Definition: A random graph is a graph of N nodes where each pair of nodes is connected by probability p.
N lableled nodes are connected with L randomly placed links. Erdös-Rényi(1959)
Each pair of N labeled nodes is connected with probability p. Gilbert (1959)
In a random network the probability that node i has exactly $k$ links is the product of three terms:
Consequently the degree distribution of a random network is:
which follows the binomial distribution. The shape of this distribution depends on the system size $N$ and the probability $p$.
The evolution of random networks The relative size of the giant component in function of the average degree $<k>$ in the Erdős-Rényi model. The figure illustrates the phase tranisition at $<k> = 1$, responsible for the emergence of a giant component with nonzero $N_G$. REAL NETWORKS ARE SUPERCRITICAL
also known as six degrees of separation, has long fascinated the general public. It states that if you choose any two individuals anywhere on Earth, you will find a path of at most six acquaintances between them.
for $<k> ≈ 1,000$, which is the estimated number of acquaintences an individual has, we expect $10^6$ individuals at distance two and about a billion, i.e. almost the whole earth’s population, at distance three from us.
从一个开始的节点出发走d步，共有节点数量：
$N(d) = 1 + <k> + <k>^2 + <k>^3 + ...+ <k>^d = \frac{<k>^{d+1}-1}{<k>-1}$
$N(d_{max}) = N$可以计算最大的距离，或者说网络直径。
若$<k>$ 远大于1，那么$N = <k>^{d_{max}}$， 因此，$d_{max} = \frac{ln N}{ln <k>}$
Let us illustrate the implications of (3.19) for social networks. Using $N ≈ 7 ×10^9$ and $<k> ≈ 10^3$, we obtain
I. de Sola Pool and M. Kochen. Contacts and Influence. Social Networks, 1: 5-51, 1978.
In lattices the path lengths are significantly longer than in a random network.
The first empirical study of the small world phenomena took place in 1967
Using Facebook’s social graph of May 2011, consisting of 721 million active users and 68 billion symmetric friendship links, researchers found:
L. Backstrom, P. Boldi, M. Rosa, J. Ugander, and S. Vigna. Four degrees of separation. In ACM Web Science 2012: Conference Proceedings, pages 45−54. ACM Press, 2012.
Local clustering coefficient
We need to estimate the expected number of links $L_i$ between the node’s $k_i$ neighbors.
In a random network the probability that two of i’s neighbors link to each other is $p$.
As there are $\frac{k_i(k_i - 1)}{2}$ possible links between the $k_i$ neighbors of node i, the expected value of $L_i$ is
Duncan Watts and Steven Strogatz proposed an extension of the random network model motivated by two observations：
(b) High Clustering： The average clustering coefficient of real networks is much high- er than expected for a random network of similar N and L
a regular lattice has high clustering but lacks the small-world phenomenon.
Numerical simulations indicate that for a range of rewiring parameters the model's average path length is low but the clustering coefficient is high
hence reproducing the coexistence of high clustering and small-world phenomena
D. J. Watts and S. H. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 393: 409–10, 1998.
The dependence of the average path length $d(p)$ and clustering coefficient $<C(p)>$ on the rewiring parameter $p$.
Note that $d(p)$ and $<C(p)>$ have been normalized by $d(0)$ and $<C(0)>$ obtained for a regular lattice.
D. J. Watts and S. H. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 393: 409–10, 1998.
度分布的n阶矩被定义为：
$ <k^n> = \sum_{k_{min}}^{k_{max}}k^np_k = \int_{k_{min}}^{k_{max}}k^np_kdk $ （1）
低阶矩具有明确的统计意义：
对于无标度网络而言，满足幂律分布：
$p(k) = Ck^{-\gamma}$ （2）
由公式（1）和（2）可以得到：
$ <k^n> = \int_{k_{min}}^{k_{max}}k^np_kdk = C \frac{k_{max}^{n- \gamma +1} - k_{min}^{n - \gamma +1}}{n - \gamma + 1} $ （3）
可以使用wolframalpha的积分计算器积分来进行简单验证，例如x^(n-r)dx从10到100积分 网页链接：http://www.wolframalpha.com/input/?i=integrate+x%5E%28n-r%29+dx+from+10+to+100
显然：
对于无标度网络而言，一般幂参数$2 < \gamma < 3$，所以：
对于n = 1的情况，即一阶矩平均度$<k^{}>$是有限的。
但对于n >= 2的情况，即$k^2$或$k^3$是无极限的。二阶和高阶矩无穷大是“无标度”的来源
对于不同的网络，平均度与标准差的取值
英国的天才 第一集 Genius of Britain: The Scientists Who Changed the World (2010) https://movie.douban.com/subject/4887415/
The complex interaction between a large number of components can be simplified into a single averaged effect of all the other individuals on any given one.
A discrete variables that is "smooth enough", i.e. increase by one unit in each period, such as time steps [1], spatial jumps [2-3], and population, can be viewed as a continuous geometric structure. Any measure on this structure, like degree [1], number of unique locations visited [2], and distance [3], can be described by differential equations.
$k_i$在一个时间步获得一个度的概率表示为$\prod (k_i) $， 那么有：
$\prod (k_i) = \frac{k_i}{\sum k_i} = \frac{k_i}{2mt}$
一个时间步，$k_i$随t的变化率可以表达为：
当我们思考一个累积概率分布的时候，我们想要的是$k_i(t) < k$的概率：$P(k_i(t) < k) $
由公式（3），可以知道：
$P(k_i(t) < k) = P( m (\frac{t}{t_i})^{0.5} < k ) = P( t_i > \frac{m^2 t}{k^2} ) = 1 - P(t_i \leqslant \frac{m^2 t}{k^2} )$（4）
在初始状态$ t = 0$, 有$m_0$个节点，那么$t_{m_0} = 0$
假设我们将节点加入的时间步是均匀的，那么$t_i$的概率是一个常数：
$P(t_i) = \frac{1}{m_0 + t}$ 公式（5）
根据均匀分布的性质，将公式（5）代入公式（4）得到：
$P(k_i(t) < k) = 1- \frac{m^2 t}{k^2} P(t_i) = 1 - \frac{m^2 t}{k^2 (m_0 + t)} $ 公式（6）
对累积概率函数求微分，就可以到达概率密度函数:
$P( k ) = \frac{\partial P(k_i(t) < k)}{\partial k} = \frac{2m^2 t}{m_0 + t} \frac{1}{k^3}$ 公式（7）
也就是说：$\gamma = 3$, 与m无关。
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