spectral analysis of sparse random graphs justin salez...
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![Page 1: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/1.jpg)
Spectral analysis of sparse random graphsJustin Salez (LPSM)
![Page 2: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/2.jpg)
Spectral graph theory
A graph G = (V ,E ) can be represented by its adjacency matrix:
Aij =
{1 if {i , j} ∈ E0 otherwise.
Eigenvalues λ1 ≥ . . . ≥ λ|V | capture essential information about G .
B It is convenient to encode them into a probability measure:
µG :=1
|V |
|V |∑k=1
δλk .
Question: how does µG typically look when G is large?
![Page 3: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/3.jpg)
Spectral graph theory
A graph G = (V ,E ) can be represented by its adjacency matrix:
Aij =
{1 if {i , j} ∈ E0 otherwise.
Eigenvalues λ1 ≥ . . . ≥ λ|V | capture essential information about G .
B It is convenient to encode them into a probability measure:
µG :=1
|V |
|V |∑k=1
δλk .
Question: how does µG typically look when G is large?
![Page 4: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/4.jpg)
Spectral graph theory
A graph G = (V ,E ) can be represented by its adjacency matrix:
Aij =
{1 if {i , j} ∈ E0 otherwise.
Eigenvalues λ1 ≥ . . . ≥ λ|V | capture essential information about G .
B It is convenient to encode them into a probability measure:
µG :=1
|V |
|V |∑k=1
δλk .
Question: how does µG typically look when G is large?
![Page 5: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/5.jpg)
Spectral graph theory
A graph G = (V ,E ) can be represented by its adjacency matrix:
Aij =
{1 if {i , j} ∈ E0 otherwise.
Eigenvalues λ1 ≥ . . . ≥ λ|V | capture essential information about G .
B It is convenient to encode them into a probability measure:
µG :=1
|V |
|V |∑k=1
δλk .
Question: how does µG typically look when G is large?
![Page 6: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/6.jpg)
Spectral graph theory
A graph G = (V ,E ) can be represented by its adjacency matrix:
Aij =
{1 if {i , j} ∈ E0 otherwise.
Eigenvalues λ1 ≥ . . . ≥ λ|V | capture essential information about G .
B It is convenient to encode them into a probability measure:
µG :=1
|V |
|V |∑k=1
δλk .
Question: how does µG typically look when G is large?
![Page 7: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/7.jpg)
Spectrum of a uniform random graph on 10000 vertices
![Page 8: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/8.jpg)
Spectrum of a uniform random graph on 10000 vertices
![Page 9: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/9.jpg)
Erdos-Renyi model with average degree 3
![Page 10: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/10.jpg)
Uniform random 3-regular graph
![Page 11: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/11.jpg)
Uniform random tree on 3000 nodes
![Page 12: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/12.jpg)
Sparse graphs: the need for a theory
Wigner’s universality is restricted to the dense regime |E | � |V |,but real-world networks are embarassingly sparse: |E | � |V |.
I Real-world graphs: beyond the semicircle law (Farkas 2001).
I Graph spectra for complex networks (Piet van Mieghem 2010).
In the sparse regime, the spectrum µG typically concentratesaround a model-dependent limit µ, about which little is known.
Main challenge: understand the fundamental decomposition
µ = µpp + µac + µsc
in terms of the geometry of the underlying graph model.
Conjectures were proposed by Bordenave, Sen, Virag (JEMS 2017).
![Page 13: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/13.jpg)
Sparse graphs: the need for a theory
Wigner’s universality is restricted to the dense regime |E | � |V |,but real-world networks are embarassingly sparse: |E | � |V |.
I Real-world graphs: beyond the semicircle law (Farkas 2001).
I Graph spectra for complex networks (Piet van Mieghem 2010).
In the sparse regime, the spectrum µG typically concentratesaround a model-dependent limit µ, about which little is known.
Main challenge: understand the fundamental decomposition
µ = µpp + µac + µsc
in terms of the geometry of the underlying graph model.
Conjectures were proposed by Bordenave, Sen, Virag (JEMS 2017).
![Page 14: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/14.jpg)
Sparse graphs: the need for a theory
Wigner’s universality is restricted to the dense regime |E | � |V |,but real-world networks are embarassingly sparse: |E | � |V |.I Real-world graphs: beyond the semicircle law (Farkas 2001).
I Graph spectra for complex networks (Piet van Mieghem 2010).
In the sparse regime, the spectrum µG typically concentratesaround a model-dependent limit µ, about which little is known.
Main challenge: understand the fundamental decomposition
µ = µpp + µac + µsc
in terms of the geometry of the underlying graph model.
Conjectures were proposed by Bordenave, Sen, Virag (JEMS 2017).
![Page 15: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/15.jpg)
Sparse graphs: the need for a theory
Wigner’s universality is restricted to the dense regime |E | � |V |,but real-world networks are embarassingly sparse: |E | � |V |.I Real-world graphs: beyond the semicircle law (Farkas 2001).
I Graph spectra for complex networks (Piet van Mieghem 2010).
In the sparse regime, the spectrum µG typically concentratesaround a model-dependent limit µ, about which little is known.
Main challenge: understand the fundamental decomposition
µ = µpp + µac + µsc
in terms of the geometry of the underlying graph model.
Conjectures were proposed by Bordenave, Sen, Virag (JEMS 2017).
![Page 16: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/16.jpg)
Sparse graphs: the need for a theory
Wigner’s universality is restricted to the dense regime |E | � |V |,but real-world networks are embarassingly sparse: |E | � |V |.I Real-world graphs: beyond the semicircle law (Farkas 2001).
I Graph spectra for complex networks (Piet van Mieghem 2010).
In the sparse regime, the spectrum µG typically concentratesaround a model-dependent limit µ, about which little is known.
Main challenge: understand the fundamental decomposition
µ = µpp + µac + µsc
in terms of the geometry of the underlying graph model.
Conjectures were proposed by Bordenave, Sen, Virag (JEMS 2017).
![Page 17: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/17.jpg)
Sparse graphs: the need for a theory
Wigner’s universality is restricted to the dense regime |E | � |V |,but real-world networks are embarassingly sparse: |E | � |V |.I Real-world graphs: beyond the semicircle law (Farkas 2001).
I Graph spectra for complex networks (Piet van Mieghem 2010).
In the sparse regime, the spectrum µG typically concentratesaround a model-dependent limit µ, about which little is known.
Main challenge: understand the fundamental decomposition
µ = µpp + µac + µsc
in terms of the geometry of the underlying graph model.
Conjectures were proposed by Bordenave, Sen, Virag (JEMS 2017).
![Page 18: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/18.jpg)
Sparse graphs: the need for a theory
Wigner’s universality is restricted to the dense regime |E | � |V |,but real-world networks are embarassingly sparse: |E | � |V |.I Real-world graphs: beyond the semicircle law (Farkas 2001).
I Graph spectra for complex networks (Piet van Mieghem 2010).
In the sparse regime, the spectrum µG typically concentratesaround a model-dependent limit µ, about which little is known.
Main challenge: understand the fundamental decomposition
µ = µpp + µac + µsc
in terms of the geometry of the underlying graph model.
Conjectures were proposed by Bordenave, Sen, Virag (JEMS 2017).
![Page 19: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/19.jpg)
Local convergence of rooted graphs
Write BR(G , o) for the ball of radius R around the vertex o in G :
Say that (Gn, on)→ (G , o) if for each fixed R and n large enough,
BR(Gn, on) ≡ BR(G , o).
G? := {locally finite, connected rooted graphs} is a Polish space.
![Page 20: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/20.jpg)
Local convergence of rooted graphs
Write BR(G , o) for the ball of radius R around the vertex o in G :
Say that (Gn, on)→ (G , o) if for each fixed R and n large enough,
BR(Gn, on) ≡ BR(G , o).
G? := {locally finite, connected rooted graphs} is a Polish space.
![Page 21: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/21.jpg)
Local convergence of rooted graphs
Write BR(G , o) for the ball of radius R around the vertex o in G :
Say that (Gn, on)→ (G , o) if for each fixed R and n large enough,
BR(Gn, on) ≡ BR(G , o).
G? := {locally finite, connected rooted graphs} is a Polish space.
![Page 22: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/22.jpg)
Local convergence of rooted graphs
Write BR(G , o) for the ball of radius R around the vertex o in G :
Say that (Gn, on)→ (G , o) if for each fixed R and n large enough,
BR(Gn, on) ≡ BR(G , o).
G? := {locally finite, connected rooted graphs} is a Polish space.
![Page 23: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/23.jpg)
Local convergence of rooted graphs
Write BR(G , o) for the ball of radius R around the vertex o in G :
Say that (Gn, on)→ (G , o) if for each fixed R and n large enough,
BR(Gn, on) ≡ BR(G , o).
G? := {locally finite, connected rooted graphs} is a Polish space.
![Page 24: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/24.jpg)
Benjamini-Schramm convergence
Goal: capture the local geometry around all vertices.
Given a finite graph G = (V ,E ), consider the empiricaldistribution of all its possible rootings:
LG :=1
|V |∑o∈V
δ(G ,o) ∈ P(G?).
Say that a sequence of finite graphs (Gn) has local weak limit L if
LGn =⇒ L
in the usual weak sense for probability measures on Polish spaces.
Intuition: L describes the local geometry around a typical vertex.
![Page 25: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/25.jpg)
Benjamini-Schramm convergence
Goal: capture the local geometry around all vertices.
Given a finite graph G = (V ,E ), consider the empiricaldistribution of all its possible rootings:
LG :=1
|V |∑o∈V
δ(G ,o) ∈ P(G?).
Say that a sequence of finite graphs (Gn) has local weak limit L if
LGn =⇒ L
in the usual weak sense for probability measures on Polish spaces.
Intuition: L describes the local geometry around a typical vertex.
![Page 26: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/26.jpg)
Benjamini-Schramm convergence
Goal: capture the local geometry around all vertices.
Given a finite graph G = (V ,E ), consider the empiricaldistribution of all its possible rootings:
LG :=1
|V |∑o∈V
δ(G ,o) ∈ P(G?).
Say that a sequence of finite graphs (Gn) has local weak limit L if
LGn =⇒ L
in the usual weak sense for probability measures on Polish spaces.
Intuition: L describes the local geometry around a typical vertex.
![Page 27: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/27.jpg)
Benjamini-Schramm convergence
Goal: capture the local geometry around all vertices.
Given a finite graph G = (V ,E ), consider the empiricaldistribution of all its possible rootings:
LG :=1
|V |∑o∈V
δ(G ,o) ∈ P(G?).
Say that a sequence of finite graphs (Gn) has local weak limit L if
LGn =⇒ L
in the usual weak sense for probability measures on Polish spaces.
Intuition: L describes the local geometry around a typical vertex.
![Page 28: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/28.jpg)
Benjamini-Schramm convergence
Goal: capture the local geometry around all vertices.
Given a finite graph G = (V ,E ), consider the empiricaldistribution of all its possible rootings:
LG :=1
|V |∑o∈V
δ(G ,o) ∈ P(G?).
Say that a sequence of finite graphs (Gn) has local weak limit L if
LGn =⇒ L
in the usual weak sense for probability measures on Polish spaces.
Intuition: L describes the local geometry around a typical vertex.
![Page 29: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/29.jpg)
Examples of local weak limits
I Gn : Random d−regular graphL : Infinite d−regular rooted tree
I Gn : Erdos-Renyi model with edge probability pn = cn
L : Galton-Watson tree with degree Poisson with mean c
I Gn : Configuration model with empirical degree distribution πL : Unimodular Galton-Watson tree with degree law π
I Gn : Uniform random treeL : Infinite Skeleton Tree (Grimmett, 1980)
I Gn : Preferential attachment graphL : Polya-point tree (Berger-Borgs-Chayes-Sabery, 2009)
Fact: uniform rooting confers to every local weak limit L apowerful form of stationarity known as unimodularity.
![Page 30: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/30.jpg)
Examples of local weak limits
I Gn : Random d−regular graph
L : Infinite d−regular rooted tree
I Gn : Erdos-Renyi model with edge probability pn = cn
L : Galton-Watson tree with degree Poisson with mean c
I Gn : Configuration model with empirical degree distribution πL : Unimodular Galton-Watson tree with degree law π
I Gn : Uniform random treeL : Infinite Skeleton Tree (Grimmett, 1980)
I Gn : Preferential attachment graphL : Polya-point tree (Berger-Borgs-Chayes-Sabery, 2009)
Fact: uniform rooting confers to every local weak limit L apowerful form of stationarity known as unimodularity.
![Page 31: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/31.jpg)
Examples of local weak limits
I Gn : Random d−regular graphL : Infinite d−regular rooted tree
I Gn : Erdos-Renyi model with edge probability pn = cn
L : Galton-Watson tree with degree Poisson with mean c
I Gn : Configuration model with empirical degree distribution πL : Unimodular Galton-Watson tree with degree law π
I Gn : Uniform random treeL : Infinite Skeleton Tree (Grimmett, 1980)
I Gn : Preferential attachment graphL : Polya-point tree (Berger-Borgs-Chayes-Sabery, 2009)
Fact: uniform rooting confers to every local weak limit L apowerful form of stationarity known as unimodularity.
![Page 32: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/32.jpg)
Examples of local weak limits
I Gn : Random d−regular graphL : Infinite d−regular rooted tree
I Gn : Erdos-Renyi model with edge probability pn = cn
L : Galton-Watson tree with degree Poisson with mean c
I Gn : Configuration model with empirical degree distribution πL : Unimodular Galton-Watson tree with degree law π
I Gn : Uniform random treeL : Infinite Skeleton Tree (Grimmett, 1980)
I Gn : Preferential attachment graphL : Polya-point tree (Berger-Borgs-Chayes-Sabery, 2009)
Fact: uniform rooting confers to every local weak limit L apowerful form of stationarity known as unimodularity.
![Page 33: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/33.jpg)
Examples of local weak limits
I Gn : Random d−regular graphL : Infinite d−regular rooted tree
I Gn : Erdos-Renyi model with edge probability pn = cn
L : Galton-Watson tree with degree Poisson with mean c
I Gn : Configuration model with empirical degree distribution πL : Unimodular Galton-Watson tree with degree law π
I Gn : Uniform random treeL : Infinite Skeleton Tree (Grimmett, 1980)
I Gn : Preferential attachment graphL : Polya-point tree (Berger-Borgs-Chayes-Sabery, 2009)
Fact: uniform rooting confers to every local weak limit L apowerful form of stationarity known as unimodularity.
![Page 34: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/34.jpg)
Examples of local weak limits
I Gn : Random d−regular graphL : Infinite d−regular rooted tree
I Gn : Erdos-Renyi model with edge probability pn = cn
L : Galton-Watson tree with degree Poisson with mean c
I Gn : Configuration model with empirical degree distribution π
L : Unimodular Galton-Watson tree with degree law π
I Gn : Uniform random treeL : Infinite Skeleton Tree (Grimmett, 1980)
I Gn : Preferential attachment graphL : Polya-point tree (Berger-Borgs-Chayes-Sabery, 2009)
Fact: uniform rooting confers to every local weak limit L apowerful form of stationarity known as unimodularity.
![Page 35: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/35.jpg)
Examples of local weak limits
I Gn : Random d−regular graphL : Infinite d−regular rooted tree
I Gn : Erdos-Renyi model with edge probability pn = cn
L : Galton-Watson tree with degree Poisson with mean c
I Gn : Configuration model with empirical degree distribution πL : Unimodular Galton-Watson tree with degree law π
I Gn : Uniform random treeL : Infinite Skeleton Tree (Grimmett, 1980)
I Gn : Preferential attachment graphL : Polya-point tree (Berger-Borgs-Chayes-Sabery, 2009)
Fact: uniform rooting confers to every local weak limit L apowerful form of stationarity known as unimodularity.
![Page 36: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/36.jpg)
Examples of local weak limits
I Gn : Random d−regular graphL : Infinite d−regular rooted tree
I Gn : Erdos-Renyi model with edge probability pn = cn
L : Galton-Watson tree with degree Poisson with mean c
I Gn : Configuration model with empirical degree distribution πL : Unimodular Galton-Watson tree with degree law π
I Gn : Uniform random tree
L : Infinite Skeleton Tree (Grimmett, 1980)
I Gn : Preferential attachment graphL : Polya-point tree (Berger-Borgs-Chayes-Sabery, 2009)
Fact: uniform rooting confers to every local weak limit L apowerful form of stationarity known as unimodularity.
![Page 37: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/37.jpg)
Examples of local weak limits
I Gn : Random d−regular graphL : Infinite d−regular rooted tree
I Gn : Erdos-Renyi model with edge probability pn = cn
L : Galton-Watson tree with degree Poisson with mean c
I Gn : Configuration model with empirical degree distribution πL : Unimodular Galton-Watson tree with degree law π
I Gn : Uniform random treeL : Infinite Skeleton Tree (Grimmett, 1980)
I Gn : Preferential attachment graphL : Polya-point tree (Berger-Borgs-Chayes-Sabery, 2009)
Fact: uniform rooting confers to every local weak limit L apowerful form of stationarity known as unimodularity.
![Page 38: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/38.jpg)
Examples of local weak limits
I Gn : Random d−regular graphL : Infinite d−regular rooted tree
I Gn : Erdos-Renyi model with edge probability pn = cn
L : Galton-Watson tree with degree Poisson with mean c
I Gn : Configuration model with empirical degree distribution πL : Unimodular Galton-Watson tree with degree law π
I Gn : Uniform random treeL : Infinite Skeleton Tree (Grimmett, 1980)
I Gn : Preferential attachment graph
L : Polya-point tree (Berger-Borgs-Chayes-Sabery, 2009)
Fact: uniform rooting confers to every local weak limit L apowerful form of stationarity known as unimodularity.
![Page 39: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/39.jpg)
Examples of local weak limits
I Gn : Random d−regular graphL : Infinite d−regular rooted tree
I Gn : Erdos-Renyi model with edge probability pn = cn
L : Galton-Watson tree with degree Poisson with mean c
I Gn : Configuration model with empirical degree distribution πL : Unimodular Galton-Watson tree with degree law π
I Gn : Uniform random treeL : Infinite Skeleton Tree (Grimmett, 1980)
I Gn : Preferential attachment graphL : Polya-point tree (Berger-Borgs-Chayes-Sabery, 2009)
Fact: uniform rooting confers to every local weak limit L apowerful form of stationarity known as unimodularity.
![Page 40: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/40.jpg)
Examples of local weak limits
I Gn : Random d−regular graphL : Infinite d−regular rooted tree
I Gn : Erdos-Renyi model with edge probability pn = cn
L : Galton-Watson tree with degree Poisson with mean c
I Gn : Configuration model with empirical degree distribution πL : Unimodular Galton-Watson tree with degree law π
I Gn : Uniform random treeL : Infinite Skeleton Tree (Grimmett, 1980)
I Gn : Preferential attachment graphL : Polya-point tree (Berger-Borgs-Chayes-Sabery, 2009)
Fact: uniform rooting confers to every local weak limit L apowerful form of stationarity known as unimodularity.
![Page 41: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/41.jpg)
The spectral convergence theorem for sparse graphs
Theorem (Bordenave-Lelarge-Abert-Thom-Virag):
1. If (Gn) admits a limit L, then (µGn) admits a weak limit µL.
2. The convergence holds in the Kolmogorov-Smirnov sense:
supλ∈R|µGn ((−∞, λ])− µL ((−∞, λ])| −−−→
n→∞0
3. We have µL = E[µ(G ,o)] where (G , o) ∼ L and
∀z ∈ C \ R,∫R
1
λ− zµ(G ,o)(dλ) = (AG − z)−1
oo .
Note: in general, the existence of (AG − z)−1 is a delicate issue...
![Page 42: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/42.jpg)
The spectral convergence theorem for sparse graphs
Theorem (Bordenave-Lelarge-Abert-Thom-Virag):
1. If (Gn) admits a limit L, then (µGn) admits a weak limit µL.
2. The convergence holds in the Kolmogorov-Smirnov sense:
supλ∈R|µGn ((−∞, λ])− µL ((−∞, λ])| −−−→
n→∞0
3. We have µL = E[µ(G ,o)] where (G , o) ∼ L and
∀z ∈ C \ R,∫R
1
λ− zµ(G ,o)(dλ) = (AG − z)−1
oo .
Note: in general, the existence of (AG − z)−1 is a delicate issue...
![Page 43: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/43.jpg)
The spectral convergence theorem for sparse graphs
Theorem (Bordenave-Lelarge-Abert-Thom-Virag):
1. If (Gn) admits a limit L, then (µGn) admits a weak limit µL.
2. The convergence holds in the Kolmogorov-Smirnov sense:
supλ∈R|µGn ((−∞, λ])− µL ((−∞, λ])| −−−→
n→∞0
3. We have µL = E[µ(G ,o)] where (G , o) ∼ L and
∀z ∈ C \ R,∫R
1
λ− zµ(G ,o)(dλ) = (AG − z)−1
oo .
Note: in general, the existence of (AG − z)−1 is a delicate issue...
![Page 44: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/44.jpg)
The spectral convergence theorem for sparse graphs
Theorem (Bordenave-Lelarge-Abert-Thom-Virag):
1. If (Gn) admits a limit L, then (µGn) admits a weak limit µL.
2. The convergence holds in the Kolmogorov-Smirnov sense:
supλ∈R|µGn ((−∞, λ])− µL ((−∞, λ])| −−−→
n→∞0
3. We have µL = E[µ(G ,o)] where (G , o) ∼ L and
∀z ∈ C \ R,∫R
1
λ− zµ(G ,o)(dλ) = (AG − z)−1
oo .
Note: in general, the existence of (AG − z)−1 is a delicate issue...
![Page 45: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/45.jpg)
The spectral convergence theorem for sparse graphs
Theorem (Bordenave-Lelarge-Abert-Thom-Virag):
1. If (Gn) admits a limit L, then (µGn) admits a weak limit µL.
2. The convergence holds in the Kolmogorov-Smirnov sense:
supλ∈R|µGn ((−∞, λ])− µL ((−∞, λ])| −−−→
n→∞0
3. We have µL = E[µ(G ,o)] where (G , o) ∼ L and
∀z ∈ C \ R,∫R
1
λ− zµ(G ,o)(dλ) = (AG − z)−1
oo .
Note: in general, the existence of (AG − z)−1 is a delicate issue...
![Page 46: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/46.jpg)
The spectral convergence theorem for sparse graphs
Theorem (Bordenave-Lelarge-Abert-Thom-Virag):
1. If (Gn) admits a limit L, then (µGn) admits a weak limit µL.
2. The convergence holds in the Kolmogorov-Smirnov sense:
supλ∈R|µGn ((−∞, λ])− µL ((−∞, λ])| −−−→
n→∞0
3. We have µL = E[µ(G ,o)] where (G , o) ∼ L and
∀z ∈ C \ R,∫R
1
λ− zµ(G ,o)(dλ) = (AG − z)−1
oo .
Note: in general, the existence of (AG − z)−1 is a delicate issue...
![Page 47: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/47.jpg)
The case of trees
T1 T2 Td
T =
1 2 d
o
(AT − z)−1oo =
−1
z +∑d
i=1(ATi− z)−1
ii
I Explicit resolution for infinite regular trees
I Recursive distributional equation for Galton-Watson trees
I In principle, this equation contains everything about µLI See the wonderful survey by Bordenave for details.
![Page 48: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/48.jpg)
The case of trees
T1 T2 Td
T =
1 2 d
o
(AT − z)−1oo =
−1
z +∑d
i=1(ATi− z)−1
ii
I Explicit resolution for infinite regular trees
I Recursive distributional equation for Galton-Watson trees
I In principle, this equation contains everything about µLI See the wonderful survey by Bordenave for details.
![Page 49: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/49.jpg)
The case of trees
T1 T2 Td
T =
1 2 d
o
(AT − z)−1oo =
−1
z +∑d
i=1(ATi− z)−1
ii
I Explicit resolution for infinite regular trees
I Recursive distributional equation for Galton-Watson trees
I In principle, this equation contains everything about µLI See the wonderful survey by Bordenave for details.
![Page 50: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/50.jpg)
The case of trees
T1 T2 Td
T =
1 2 d
o
(AT − z)−1oo =
−1
z +∑d
i=1(ATi− z)−1
ii
I Explicit resolution for infinite regular trees
I Recursive distributional equation for Galton-Watson trees
I In principle, this equation contains everything about µLI See the wonderful survey by Bordenave for details.
![Page 51: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/51.jpg)
The case of trees
T1 T2 Td
T =
1 2 d
o
(AT − z)−1oo =
−1
z +∑d
i=1(ATi− z)−1
ii
I Explicit resolution for infinite regular trees
I Recursive distributional equation for Galton-Watson trees
I In principle, this equation contains everything about µL
I See the wonderful survey by Bordenave for details.
![Page 52: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/52.jpg)
The case of trees
T1 T2 Td
T =
1 2 d
o
(AT − z)−1oo =
−1
z +∑d
i=1(ATi− z)−1
ii
I Explicit resolution for infinite regular trees
I Recursive distributional equation for Galton-Watson trees
I In principle, this equation contains everything about µLI See the wonderful survey by Bordenave for details.
![Page 53: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/53.jpg)
The eigenspace Eλ := ker(AG − λ)
Given λ ∈ R and a locally finite graph G = (V ,E ), consider
S :=⋃f ∈Eλ
support(f ).
Lemma (Finite trees). If G is a finite tree, then λ is a simpleeigenvalue of each connected component of S. Consequently,
dim(Eλ) = |S| − |E (S)| − |∂S|.
Theorem (Main result). On a unimodular random tree, theconnected components of S are almost-surely finite. Moreover,
µL({λ}) = P (o ∈ S)− 1
2E[degS(o)1(o∈S)
]− P (o ∈ ∂S) .
![Page 54: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/54.jpg)
The eigenspace Eλ := ker(AG − λ)
Given λ ∈ R and a locally finite graph G = (V ,E ), consider
S :=⋃f ∈Eλ
support(f ).
Lemma (Finite trees). If G is a finite tree, then λ is a simpleeigenvalue of each connected component of S. Consequently,
dim(Eλ) = |S| − |E (S)| − |∂S|.
Theorem (Main result). On a unimodular random tree, theconnected components of S are almost-surely finite. Moreover,
µL({λ}) = P (o ∈ S)− 1
2E[degS(o)1(o∈S)
]− P (o ∈ ∂S) .
![Page 55: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/55.jpg)
The eigenspace Eλ := ker(AG − λ)
Given λ ∈ R and a locally finite graph G = (V ,E ), consider
S :=⋃f ∈Eλ
support(f ).
Lemma (Finite trees). If G is a finite tree, then λ is a simpleeigenvalue of each connected component of S.
Consequently,
dim(Eλ) = |S| − |E (S)| − |∂S|.
Theorem (Main result). On a unimodular random tree, theconnected components of S are almost-surely finite. Moreover,
µL({λ}) = P (o ∈ S)− 1
2E[degS(o)1(o∈S)
]− P (o ∈ ∂S) .
![Page 56: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/56.jpg)
The eigenspace Eλ := ker(AG − λ)
Given λ ∈ R and a locally finite graph G = (V ,E ), consider
S :=⋃f ∈Eλ
support(f ).
Lemma (Finite trees). If G is a finite tree, then λ is a simpleeigenvalue of each connected component of S. Consequently,
dim(Eλ) = |S| − |E (S)| − |∂S|.
Theorem (Main result). On a unimodular random tree, theconnected components of S are almost-surely finite. Moreover,
µL({λ}) = P (o ∈ S)− 1
2E[degS(o)1(o∈S)
]− P (o ∈ ∂S) .
![Page 57: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/57.jpg)
The eigenspace Eλ := ker(AG − λ)
Given λ ∈ R and a locally finite graph G = (V ,E ), consider
S :=⋃f ∈Eλ
support(f ).
Lemma (Finite trees). If G is a finite tree, then λ is a simpleeigenvalue of each connected component of S. Consequently,
dim(Eλ) = |S| − |E (S)| − |∂S|.
Theorem (Main result). On a unimodular random tree, theconnected components of S are almost-surely finite.
Moreover,
µL({λ}) = P (o ∈ S)− 1
2E[degS(o)1(o∈S)
]− P (o ∈ ∂S) .
![Page 58: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/58.jpg)
The eigenspace Eλ := ker(AG − λ)
Given λ ∈ R and a locally finite graph G = (V ,E ), consider
S :=⋃f ∈Eλ
support(f ).
Lemma (Finite trees). If G is a finite tree, then λ is a simpleeigenvalue of each connected component of S. Consequently,
dim(Eλ) = |S| − |E (S)| − |∂S|.
Theorem (Main result). On a unimodular random tree, theconnected components of S are almost-surely finite. Moreover,
µL({λ}) = P (o ∈ S)− 1
2E[degS(o)1(o∈S)
]− P (o ∈ ∂S) .
![Page 59: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/59.jpg)
Consequences for the pure-point support
Σpp(L) := {λ ∈ R : µL({λ}) > 0}.
Corollary. If L is supported on trees, then
Σpp(L) ⊆ {eigenvalues of finite trees} =: A.
Corollary. Σpp(L) = A for many “natural” limits L, including:
I The Poisson-Galton-Watson tree.
I Unimodular Galton-Watson trees with supp(π) = Z+.
I Their “conditioned on non-extinction” versions.
I The Infinite Skeleton tree.
Remark. A is dense in R, so these graphs have “rough” spectrum.
![Page 60: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/60.jpg)
Consequences for the pure-point support
Σpp(L) := {λ ∈ R : µL({λ}) > 0}.
Corollary. If L is supported on trees, then
Σpp(L) ⊆ {eigenvalues of finite trees} =: A.
Corollary. Σpp(L) = A for many “natural” limits L, including:
I The Poisson-Galton-Watson tree.
I Unimodular Galton-Watson trees with supp(π) = Z+.
I Their “conditioned on non-extinction” versions.
I The Infinite Skeleton tree.
Remark. A is dense in R, so these graphs have “rough” spectrum.
![Page 61: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/61.jpg)
Consequences for the pure-point support
Σpp(L) := {λ ∈ R : µL({λ}) > 0}.
Corollary. If L is supported on trees, then
Σpp(L) ⊆ {eigenvalues of finite trees} =: A.
Corollary. Σpp(L) = A for many “natural” limits L, including:
I The Poisson-Galton-Watson tree.
I Unimodular Galton-Watson trees with supp(π) = Z+.
I Their “conditioned on non-extinction” versions.
I The Infinite Skeleton tree.
Remark. A is dense in R, so these graphs have “rough” spectrum.
![Page 62: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/62.jpg)
Consequences for the pure-point support
Σpp(L) := {λ ∈ R : µL({λ}) > 0}.
Corollary. If L is supported on trees, then
Σpp(L) ⊆ {eigenvalues of finite trees} =: A.
Corollary. Σpp(L) = A for many “natural” limits L, including:
I The Poisson-Galton-Watson tree.
I Unimodular Galton-Watson trees with supp(π) = Z+.
I Their “conditioned on non-extinction” versions.
I The Infinite Skeleton tree.
Remark. A is dense in R, so these graphs have “rough” spectrum.
![Page 63: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/63.jpg)
Consequences for the pure-point support
Σpp(L) := {λ ∈ R : µL({λ}) > 0}.
Corollary. If L is supported on trees, then
Σpp(L) ⊆ {eigenvalues of finite trees} =: A.
Corollary. Σpp(L) = A for many “natural” limits L, including:
I The Poisson-Galton-Watson tree.
I Unimodular Galton-Watson trees with supp(π) = Z+.
I Their “conditioned on non-extinction” versions.
I The Infinite Skeleton tree.
Remark. A is dense in R, so these graphs have “rough” spectrum.
![Page 64: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/64.jpg)
Consequences for the pure-point support
Σpp(L) := {λ ∈ R : µL({λ}) > 0}.
Corollary. If L is supported on trees, then
Σpp(L) ⊆ {eigenvalues of finite trees} =: A.
Corollary. Σpp(L) = A for many “natural” limits L, including:
I The Poisson-Galton-Watson tree.
I Unimodular Galton-Watson trees with supp(π) = Z+.
I Their “conditioned on non-extinction” versions.
I The Infinite Skeleton tree.
Remark. A is dense in R, so these graphs have “rough” spectrum.
![Page 65: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/65.jpg)
Consequences for the pure-point support
Σpp(L) := {λ ∈ R : µL({λ}) > 0}.
Corollary. If L is supported on trees, then
Σpp(L) ⊆ {eigenvalues of finite trees} =: A.
Corollary. Σpp(L) = A for many “natural” limits L, including:
I The Poisson-Galton-Watson tree.
I Unimodular Galton-Watson trees with supp(π) = Z+.
I Their “conditioned on non-extinction” versions.
I The Infinite Skeleton tree.
Remark. A is dense in R, so these graphs have “rough” spectrum.
![Page 66: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/66.jpg)
Consequences for the pure-point support
Σpp(L) := {λ ∈ R : µL({λ}) > 0}.
Corollary. If L is supported on trees, then
Σpp(L) ⊆ {eigenvalues of finite trees} =: A.
Corollary. Σpp(L) = A for many “natural” limits L, including:
I The Poisson-Galton-Watson tree.
I Unimodular Galton-Watson trees with supp(π) = Z+.
I Their “conditioned on non-extinction” versions.
I The Infinite Skeleton tree.
Remark. A is dense in R, so these graphs have “rough” spectrum.
![Page 67: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/67.jpg)
Pure-point spectrum and degrees
Write τ(λ) for the size of the smallest tree having λ as eigenvalue.
Corollary. If L is supported on trees with degrees in {δ, . . . ,∆},
Σpp(L) ⊆{λ : τ(λ) <
∆− 2
δ − 2
}.
In particular,
I If ∆−2δ−2 ≤ 2, then Σpp(L) ⊆ {0}.
I if ∆−2δ−2 ≤ 3 then Σpp(L) ⊆ {−1, 0,+1}.
I If ∆−2δ−2 ≤ 4 then Σpp(L) ⊆ {−
√2,−1, 0,+1,
√2}.
![Page 68: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/68.jpg)
Pure-point spectrum and degrees
Write τ(λ) for the size of the smallest tree having λ as eigenvalue.
Corollary. If L is supported on trees with degrees in {δ, . . . ,∆},
Σpp(L) ⊆{λ : τ(λ) <
∆− 2
δ − 2
}.
In particular,
I If ∆−2δ−2 ≤ 2, then Σpp(L) ⊆ {0}.
I if ∆−2δ−2 ≤ 3 then Σpp(L) ⊆ {−1, 0,+1}.
I If ∆−2δ−2 ≤ 4 then Σpp(L) ⊆ {−
√2,−1, 0,+1,
√2}.
![Page 69: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/69.jpg)
Pure-point spectrum and degrees
Write τ(λ) for the size of the smallest tree having λ as eigenvalue.
Corollary. If L is supported on trees with degrees in {δ, . . . ,∆},
Σpp(L) ⊆{λ : τ(λ) <
∆− 2
δ − 2
}.
In particular,
I If ∆−2δ−2 ≤ 2, then Σpp(L) ⊆ {0}.
I if ∆−2δ−2 ≤ 3 then Σpp(L) ⊆ {−1, 0,+1}.
I If ∆−2δ−2 ≤ 4 then Σpp(L) ⊆ {−
√2,−1, 0,+1,
√2}.
![Page 70: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/70.jpg)
Pure-point spectrum and degrees
Write τ(λ) for the size of the smallest tree having λ as eigenvalue.
Corollary. If L is supported on trees with degrees in {δ, . . . ,∆},
Σpp(L) ⊆{λ : τ(λ) <
∆− 2
δ − 2
}.
In particular,
I If ∆−2δ−2 ≤ 2, then Σpp(L) ⊆ {0}.
I if ∆−2δ−2 ≤ 3 then Σpp(L) ⊆ {−1, 0,+1}.
I If ∆−2δ−2 ≤ 4 then Σpp(L) ⊆ {−
√2,−1, 0,+1,
√2}.
![Page 71: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/71.jpg)
Pure-point spectrum and degrees
Write τ(λ) for the size of the smallest tree having λ as eigenvalue.
Corollary. If L is supported on trees with degrees in {δ, . . . ,∆},
Σpp(L) ⊆{λ : τ(λ) <
∆− 2
δ − 2
}.
In particular,
I If ∆−2δ−2 ≤ 2, then Σpp(L) ⊆ {0}.
I if ∆−2δ−2 ≤ 3 then Σpp(L) ⊆ {−1, 0,+1}.
I If ∆−2δ−2 ≤ 4 then Σpp(L) ⊆ {−
√2,−1, 0,+1,
√2}.
![Page 72: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/72.jpg)
Pure-point spectrum and degrees
Write τ(λ) for the size of the smallest tree having λ as eigenvalue.
Corollary. If L is supported on trees with degrees in {δ, . . . ,∆},
Σpp(L) ⊆{λ : τ(λ) <
∆− 2
δ − 2
}.
In particular,
I If ∆−2δ−2 ≤ 2, then Σpp(L) ⊆ {0}.
I if ∆−2δ−2 ≤ 3 then Σpp(L) ⊆ {−1, 0,+1}.
I If ∆−2δ−2 ≤ 4 then Σpp(L) ⊆ {−
√2,−1, 0,+1,
√2}.
![Page 73: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/73.jpg)
Pure-point spectrum and degrees
Write τ(λ) for the size of the smallest tree having λ as eigenvalue.
Corollary. If L is supported on trees with degrees in {δ, . . . ,∆},
Σpp(L) ⊆{λ : τ(λ) <
∆− 2
δ − 2
}.
In particular,
I If ∆−2δ−2 ≤ 2, then Σpp(L) ⊆ {0}.
I if ∆−2δ−2 ≤ 3 then Σpp(L) ⊆ {−1, 0,+1}.
I If ∆−2δ−2 ≤ 4 then Σpp(L) ⊆ {−
√2,−1, 0,+1,
√2}.
![Page 74: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/74.jpg)
Pure-point spectrum and isoperimetric profile
Anchored isoperimetric constant:
ι?(G , o) := limn→∞
inf
{|∂S ||S |
: o ∈ S ,S connected, n ≤ |S | <∞}.
Corollary. If L is supported on trees with anchored isoperimetricconstant ≥ ε and degrees in {2, . . . ,∆}, then
Σpp(L) ⊆{λ : τ(λ) <
∆2
ε
}.
Remark. The anchored isoperimetric constant of a GWTconditioned on non-extinction is positive (Chen & Peres, 2004).
![Page 75: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/75.jpg)
Pure-point spectrum and isoperimetric profile
Anchored isoperimetric constant:
ι?(G , o) := limn→∞
inf
{|∂S ||S |
: o ∈ S , S connected, n ≤ |S | <∞}.
Corollary. If L is supported on trees with anchored isoperimetricconstant ≥ ε and degrees in {2, . . . ,∆}, then
Σpp(L) ⊆{λ : τ(λ) <
∆2
ε
}.
Remark. The anchored isoperimetric constant of a GWTconditioned on non-extinction is positive (Chen & Peres, 2004).
![Page 76: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/76.jpg)
Pure-point spectrum and isoperimetric profile
Anchored isoperimetric constant:
ι?(G , o) := limn→∞
inf
{|∂S ||S |
: o ∈ S , S connected, n ≤ |S | <∞}.
Corollary. If L is supported on trees with anchored isoperimetricconstant ≥ ε and degrees in {2, . . . ,∆}, then
Σpp(L) ⊆{λ : τ(λ) <
∆2
ε
}.
Remark. The anchored isoperimetric constant of a GWTconditioned on non-extinction is positive (Chen & Peres, 2004).
![Page 77: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/77.jpg)
Pure-point spectrum and isoperimetric profile
Anchored isoperimetric constant:
ι?(G , o) := limn→∞
inf
{|∂S ||S |
: o ∈ S , S connected, n ≤ |S | <∞}.
Corollary. If L is supported on trees with anchored isoperimetricconstant ≥ ε and degrees in {2, . . . ,∆}, then
Σpp(L) ⊆{λ : τ(λ) <
∆2
ε
}.
Remark. The anchored isoperimetric constant of a GWTconditioned on non-extinction is positive (Chen & Peres, 2004).
![Page 78: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/78.jpg)
A striking dichotomy in the Galton-Watson case
Corollary. When L is the unimodular GWT with degreedistribution π ∈ P(Z+), we have the following dichotomy:
I If π1 = 0 (no leaves) then Σpp(L) is a finite set.
I If π1 > 0 then Σpp(L) is dense in [−2√
∆− 1,+2√
∆− 1].
This was conjectured by Bordenave, Sen & Virag (JEMS 2017).
![Page 79: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/79.jpg)
A striking dichotomy in the Galton-Watson case
Corollary. When L is the unimodular GWT with degreedistribution π ∈ P(Z+), we have the following dichotomy:
I If π1 = 0 (no leaves) then Σpp(L) is a finite set.
I If π1 > 0 then Σpp(L) is dense in [−2√
∆− 1,+2√
∆− 1].
This was conjectured by Bordenave, Sen & Virag (JEMS 2017).
![Page 80: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/80.jpg)
A striking dichotomy in the Galton-Watson case
Corollary. When L is the unimodular GWT with degreedistribution π ∈ P(Z+), we have the following dichotomy:
I If π1 = 0 (no leaves) then Σpp(L) is a finite set.
I If π1 > 0 then Σpp(L) is dense in [−2√
∆− 1,+2√
∆− 1].
This was conjectured by Bordenave, Sen & Virag (JEMS 2017).
![Page 81: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/81.jpg)
A striking dichotomy in the Galton-Watson case
Corollary. When L is the unimodular GWT with degreedistribution π ∈ P(Z+), we have the following dichotomy:
I If π1 = 0 (no leaves) then Σpp(L) is a finite set.
I If π1 > 0 then Σpp(L) is dense in [−2√
∆− 1,+2√
∆− 1].
This was conjectured by Bordenave, Sen & Virag (JEMS 2017).
![Page 82: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/82.jpg)
A striking dichotomy in the Galton-Watson case
Corollary. When L is the unimodular GWT with degreedistribution π ∈ P(Z+), we have the following dichotomy:
I If π1 = 0 (no leaves) then Σpp(L) is a finite set.
I If π1 > 0 then Σpp(L) is dense in [−2√
∆− 1,+2√
∆− 1].
This was conjectured by Bordenave, Sen & Virag (JEMS 2017).
![Page 83: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/83.jpg)
Three specific open problems
µL = µpp + µac + µsc
I For which degree π does UGWT(π) satisfy µpp(R) = 0 ?
I Does the Infinite Skeleton Tree satisfy µpp(R) = 1 ?
I When π = Poisson(c), does µac(R) > 0 as soon as c > 1 ?
![Page 84: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/84.jpg)
Three specific open problems
µL = µpp + µac + µsc
I For which degree π does UGWT(π) satisfy µpp(R) = 0 ?
I Does the Infinite Skeleton Tree satisfy µpp(R) = 1 ?
I When π = Poisson(c), does µac(R) > 0 as soon as c > 1 ?
![Page 85: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/85.jpg)
Three specific open problems
µL = µpp + µac + µsc
I For which degree π does UGWT(π) satisfy µpp(R) = 0 ?
I Does the Infinite Skeleton Tree satisfy µpp(R) = 1 ?
I When π = Poisson(c), does µac(R) > 0 as soon as c > 1 ?
![Page 86: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/86.jpg)
Three specific open problems
µL = µpp + µac + µsc
I For which degree π does UGWT(π) satisfy µpp(R) = 0 ?
I Does the Infinite Skeleton Tree satisfy µpp(R) = 1 ?
I When π = Poisson(c), does µac(R) > 0 as soon as c > 1 ?
![Page 87: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/87.jpg)
Three specific open problems
µL = µpp + µac + µsc
I For which degree π does UGWT(π) satisfy µpp(R) = 0 ?
I Does the Infinite Skeleton Tree satisfy µpp(R) = 1 ?
I When π = Poisson(c), does µac(R) > 0 as soon as c > 1 ?
![Page 88: Spectral analysis of sparse random graphs Justin Salez (LPSM)djalil.chafai.net/wiki/_media/kyoto2018:salez.pdf · in the usual weak sense for probability measures on Polish spaces](https://reader034.vdocuments.net/reader034/viewer/2022042806/5f6f9c4bdb24710a364d0067/html5/thumbnails/88.jpg)
Thank you !