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˝ os-R´ enyi random graphs • vertex set { 1 , . . . n } G ( n, p ) • vertices linked by an edge independently with probability p

Erd˝ os-R´ enyi random graphs • vertex set { 1 , . . . n } G ( n, p ) • vertices linked by an edge independently with probability p • symmetric Adjacency matrix A • if i � = j , P ( A i,j = 1) = 1 − P ( A i,j = 0) = p • for every i , A i,i = 0

Erd˝ os-R´ enyi random graphs • vertex set { 1 , . . . n } G ( n, p ) • vertices linked by an edge independently with probability p • symmetric Adjacency matrix A • if i � = j , P ( A i,j = 1) = 1 − P ( A i,j = 0) = p • for every i , A i,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...

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 n = 1 � µ c δ λ : empirical spectral distribution of G ( n, c/n ) n λ ∈ Sp( c − 1 / 2 A )

´ Etat de l’art n = 1 � µ c δ λ : empirical spectral distribution of G ( n, c/n ) n λ ∈ Sp( c − 1 / 2 A ) Fact : as n → ∞ , µ c n converges weakly to a probability measure µ c

´ Etat de l’art n = 1 � µ c δ λ : empirical spectral distribution of G ( n, c/n ) n λ ∈ Sp( c − 1 / 2 A ) Fact : as n → ∞ , µ c n converges weakly to a probability measure µ c Known properties of µ c : • if c → ∞ , µ c converges weakly to Wigner semi-circle law • unbounded support • dense set of atoms

´ Etat de l’art n = 1 � µ c δ λ : empirical spectral distribution of G ( n, c/n ) n λ ∈ Sp( c − 1 / 2 A ) Fact : as n → ∞ , µ c n converges weakly to a probability measure µ c Known properties of µ c : • if c → ∞ , µ c converges weakly to Wigner semi-circle law • unbounded support • dense set of atoms • µ c ( { 0 } ) known explicitly [Bordenave, Lelarge, Salez 2011]

´ Etat de l’art n = 1 � µ c δ λ : empirical spectral distribution of G ( n, c/n ) n λ ∈ Sp( c − 1 / 2 A ) Fact : as n → ∞ , µ c n converges weakly to a probability measure µ c Known properties of µ c : • if c → ∞ , µ c converges weakly to Wigner semi-circle law • unbounded support • dense set of atoms • µ c ( { 0 } ) known explicitly [Bordenave, Lelarge, Salez 2011] • µ c is not purely atomic iif c > 1 [Bordenave, Sen, Vir´ ag 2013]

Asymptotic expansion of the spectrum

Asymptotic expansion of the spectrum | x | k | dµ ( x ) | < ∞ , denote m k ( µ ) = x k dµ ( x ) � � If µ is a (signed) mesure and

Asymptotic expansion of the spectrum | x | k | dµ ( x ) | < ∞ , denote m k ( µ ) = x k dµ ( x ) � � If µ is a (signed) mesure and Theorem: For every k ≥ 0 and as c → ∞ � 1 � m k ( µ c ) = m k ( σ ) + 1 c m k ( σ { 1 } ) + o c where σ is the semi-circle law having density 1 � 4 − x 2 1 | x | < 2 2 π and σ { 1 } is a measure with total mass 0 and density x 4 − 4 x 2 + 2 1 √ 1 | x | < 2 . 2 π 4 − x 2

Asymptotic expansion of the spectrum – numerical simulations 100 matrices of size 10000 with c = 20 Histogram of µ c n Density of σ

Asymptotic expansion of the spectrum – numerical simulations 100 matrices of size 10000 with c = 20 Histogram of c ( µ c 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 : � 1 � m k ( µ c ) = m k ( σ ) + 1 c m k ( σ { 1 } ) + 1 c 2 d k + o c 2 where the numbers d k 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 : � 1 � m k ( µ c ) = m k ( σ ) + 1 c m k ( σ { 1 } ) + 1 c 2 d k + o c 2 where the numbers d k are NOT the moments of a measure! The asymptotic expansion must take into account the fact that � 1 � µ c ( R \ [ − 2; 2]) = O . c 2

Asymptotic expansion of the spectrum: second order (I) Proposition: For every k � 0 we have the following asymtotic expansion in c : � 1 � m k ( µ c ) = m k ( σ ) + 1 c m k ( σ { 1 } ) + 1 c 2 d k + o c 2 where the numbers d k are NOT the moments of a measure! The asymptotic expansion must take into account the fact that � 1 � µ c ( R \ [ − 2; 2]) = O . c 2 Dilation operator Λ α for measures defined by Λ α ( 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 : � 1 � m k ( µ c ) = m k ( σ ) + 1 c m k ( σ { 1 } ) + 1 c 2 d k + o c 2 where the numbers d k are NOT the moments of a measure! The asymptotic expansion must take into account the fact that � 1 � µ c ( R \ [ − 2; 2]) = O . c 2 Dilation operator Λ α for measures defined by Λ α ( µ )( A ) = µ ( A/α ) for a measure µ and a Borel set A . For example, Λ α ( σ ) is supported on [ − 2 α ; 2 α ] .

Asymptotic expansion of the spectrum: second order (II) Theorem: For every k ≥ 0 and as c → ∞ � 1 � � �� � σ + 1 σ { 1 } + 1 σ { 2 } m k ( µ c ) = m k Λ 1+ 1 c ˆ c 2 ˆ + o c 2 2 c σ { 1 } is a measure with null total mass and density where ˆ − x 4 − 5 x 2 + 4 √ 4 − x 2 1 | x | < 2 2 π σ { 2 } is a measure with null total mass and density and where ˆ − 2 x 8 − 17 x 6 + 46 x 4 − 325 8 x 2 + 21 4 √ 1 | x | < 2 . 4 − x 2 π

Second order – numerical simulations 100 matrices of size 10000 with c = 20 σ { 1 } � � Density of Λ 1+ 1 ˆ 2 c � � µ c Histogram of c n − Λ 1+ 1 2 c ( σ )

Second order – numerical simulations 100 matrices of size 10000 with c = 20 Histogram of c 2 � σ { 1 } �� σ + 1 σ { 2 } � µ c � � n − Λ 1+ 1 c ˆ Density of Λ 1+ 1 ˆ 2 c 2 c

Edge of the Spectrum � 1 � � �� � σ + 1 σ { 1 } + 1 m k ( µ c ) = m k σ { 2 } Λ 1+ 1 c ˆ c 2 ˆ + o c 2 2 c The measure on the right hand side is supported on [ − 2 − 1 /c ; 2 + 1 /c ] .

Edge of the Spectrum � 1 � � �� � σ + 1 σ { 1 } + 1 m k ( µ c ) = m k σ { 2 } Λ 1+ 1 c ˆ c 2 ˆ + o c 2 2 c The measure on the right hand side is supported on [ − 2 − 1 /c ; 2 + 1 /c ] . This suggests that for ε > 0 , as c → ∞ , � 1 �� � � �� � −∞ ; − 2 − 1 + ε 2 + 1 + ε µ c ∪ ; + ∞ = o . c 2 c c

Edge of the Spectrum � 1 � � �� � σ + 1 σ { 1 } + 1 m k ( µ c ) = m k σ { 2 } Λ 1+ 1 c ˆ c 2 ˆ + o c 2 2 c The measure on the right hand side is supported on [ − 2 − 1 /c ; 2 + 1 /c ] . This suggests that for ε > 0 , as c → ∞ , � 1 �� � � �� � −∞ ; − 2 − 1 + ε 2 + 1 + ε µ c ∪ ; + ∞ = o . c 2 c c Thank you!