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Ising models on power-law random graphs

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Ising models on power-law random graphs Powered By Docstoc
					     Ising models on power-law
     random graphs
     Sander Dommers

     Joint work with:
        Cristian Giardinà
        Remco van der Hofstad


Random Graphs and the Brain
May 11, 2011
                                Where innovation starts
     Introduction                                                  2/17

     The brain is a complex network of neurons.




     Other examples of complex networks include social networks,
     information networks, technological networks.

/   department of mathematics and computer science
     Ising model                                                              3/17


     Ising model: paradigm model in statistical physics for cooperative
     behavior.

     When studied on complex networks it can model for example opinion
     spreading in society.

     We will model complex networks with power-law random graphs.

     What are effects of structure of complex networks on behavior of Ising
     model?




/   department of mathematics and computer science
     Definition of the Ising model                                                   4/17


     On a graph Gn , the ferromagnetic Ising model is given by the following
     Boltzmann distributions over σ ∈ {−1, +1}n ,
                                                               
                              1                                
                µ(σ ) =              exp β     σi σj + B     σi ,
                          Zn (β, B )
                                                     (i ,j )∈En
                                                               
                                                                    i ∈[n]


     where
             β ≥ 0 is the inverse temperature;
             B is the external magnetic field;
             Zn (β, B ) is a normalization factor (the partition function), i.e.,
                                                                       
                                                                       
                   Zn (β, B ) =           exp β         σi σj + B     σi .
                                        nσ ∈{−1,1}     (i ,j )∈En
                                                                       
                                                                      i ∈[n]




/   department of mathematics and computer science
     Power-law random graphs                                                       5/17


     In the configuration model a graph Gn = (Vn = [n], En ) is constructed as
     follows.


             Let D have a certain distribution (the degree distribution);
             Assign Di half-edges to each vertex i ∈ [n], where Di are i.i.d. like D
             (Add one half-edge to last vertex when the total number of
             half-edges is odd);
             Attach first half-edge to another half-edge uniformly at random;
             Continue until all half-edges are connected.


     Special attention to power-law degree sequences, i.e.,

                                      P[D ≥ k ] ≤ ck −(τ −1) ,   τ > 2.


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     Local structure configuration model for τ > 2                             6/17


     Start from random vertex i which has degree Di .

     Look at neighbors of vertex i , probability such a neighbor has degree
     k + 1 is approximately,

        (k + 1)          j ∈[n]   1{Dj =k +1}
                        j ∈[n]   Dj




/   department of mathematics and computer science
     Local structure configuration model for τ > 2                                              6/17


     Start from random vertex i which has degree Di .

     Look at neighbors of vertex i , probability such a neighbor has degree
     k + 1 is approximately,

        (k + 1)          j ∈[n]   1{Dj =k +1} /n          (k + 1)P[D = k + 1]
                                                     −→                       ,   for τ > 2.
                        j ∈[n]   Dj /n                           E[D ]




/   department of mathematics and computer science
     Local structure configuration model for τ > 2                                              6/17


     Start from random vertex i which has degree Di .

     Look at neighbors of vertex i , probability such a neighbor has degree
     k + 1 is approximately,

        (k + 1)          j ∈[n]   1{Dj =k +1} /n          (k + 1)P[D = k + 1]
                                                     −→                       ,   for τ > 2.
                        j ∈[n]   Dj /n                           E[D ]

     Let K have distribution (the forward degree distribution),
                                                      (k + 1)P[D = k + 1]
                                    P[K = k ] =                           .
                                                             E[D ]
     Locally tree-like structure: a branching process with offspring D in first
     generation and K in further generations. Also, uniformly sparse.


/   department of mathematics and computer science
     Pressure in thermodynamic limit (E[K ] < ∞)                                                       7/17



     Theorem (Dembo, Montanari, ’10)
     For a locally tree-like and uniformly sparse graph sequence {Gn }n≥1 with
     E[K ] < ∞, the pressure per particle,
                                                          1
                                           ψn (β, B ) =     log Zn (β, B ),
                                                          n
     converges, for n → ∞, to
                         E[D ]               E[D ]
     ϕh (β, B ) ≡              log cosh(β) −       E[ log(1 + tanh(β) tanh(h1 ) tanh(h2 ))]
                          2                   2
                                                           D
                                        + E log e B              1 + tanh(β) tanh(hi )
                                                          i =1
                                                                     D
                                                          +e   −B
                                                                           1 − tanh(β) tanh(hi )   .
                                                                    i =1
/   department of mathematics and computer science
     Pressure in thermodynamic limit (E[D ] < ∞)                                                       8/17



     Theorem (DGvdH, ’10)
     Let τ > 2. Then, in the configuration model, the pressure per particle,
                                                          1
                                           ψn (β, B ) =     log Zn (β, B ),
                                                          n
     converges almost surely, for n → ∞, to
                         E[D ]               E[D ]
     ϕh (β, B ) ≡              log cosh(β) −       E[ log(1 + tanh(β) tanh(h1 ) tanh(h2 ))]
                          2                   2
                                                           D
                                        + E log e     B
                                                                 1 + tanh(β) tanh(hi )
                                                          i =1
                                                                     D
                                                          +e   −B
                                                                           1 − tanh(β) tanh(hi )   .
                                                                    i =1


/   department of mathematics and computer science
     Tree recursion                                                                         9/17


     Proposition
     Let Kt be i.i.d. like K and B > 0. Then, the recursion
                                                     Kt
                                                            atanh(tanh(β) tanh(hi(t ) )),
                                        d
                             h (t +1) = B +
                                                     i =1

     has a unique fixed point hβ .
                              ∗


     Interpretation: the effective field of a vertex in a tree expressed in that of
     its neighbors.

     Uniqueness shown by showing that effect of boundary conditions on
     generation t vanishes for t → ∞.




/   department of mathematics and computer science
     Correlation inequalities                                                10/17



     Lemma (Griffiths, ’67, Kelly, Sherman, ’68)
     For a ferromagnet with positive external field, the magnetization at a
     vertex will not decrease, when
             The number of edges increases;
             The external magnetic field increases;
             The temperature decreases.

     Lemma (Griffiths, Hurst, Sherman, ’70)
     For a ferromagnet with positive external field, the magnetization is
     concave in the external fields, i.e.,

                                                       ∂2
                                                            mj (B ) ≤ 0.
                                                     ∂Bk ∂B


/   department of mathematics and computer science
     Outline of the proof                                                                                       11/17




      lim ψn (β, B )
     n→∞


                                                                 ε                            β
                                                                      ∂                            ∂
             = lim lim ψn (0, B ) +                                     ψn (β , B )dβ +              ψn (β , B )dβ
                   ε↓0 n→∞                                   0       ∂β                   ε       ∂β

                                                         β
                                                              ∂
             = ϕh (0, B ) + 0 + lim                             ϕ(β , B )dβ
                                            ε↓0      ε       ∂β

             = ϕh (β, B ).




/   department of mathematics and computer science
     Outline of the proof                                                                                       11/17




      lim ψn (β, B )
     n→∞


                                                                 ε                            β
                                                                      ∂                            ∂
             = lim lim ψn (0, B ) +                                     ψn (β , B )dβ +              ψn (β , B )dβ
                   ε↓0 n→∞                                   0       ∂β                   ε       ∂β

                                                         β
                                                              ∂
             = ϕh (0, B ) + 0 + lim                             ϕ(β , B )dβ
                                            ε↓0      ε       ∂β

             = ϕh (β, B ).




/   department of mathematics and computer science
     Outline of the proof                                                                                       11/17




      lim ψn (β, B )
     n→∞


                                                                 ε                            β
                                                                      ∂                            ∂
             = lim lim ψn (0, B ) +                                     ψn (β , B )dβ +              ψn (β , B )dβ
                   ε↓0 n→∞                                   0       ∂β                   ε       ∂β

                                                         β
                                                              ∂
             = ϕh (0, B ) + 0 + lim                             ϕ(β , B )dβ
                                            ε↓0      ε       ∂β

             = ϕh (β, B ).




/   department of mathematics and computer science
     Outline of the proof                                                                                       11/17




      lim ψn (β, B )
     n→∞


                                                                 ε                            β
                                                                      ∂                            ∂
             = lim lim ψn (0, B ) +                                     ψn (β , B )dβ +              ψn (β , B )dβ
                   ε↓0 n→∞                                   0       ∂β                   ε       ∂β

                                                         β
                                                              ∂
             = ϕh (0, B ) + 0 + lim                             ϕ(β , B )dβ
                                            ε↓0      ε       ∂β

             = ϕh (β, B ).




/   department of mathematics and computer science
     Internal energy                                                                                               12/17




                      ∂              1                                            |En |   (i ,j )∈En   σi σj   µ
                        ψn (β, B ) =                              σi σj   µ
                                                                              =
                     ∂β              n                                             n           |En |
                                                     (i ,j )∈En
                                                     E[D ]
                                          −→               E σi σj            µ
                                                      2




/   department of mathematics and computer science
     Internal energy                                                                                               12/17




                      ∂              1                                            |En |   (i ,j )∈En   σi σj   µ
                        ψn (β, B ) =                              σi σj   µ
                                                                              =
                     ∂β              n                                             n           |En |
                                                     (i ,j )∈En
                                                     E[D ]
                                          −→               E σi σj            µ
                                                      2




                                   E[D ]                                  E[D ]
                                         E σi σj            µ
                                                                  −→            E σi σj        e
                                    2                                      2



/   department of mathematics and computer science
     Derivative of ϕ                                                                             13/17



                                        ∂               E[D ]
                                          ϕhβ (β, B ) =
                                            ∗                 E σi σj         .
                                       ∂β                2               e



                       E[D ]               E[D ]
ϕh (β, B ) =                 log cosh(β) −       E[ log(1 + tanh(β) tanh(h1 ) tanh(h2 ))]
                        2                   2
                          D                                        D
+ E log e B                      1 + tanh(β) tanh(hi ) + e −B                1 − tanh(β) tanh(hi )
                         i =1                                     i =1


             Show that we can ignore dependence of hβ on β;
                                                         ∗

             (Interpolation techniques. Split analysis into two parts, one for
             small degrees and one for large degrees)

             Compute the derivative with assuming β fixed in hβ .
                                                             ∗



/   department of mathematics and computer science
     Thermodynamic quantities                                                                            14/17


     Corollary
     Let τ > 2. Then, in the configuration model, a.s.:
     The magnetization is given by
                                                 n
                                1                                    ∂
                 m(β, B ) ≡ lim                        σi   µ
                                                                =      ϕh ∗ (β, B ) = E σ0   νD +1
                                                                                                     .
                           n→∞ n                                    ∂B
                                                i =1




     The susceptibility is given by

                                                  ∂Mn (β, B )    ∂2
                            χ(β, B ) ≡ lim                    =      ϕh ∗ (β, B ).
                                              n→∞    ∂B         ∂B 2
/   department of mathematics and computer science
     Critical temperature                                                                   15/17


     Define the magnetization on Gn as
                                                                  n
                                                             1
                                              mn (β, B ) =              σi µ .
                                                             n
                                                                 i =1

     Then, the spontaneous magnetization,

                                                                 = 0,            β < βc ;
                         m(β, 0+) = lim m(β, B )
                                              B ↓0               > 0,            β > βc .

     The critical inverse temperature βc is given by

                                                 E[K ](tanh βc ) = 1.

     Note that, for τ ∈ (2, 3), we have E[K ] = ∞, so that βc = 0.


/   department of mathematics and computer science
     Critical exponents                                                                       16/17


     Predictions by physicists (e.g. Leone, Vázquez, Vespignani, Zecchina,
     ’02).

     Critical behavior of magnetization m, and susceptibility χ .


                              m(β, 0+ ), β ↓ βc      m(βc , B ), B ↓ 0   χ (β, 0+ ), β ↓ βc

          τ >5                   ∼ (β − βc )1/2           ∼ B 1/3         ∼ (β − βc )−1

          τ ∈ (3, 5)          ∼ (β − βc )1/(τ −3)       ∼ B 1/(τ −2)

          τ ∈ (2, 3)          ∼ (β − βc )1/(3−τ )         ∼ B1             ∼ (β − βc )1




/   department of mathematics and computer science
     Critical exponents                                                                       16/17


     Predictions by physicists (e.g. Leone, Vázquez, Vespignani, Zecchina,
     ’02).

     Critical behavior of magnetization m, and susceptibility χ .


                              m(β, 0+ ), β ↓ βc      m(βc , B ), B ↓ 0   χ (β, 0+ ), β ↓ βc

          τ >5                   ∼ (β − βc )1/2           ∼ B 1/3         ∼ (β − βc )−1

          τ ∈ (3, 5)          ∼ (β − βc )1/(τ −3)       ∼ B 1/(τ −2)

          τ ∈ (2, 3)          ∼ (β − βc )1/(3−τ )         ∼ B1             ∼ (β − βc )1




/   department of mathematics and computer science
     Distances in power-law random graphs                                     17/17


     Let Hn be the graph distance between two uniformly chosen connected
     vertices in the configuration model. Then:
             For τ > 3 and E[K ] > 1 (vdH, Hooghiemstra, Van Mieghem, ’05),

                                                      Hn ∼ log n,

             For τ ∈ (2, 3) (vdH, Hooghiemstra, Znamenski, ’07),

                                                     Hn ∼ log log n;




/   department of mathematics and computer science
     Distances in power-law random graphs                                       17/17


     Let Hn be the graph distance between two uniformly chosen connected
     vertices in the configuration model. Then:
             For τ > 3 and E[K ] > 1 (vdH, Hooghiemstra, Van Mieghem, ’05),

                                                      Hn ∼ log n,

             For τ ∈ (2, 3) (vdH, Hooghiemstra, Znamenski, ’07),

                                                     Hn ∼ log log n;

     For τ > 3 and τ ∈ (2, 3) similar results hold for the diameter of linear
     preferential attachment models (D, vdH, Hooghiemstra, ’10).




/   department of mathematics and computer science

				
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