# Ising Model

Consider the Ising model on a $d$-dimensional cubic lattice. The Hamiltonian is given by

$H = -J \sum_{\langle i,j\rangle} s_i s_{j} - h \sum_i s_i$
where the $\sum_{\langle i,j\rangle}$ indicates summation over the pair of nearest neighbors $\langle i,j\rangle$, and $s_i = \pm 1$ and $s_j$ are neighboring Ising spins.

Let us transform our spin variable by introducing the fluctuation from its mean value $m_i \equiv \langle s_i\rangle$. We may rewrite the Hamiltonian:

$H = -J \sum_{\langle i,j \rangle} (m_i + \delta s_i ) (m_j + \delta s_j) - h \sum_i s_i$
where we define $\delta s_i \equiv s_i - m_i$; this is the fluctuation of the spin. If we expand the right hand side, we obtain one term that is entirely dependent on the mean values of the spins, and independent of the spin configurations. This is the trivial term, which does not affect the statistical properties of the system. The next term is the one involving the product of the mean value of the spin and the fluctuation value. Finally, the last term involves a product of two fluctuation values.

The mean-field approximation consists of neglecting this second order fluctuation term. These fluctuations are enhanced at low dimensions, making MFT a better approximation for high dimensions.

$H \approx H^{MF} \equiv -J \sum_{\langle i,j \rangle} (m_i m_j +m_i \delta s_j + m_j \delta s_i ) - h \sum_i s_i$
Again, the summand can be reexpanded. In addition, we expect that the mean value of each spin is site-independent, since the Ising chain is translationally invariant. This yields
$H^{MF} = -J \sum_{\langle i,j \rangle} \left( m^2 + 2m(s_i-m) \right) - h \sum_i s_i$
The summation over neighboring spins can be rewritten as $\sum_{\langle i,j\rangle} = \frac{1}{2} \sum_i \sum_{j\in nn(i)}$ where $nn(i)$ means 'nearest-neighbor of $i$' and the $1/2$ prefactor avoids double-counting, since each bond participates in two spins. Simplifying leads to the final expression
$H^{MF}= \frac{J m^2 N z}{2}- \underbrace{(h+m J z)}_{h^{\mathrm{eff}}} \sum_i s_i$
where $z$ is the coordination number. At this point, the Ising Hamiltonian has been decoupled into a sum of one-body Hamiltonians with an effective mean-field $h^{\mathrm{eff}}=h+J z m$ which is the sum of the external field $h$ and of the mean-field induced by the neighboring spins. It is worth noting that this mean field directly depends on the number of nearest neighbors and thus on the dimension of the system (for instance, for a hypercubic lattice of dimension $d$, $z = 2 d$).

Substituting this Hamiltonian into the partition function, and solving the effective 1D problem, we obtain

$Z = e^{-\beta J m^2 N z /2} \left[2]^{N}$
where $N$ is the number of lattice sites. This is a closed and exact expression for the partition function of the system. We may obtain the free energy of the system, and calculate critical exponents. In particular, we can obtain the magnetization $m$ as a function of $h^{\mathrm{eff}}$.

We thus have two equations between $m$ and $h^{\mathrm{eff}}$, allowing us to determine $m$ as a function of temperature. This leads to the following observation:

• for temperatures greater than a certain value $T_c$, the only solution is $m=0$. The system is paramagnetic.
• for $T < T_c$, there are two non-zero solutions: $m = \pm m_0$. The system is ferromagnetic.
$T_c$ is given by the following relation: $T_c = \frac{J z}{k_B}$. This shows that MFT can account for the ferromagnetic phase transition.

# BA Model

## 模型设定

• 初始状态有$m_0$个节点
• 1. 增长原则：每次加入一个节点i （加入时间记为$t_i$）, 每个节点的加入带来m条边，2m个度的增加
• 其中老节点分到的度数是m，新加入的那一个节点分到的度数为m
• 那么到时间t的时候，网络的总节点数是$m_0 + t$，网络的总度数为$2mt$。
• 2. 优先链接原则：每一次从m条边中占有一条边的概率正比于节点的度$k_i$
• 那么显然，加入的越早（$t_i$越小）越容易获得更多的链接数。
• 从时间0开始，每一个时间步系统中的节点度$k_i$是不断增加的。

## 度的增长/时间依赖性

$k_i$在一个时间步获得一个度的概率表示为$\prod (k_i)$， 那么有：

$\prod (k_i) = \frac{k_i}{\sum k_i} = \frac{k_i}{2mt}$

$\frac{\partial k_i}{\partial t} = \Delta k \prod (k_i) = m \frac{k_i}{2mt} = \frac{k_i}{2t}$

$\frac{\partial k_i}{k_i} = \frac{\partial t}{2t}$

$\int \frac{1}{k_i} d k_i = \int \frac{1}{2t} dt$

$k_i =(Ct) ^ {0.5}$ 公式（1）

$k_i(t_i) = m$ 代入公式（1）

$k_i = m (\frac{t}{t_i})^{0.5}$ 公式（3）

## 累积概率分布

$P(k_i(t) < k) = P( m (\frac{t}{t_i})^{0.5} < k ) = P( t_i > \frac{m^2 t}{k^2} ) = 1 - P(t_i \leqslant \frac{m^2 t}{k^2} )$ 公式（4）

$P(t_i) = \frac{1}{m_0 + t}$ 公式（5）

### 均匀分布的性质

• 设连续型随机变量X的概率密度函数为 $f(x)=1/(b-a)，a≤x≤b$, 则称随机变量X服从[a,b]上的均匀分布，记为X~U[a,b]。
• 若[x1,x2]是[a,b]的任一子区间，则 $P{x_1≤x≤x_2}=(x_2-x_1)/(b-a)$

$P(k_i(t) < k) = 1 - \frac{m^2 t}{k^2 (m_0 + t)}$ 公式（6）

$P( k ) = \frac{\partial P(k_i(t) < k)}{\partial k} = \frac{2m^2 t}{m_0 + k} \frac{1}{k^3}$ 公式（7）

# Continuum theory of BA Model

Barabasi (1999) Emergence of scaling in random networks.Science-509-12.[1]

Barabasi (1999) Mean-field theory for scale-free random networks. PA.[2]

Albert & Barabasi (2002) Statistical mechanics of complex networks. RMP.[3]

Barabasi将采用平均场的方法称为Continuum theory:

The continuum approach introduced by Baraba´si and Albert (1999)[4] and Baraba´si, Albert, and Jeong (1999)[5] calculates the time dependence of the degree $k_i$ of a given node i. This degree will increase every time a new node enters the system and links to node i, the probability of this process being $\prod(k_i)$. Assuming that $k_i$ is a continuous real variable, the rate at which $k_i$ changes is expected to be proportional to $\prod(k_i)$. Consequently $k_i$ satisfies the dynamical equation:

# 参考文献

• Barabasi (1999) Emergence of scaling in random networks.Science-509-12.pdf
• 2.0 2.1 Barabasi (1999) Mean-field theory for scale-free random networks. PA.pdf
• Albert & Barabasi (2002) Statistical mechanics of complex networks. RMP.pdf