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chengjun committed Jul 29, 2024
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94 changes: 48 additions & 46 deletions notebook/.ipynb_checkpoints/05-01-network-science-intro-checkpoint.ipynb
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67 changes: 34 additions & 33 deletions notebook/.ipynb_checkpoints/05-03-network-science-models-checkpoint.ipynb
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"source": [
"In a random network the probability that **node i** has exactly **$k$** links is the product of three terms:\n",
"\n",
"> $k_{max} = N-1$, 节点i的边的最大数量是N-1\n",
"* The probability of node i having k links: **$p^k$**.\n",
"* The probability that the remaining (N-1-k) links are missing: $(1-p)^{N-1-k}$\n",
"* The number of ways we can select $k$ links from $N- 1$ potential links a node can have: $C_{N-1}^k$\n",
"\n",
"> $k_{max} = N-1$, 节点i的边的最大数量是N-1\n",
"\n",
"* The probability of node i having k links, or **$p^k$**.\n",
"* The probability that the remaining (N-1-k) links are missing, or $(1-p)^{N-1-k}$\n",
"* The number of ways we can select $k$ links from $N- 1$ potential links a node can have, or $C_{N-1}^k$\n",
"\n",
"Consequently the degree distribution of a random network is:\n",
"Consequently, the degree distribution of a random network is:\n",
"\n",
"$p_k = C_{N-1}^k p^k (1-p)^{N-1-k}$\n",
"\n",
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"## WS模型 （1998）\n",
"\n",
"Duncan Watts and Steven Strogatz proposed an extension of the random network model motivated by two observations：\n",
"- (a) Small World Property： In real networks the average distance between two nodes depends logarithmically on N\n",
"- (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 \n",
"- **Small World Property**： In real networks the average distance between two nodes depends logarithmically on N\n",
"- **High Clustering**： The average clustering coefficient of real networks is much higher than expected for a random network of similar N and L \n",
"\n",
"- a regular lattice has high clustering but lacks the small-world phenomenon.\n",
"- and a random network has low clustering, but displays the small-world property.\n"
" - a regular lattice has high clustering but lacks the small-world phenomenon.\n",
" - a random network has low clustering, but displays the small-world property.\n"
]
},
{
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}
},
"source": [
"> Numerical simulations indicate that for a range of rewiring parameters the model's average path length is low but the clustering coefficient is high\n",
"<div><img src=img/net28.png width='1300px' align = 'left'\\></div>\n",
"\n",
"hence reproducing the coexistence of high clustering and small-world phenomena\n",
"\n",
"D. J. Watts and S. H. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 393: 409–10, 1998."
"Watts-Strogatz Model"
]
},
{
Expand All @@ -501,14 +500,11 @@
}
},
"source": [
"- The dependence of the average path length $d(p)$ and clustering coefficient $<C(p)>$ on the **rewiring parameter $p$**. \n",
"Numerical simulations indicate that \n",
"- for a range of rewiring parameters the model's average path length is low but the clustering coefficient is high\n",
"\n",
"- hence reproducing the coexistence of high clustering and small-world phenomena\n",
"\n",
"- Note that $d(p)$ and $<C(p)>$ have been normalized by $d(0)$ and $<C(0)>$ obtained for a regular lattice. \n",
"- The rapid drop in $d(p)$ signals the onset of the small-world phenomenon. \n",
" - During this drop, $<C(p)>$ remains high. \n",
" - Hence in the range $0.001<p<0.1$ short path lengths and high clustering co-exist in the network.\n",
" \n",
" \n",
"D. J. Watts and S. H. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 393: 409–10, 1998."
]
},
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}
},
"source": [
"<div><img src=img/net28.png width='1300px' align = 'left'\\></div>\n",
"\n",
"- The dependence of the average path length $d(p)$ and clustering coefficient $<C(p)>$ on the **rewiring parameter $p$**. \n",
"\n",
"Watts-Strogatz Model"
"- Note that $d(p)$ and $<C(p)>$ have been normalized by $d(0)$ and $<C(0)>$ obtained for a regular lattice. \n",
"- The rapid drop in $d(p)$ signals the onset of the small-world phenomenon. \n",
" - During this drop, $<C(p)>$ remains high. \n",
" - Hence in the range $0.001<p<0.1$ short path lengths and high clustering co-exist in the network.\n",
" \n",
" \n",
"D. J. Watts and S. H. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 393: 409–10, 1998."
]
},
{
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"\n",
"Barabasi (1999) Emergence of scaling in random networks.Science-509-12\n",
"\n",
"### Scaling-Free\n",
"### Scale-Free\n",
"\n",
"![image.png](img/net30.png)\n",
"\n",
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"source": [
"显然：\n",
"\n",
"* 当$n - \\gamma +1 <= 0$时，随着$k_{max}$增加，$k_{max}^{n- \\gamma +1} \\rightarrow 0$。所有满足$n <= \\gamma -1$的n阶矩都是有限的。\n",
"* 当$n - \\gamma +1 <= 0$时，随着$k_{max}$增加，$k_{max}^{n- \\gamma +1} \\rightarrow 0$。所有满足$n <= \\gamma -1$的n阶矩都是有极限的。\n",
"* 当$n - \\gamma +1 >0 $时，随着$k_{max}增加，$$k_{max}^{n- \\gamma +1} \\rightarrow \\infty$。所有满足$n > \\gamma -1$的n阶矩都是无极限的。\n",
"\n",
"对于无标度网络而言，一般幂参数$2 < \\gamma < 3$，所以：\n",
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}
},
"source": [
"## 连续平均场\"Continuum Mean-Field\" \n",
"## 连续平均场 \"Continuum Mean-Field\" \n",
"\n",
"\"Mean-Field\": many -> one\n",
"\n",
Expand All @@ -738,13 +739,13 @@
},
"source": [
"### 模型设定\n",
"* 初始状态有$m_0$个节点\n",
"* 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\n",
"** 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.\n",
"** 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$2mt$。\n",
"* 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$k_i$\n",
"** 那么显然，加入的越早（$t_i$越小）越容易获得更多的链接数。\n",
"** 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。\n"
"- 初始状态有$m_0$个节点\n",
"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\n",
" - 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.\n",
" - 时间t的时候，网络的总节点数是$m_0 + t$，网络的总度数为$2mt$。\n",
"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$k_i$\n",
" - 加入的越早（$t_i$越小）越容易获得更多的链接数。\n",
" - 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。\n"
]
},
{
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"width": "504px"
},
"toc_section_display": false,
"toc_window_display": true
"toc_window_display": false
}
},
"nbformat": 4,
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