(x) = 0 else

σ[W , µn, An] ≈

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scaling of the entropy

n lim σ[W , µn, An] = ν n→∞ log(n)

H(W (x, y)) = (x + y)W (x, y) − (1 − W (x, y)) log(1 − W (x, y))

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn W (Rn, Rn) → 0

( γ(x−Rn) γe if x ≤ Rn µn(x) = 0 else

σ[W , µn, An] ≈

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scaling of the entropy

n lim σ[W , µn, An] = ν n→∞ log(n)

H(W (x, y)) = (x + y)W (x, y) − (1 − W (x, y)) log(1 − W (x, y))

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn W (Rn, Rn) → 0

σ[W , µn, An] ≈

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scaling of the entropy

n lim σ[W , µn, An] = ν n→∞ log(n)

H(W (x, y)) = (x + y)W (x, y) − (1 − W (x, y)) log(1 − W (x, y))

( γ(x−Rn) γe if x ≤ Rn µn(x) = 0 else

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn σ[W , µn, An] ≈

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scaling of the entropy

n lim σ[W , µn, An] = ν n→∞ log(n)

H(W (x, y)) = (x + y)W (x, y) − (1 − W (x, y)) log(1 − W (x, y))

( γ(x−Rn) γe if x ≤ Rn µn(x) = W (Rn, Rn) → 0 0 else

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scaling of the entropy

n lim σ[W , µn, An] = ν n→∞ log(n)

H(W (x, y)) = (x + y)W (x, y) − (1 − W (x, y)) log(1 − W (x, y))

( γ(x−Rn) γe if x ≤ Rn µn(x) = W (Rn, Rn) → 0 0 else

ZZ σ[W , µn, An] ≈ (x + y)W (x, y) dµn(x) dµn(y) 2 An

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Scaling of the entropy

n lim σ[W , µn, An] = ν n→∞ log(n)

H(W (x, y)) = (x + y)W (x, y) − (1 − W (x, y)) log(1 − W (x, y))

( γ(x−Rn) γe if x ≤ Rn µn(x) = W (Rn, Rn) → 0 0 else

2R log(n) σ[W , µ , A ] ≈ n e−2Rn + O(n−1) = ν + O(n−1) n n β2 n

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn n 2

σ[W , µn, An] ≈

S[G ] H.3 lim n − 1 = 0 n→∞ n 2 σ[W , µn, An]

Partition An in to mn intervals It

1 ZZ Wf(x, y) = W (x, y) dµn(y) dµn(x) if (x, y) ∈ It × Is µts It ×Is

log(mn) n → 0 σ[W , µn, An] − σ[Wf, µn, An] → 0 log(n) log(n)

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Convergence graphon entropy

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn n 2

σ[W , µn, An] ≈

Partition An in to mn intervals It

1 ZZ Wf(x, y) = W (x, y) dµn(y) dµn(x) if (x, y) ∈ It × Is µts It ×Is

log(mn) n → 0 σ[W , µn, An] − σ[Wf, µn, An] → 0 log(n) log(n)

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Convergence graphon entropy

S[G ] H.3 lim n − 1 = 0 n→∞ n 2 σ[W , µn, An]

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn n 2

σ[W , µn, An] ≈

n ≤ n log(mn) + σ[Wf, µn, An] 2

Partition An in to mn intervals It

1 ZZ Wf(x, y) = W (x, y) dµn(y) dµn(x) if (x, y) ∈ It × Is µts It ×Is

log(mn) n → 0 σ[W , µn, An] − σ[Wf, µn, An] → 0 log(n) log(n)

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Convergence graphon entropy

S[G ] H.3 lim n − 1 = 0 n→∞ n 2 σ[W , µn, An]

n σ[W , µ , A ] ≤ S[G ] 2 n n n

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn n 2

σ[W , µn, An] ≈

n ≤ n log(mn) + σ[Wf, µn, An] 2

log(mn) n → 0 σ[W , µn, An] − σ[Wf, µn, An] → 0 log(n) log(n)

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Convergence graphon entropy

S[G ] H.3 lim n − 1 = 0 n→∞ n 2 σ[W , µn, An]

n σ[W , µ , A ] ≤ S[G ] 2 n n n

Partition An in to mn intervals It

1 ZZ Wf(x, y) = W (x, y) dµn(y) dµn(x) if (x, y) ∈ It × Is µts It ×Is

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn n 2

σ[W , µn, An] ≈

log(mn) n → 0 σ[W , µn, An] − σ[Wf, µn, An] → 0 log(n) log(n)

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Convergence graphon entropy

S[G ] H.3 lim n − 1 = 0 n→∞ n 2 σ[W , µn, An]

n n σ[W , µn, An] ≤ S[Gn] ≤ n log(mn) + σ[Wf, µn, An] 2 2

Partition An in to mn intervals It

1 ZZ Wf(x, y) = W (x, y) dµn(y) dµn(x) if (x, y) ∈ It × Is µts It ×Is

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn n 2

log(mn) n → 0 σ[W , µn, An] − σ[Wf, µn, An] → 0 log(n) log(n)

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Convergence graphon entropy

S[Gn] log(n) H.3 lim − 1 = 0 σ[W , µn, An] ≈ n→∞ n 2 σ[W , µn, An] n

n n σ[W , µn, An] ≤ S[Gn] ≤ n log(mn) + σ[Wf, µn, An] 2 2

Partition An in to mn intervals It

1 ZZ Wf(x, y) = W (x, y) dµn(y) dµn(x) if (x, y) ∈ It × Is µts It ×Is

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn log(mn) n → 0 σ[W , µn, An] − σ[Wf, µn, An] → 0 log(n) log(n)

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Convergence graphon entropy

S[Gn] n H.3 lim − 1 = 0 σ[W , µn, An] ≈ n log(n) n→∞ n 2 σ[W , µn, An] 2

n n σ[W , µn, An] ≤ S[Gn] ≤ n log(mn) + σ[Wf, µn, An] 2 2

Partition An in to mn intervals It

1 ZZ Wf(x, y) = W (x, y) dµn(y) dµn(x) if (x, y) ∈ It × Is µts It ×Is

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn log(mn) n → 0 σ[W , µn, An] − σ[Wf, µn, An] → 0 log(n) log(n)

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Convergence graphon entropy

S[Gn] n H.3 lim − 1 = 0 σ[W , µn, An] ≈ n log(n) n→∞ n 2 σ[W , µn, An] 2

S[G ] log(m ) σ[W , µ , A ] n n f n n n − 1 ≤ + 2 σ[W , µn, An] log(n) σ[W , µn, An]

Partition An in to mn intervals It

1 ZZ Wf(x, y) = W (x, y) dµn(y) dµn(x) if (x, y) ∈ It × Is µts It ×Is

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Convergence graphon entropy

S[Gn] n H.3 lim − 1 = 0 σ[W , µn, An] ≈ n log(n) n→∞ n 2 σ[W , µn, An] 2

S[G ] log(m ) σ[W , µ , A ] n n f n n n − 1 ≤ + 2 σ[W , µn, An] log(n) σ[W , µn, An]

Partition An in to mn intervals It

1 ZZ Wf(x, y) = W (x, y) dµn(y) dµn(x) if (x, y) ∈ It × Is µts It ×Is

log(mn) n → 0 σ[W , µn, An] − σ[Wf, µn, An] → 0 log(n) log(n)

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Convergence graphon entropy

S[Gn] n H.3 lim − 1 = 0 σ[W , µn, An] ≈ n log(n) n→∞ n 2 σ[W , µn, An] 2

S[G ] log(m ) σ[W , µ , A ] n n f n n n − 1 ≤ + 2 σ[W , µn, An] log(n) σ[W , µn, An]

Partition An in to mn intervals It

1 ZZ Wf(x, y) = W (x, y) dµn(y) dµn(x) if (x, y) ∈ It × Is µts It ×Is

log(mn) n → 0 σ[W , µn, An] − σ[Wf, µn, An] → 0 log(n) log(n)

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn • Deﬁned hypersoft entropy maximization problem for degrees • Showed model that solves this problem for scale-free degree distributions • Explained relation to Hyperbolic Random Graph model

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Summary

Current research:

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn • Showed model that solves this problem for scale-free degree distributions • Explained relation to Hyperbolic Random Graph model

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Summary

Current research: • Deﬁned hypersoft entropy maximization problem for degrees

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn • Explained relation to Hyperbolic Random Graph model

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Summary

Current research: • Deﬁned hypersoft entropy maximization problem for degrees • Showed model that solves this problem for scale-free degree distributions

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Summary

Current research: • Deﬁned hypersoft entropy maximization problem for degrees • Showed model that solves this problem for scale-free degree distributions • Explained relation to Hyperbolic Random Graph model

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn • Extend sparse entropy maximization to general case • Look at diﬀerent topological constraints (triangles, clustering) • Relation between the max entropy graphon ensemble and limits of sparse graphs • Establish link between geometry and sparse graph limits

Future research: • Extend graphon entropy convergence to general case

Paper: P. van der Hoorn, G. Lippner and D. Krioukov Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution arXiv:1705.10261

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Future research

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Future research Future research: • Extend graphon entropy convergence to general case

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn • Look at diﬀerent topological constraints (triangles, clustering) • Relation between the max entropy graphon ensemble and limits of sparse graphs • Establish link between geometry and sparse graph limits Paper: P. van der Hoorn, G. Lippner and D. Krioukov Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution arXiv:1705.10261

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Future research Future research: • Extend graphon entropy convergence to general case • Extend sparse entropy maximization to general case

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn • Relation between the max entropy graphon ensemble and limits of sparse graphs • Establish link between geometry and sparse graph limits Paper: P. van der Hoorn, G. Lippner and D. Krioukov Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution arXiv:1705.10261

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn • Establish link between geometry and sparse graph limits Paper: P. van der Hoorn, G. Lippner and D. Krioukov Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution arXiv:1705.10261

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Paper: P. van der Hoorn, G. Lippner and D. Krioukov Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution arXiv:1705.10261

Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Future research Future research: • Extend graphon entropy convergence to general case • Extend sparse entropy maximization to general case • Look at diﬀerent topological constraints (triangles, clustering) • Relation between the max entropy graphon ensemble and limits of sparse graphs • Establish link between geometry and sparse graph limits

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn Introduction Graphs with given degrees Graphons and dense graphs Hypersoft Conﬁguration Model

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn