Spectra of large diluted but bushy random graphs Laurent M enard - - PowerPoint PPT Presentation

Spectra of large diluted but bushy random graphs

Laurent M´ enard Joint work with Nathana¨ el Enriquez Modal’X Universit´ e Paris Ouest

Erd˝

• s-R´

enyi random graphs G(n, p)

• vertex set {1, . . . n}
• vertices linked by an edge independently with

probability p

Erd˝

• s-R´

enyi random graphs G(n, p)

• vertex set {1, . . . n}
• vertices linked by an edge independently with

probability p

• symmetric
• if i = j, P(Ai,j = 1) = 1 − P(Ai,j = 0) = p
• for every i, Ai,i = 0

Adjacency matrix A

Erd˝

• s-R´

enyi random graphs G(n, p)

• vertex set {1, . . . n}
• vertices linked by an edge independently with

probability p

• symmetric
• if i = j, P(Ai,j = 1) = 1 − P(Ai,j = 0) = p
• for every i, Ai,i = 0

What does the spectrum of A look like ?

• if np → 0, single atom mass at 0
• if np → ∞, semi circle law
• if np → c > 0, not much is known...

Adjacency matrix A

Numerical simulations on diluted graphs with 5000 vertices c = 0, 5

Numerical simulations on diluted graphs with 5000 vertices c = 0, 5 (zoomed in)

Numerical simulations on diluted graphs with 5000 vertices c = 1

Numerical simulations on diluted graphs with 5000 vertices c = 1 (zoomed in)

Numerical simulations on diluted graphs with 5000 vertices c = 1, 5

Numerical simulations on diluted graphs with 5000 vertices c = 1, 5 (zoomed in)

Numerical simulations on diluted graphs with 5000 vertices c = 2

Numerical simulations on diluted graphs with 5000 vertices c = 2 (zoomed in)

Numerical simulations on diluted graphs with 5000 vertices c = 2, 5

Numerical simulations on diluted graphs with 5000 vertices c = 2, 5 (zoomed in)

Numerical simulations on diluted graphs with 5000 vertices c = 2, 8

Numerical simulations on diluted graphs with 5000 vertices c = 2, 8 (zoomed in)

Numerical simulations on diluted graphs with 5000 vertices c = 3

Numerical simulations on diluted graphs with 5000 vertices c = 3 (zoomed in)

Numerical simulations on diluted graphs with 5000 vertices c = 4

Numerical simulations on diluted graphs with 5000 vertices c = 5

Numerical simulations on diluted graphs with 5000 vertices c = 10

Numerical simulations on diluted graphs with 5000 vertices c = 20

´ Etat de l’art µc

n = 1

n

• λ∈Sp(c−1/2A)

δλ : empirical spectral distribution of G(n, c/n)

´ Etat de l’art µc

n = 1

n

• λ∈Sp(c−1/2A)

δλ : empirical spectral distribution of G(n, c/n) Fact : as n → ∞, µc

n converges weakly to a probability measure µc

´ Etat de l’art µc

n = 1

n

• λ∈Sp(c−1/2A)

δλ : empirical spectral distribution of G(n, c/n) Fact : as n → ∞, µc

n converges weakly to a probability measure µc

Known properties of µc:

• unbounded support
• dense set of atoms
• if c → ∞, µc converges weakly to Wigner semi-circle law

´ Etat de l’art µc

n = 1

n

• λ∈Sp(c−1/2A)

δλ : empirical spectral distribution of G(n, c/n) Fact : as n → ∞, µc

n converges weakly to a probability measure µc

Known properties of µc:

• unbounded support
• dense set of atoms
• µc ({0}) known explicitly
• if c → ∞, µc converges weakly to Wigner semi-circle law

[Bordenave, Lelarge, Salez 2011]

´ Etat de l’art µc

n = 1

n

• λ∈Sp(c−1/2A)

δλ : empirical spectral distribution of G(n, c/n) Fact : as n → ∞, µc

n converges weakly to a probability measure µc

Known properties of µc:

• unbounded support
• dense set of atoms
• µc ({0}) known explicitly
• µc is not purely atomic iif c > 1
• if c → ∞, µc converges weakly to Wigner semi-circle law

[Bordenave, Lelarge, Salez 2011] [Bordenave, Sen, Vir´ ag 2013]

Asymptotic expansion of the spectrum

Asymptotic expansion of the spectrum If µ is a (signed) mesure and

• |x|k|dµ(x)| < ∞, denote mk(µ) =
• xkdµ(x)

Asymptotic expansion of the spectrum Theorem: For every k ≥ 0 and as c → ∞ mk(µc) = mk(σ) + 1 c mk(σ{1}) + o 1 c

• where σ is the semi-circle law having density 1

2π

• 4 − x21|x|<2

and σ{1} is a measure with total mass 0 and density 1 2π x4 − 4x2 + 2 √ 4 − x2 1|x|<2. If µ is a (signed) mesure and

• |x|k|dµ(x)| < ∞, denote mk(µ) =
• xkdµ(x)

Asymptotic expansion of the spectrum – numerical simulations 100 matrices of size 10000 with c = 20 Histogram of µc

n

Density of σ

Asymptotic expansion of the spectrum – numerical simulations 100 matrices of size 10000 with c = 20 Histogram of c (µc

n − σ)

Density of σ{1}

Asymptotic expansion of the spectrum: second order (I) Proposition: For every k 0 we have the following asymtotic expansion in c: mk(µc) = mk(σ) + 1 c mk(σ{1}) + 1 c2 dk + o 1 c2

• where the numbers dk are NOT the moments of a measure!

Asymptotic expansion of the spectrum: second order (I) Proposition: For every k 0 we have the following asymtotic expansion in c: mk(µc) = mk(σ) + 1 c mk(σ{1}) + 1 c2 dk + o 1 c2

• where the numbers dk are NOT the moments of a measure!

The asymptotic expansion must take into account the fact that µc (R \ [−2; 2]) = O 1 c2

• .

Asymptotic expansion of the spectrum: second order (I) Proposition: For every k 0 we have the following asymtotic expansion in c: mk(µc) = mk(σ) + 1 c mk(σ{1}) + 1 c2 dk + o 1 c2

• where the numbers dk are NOT the moments of a measure!

The asymptotic expansion must take into account the fact that µc (R \ [−2; 2]) = O 1 c2

• .

Dilation operator Λα for measures defined by Λα(µ)(A) = µ (A/α) for a measure µ and a Borel set A.

Asymptotic expansion of the spectrum: second order (I) Proposition: For every k 0 we have the following asymtotic expansion in c: mk(µc) = mk(σ) + 1 c mk(σ{1}) + 1 c2 dk + o 1 c2

• where the numbers dk are NOT the moments of a measure!

The asymptotic expansion must take into account the fact that µc (R \ [−2; 2]) = O 1 c2

• .

Dilation operator Λα for measures defined by Λα(µ)(A) = µ (A/α) for a measure µ and a Borel set A. For example, Λα(σ) is supported on [−2α; 2α].

Asymptotic expansion of the spectrum: second order (II) Theorem: For every k ≥ 0 and as c → ∞ mk(µc) = mk

• Λ1+ 1

2c

• σ + 1

c ˆ σ{1} + 1 c2 ˆ σ{2}

• + o

1 c2

• where ˆ

σ{1} is a measure with null total mass and density −x4 − 5x2 + 4 2π √ 4 − x2 1|x|<2 and where ˆ σ{2} is a measure with null total mass and density −2x8 − 17x6 + 46x4 − 325

8 x2 + 21 4

π √ 4 − x2 1|x|<2.

Second order – numerical simulations 100 matrices of size 10000 with c = 20 Density of Λ1+ 1

2c

• ˆ

σ{1} Histogram of c

• µc

n − Λ1+ 1

2c (σ)

Second order – numerical simulations 100 matrices of size 10000 with c = 20 Density of Λ1+ 1

2c

• ˆ

σ{2} Histogram of c2 µc

n − Λ1+ 1

2c

• σ + 1

c ˆ

σ{1}

Edge of the Spectrum mk(µc) = mk

• Λ1+ 1

2c

• σ + 1

c ˆ σ{1} + 1 c2 ˆ σ{2}

• + o

1 c2

• The measure on the right hand side is supported on [−2 − 1/c; 2 + 1/c].

Edge of the Spectrum mk(µc) = mk

• Λ1+ 1

2c

• σ + 1

c ˆ σ{1} + 1 c2 ˆ σ{2}

• + o

1 c2

• The measure on the right hand side is supported on [−2 − 1/c; 2 + 1/c].

This suggests that for ε > 0, as c → ∞, µc

• −∞; −2 − 1 + ε

c

• ∪
• 2 + 1 + ε

c ; +∞

• = o

1 c2

• .

Edge of the Spectrum mk(µc) = mk

• Λ1+ 1

2c

• σ + 1

c ˆ σ{1} + 1 c2 ˆ σ{2}

• + o

1 c2

• The measure on the right hand side is supported on [−2 − 1/c; 2 + 1/c].

This suggests that for ε > 0, as c → ∞, µc

• −∞; −2 − 1 + ε

c

• ∪
• 2 + 1 + ε

c ; +∞

• = o

1 c2

• .

Thank you!