# Ising models on power-law random graphs by liuhongmei

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```									     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.

/   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}
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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