轉自：?http://www.flickr.com/photos/caseorganic/4510691991/in/set-72157624621620243

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小圖

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大圖

Network Models - Random network, Scale-free network, Hierarchical network

隨機網絡

The Erd?s–Rényi (ER) model of a random network14 (see figure, part A) starts with N nodes and connects each pair of nodes with probability p,
which creates a graph with approximately pN(N–1)/2 randomly placed links (see figure, part Aa). The node degrees follow a Poisson distribution
(see figure, part Ab), which indicates that most nodes have approximately the same number of links (close to the average degree ). The tail
(high k region) of the degree distribution P(k) decreases exponentially, which indicates that nodes that significantly deviate from the average are
extremely rare. The clustering coefficient is independent of a node’s degree, so C(k) appears as a horizontal line if plotted as a function of k (see
figure, part Ac). The mean path length is proportional to the logarithm of the network size, l ~ log N, which indicates that it is characterized by the
small-world property.

無標度網絡

Scale-free networks (see figure, part B) are characterized by a power-law degree distribution; the probability that a node has k links follows
P(k) ~ k –γ, where γ is the degree exponent. The probability that a node is highly connected is statistically more significant than in a random graph,
the network’s properties often being determined by a relatively small number of highly connected nodes that are known as hubs (see figure, part
Ba; blue nodes). In the Barabási–Albert model of a scale-free network15, at each time point a node with M links is added to the network, which
connects to an already existing node I with probability ΠI = kI/ΣJkJ, where kI is the degree of node I (FIG. 3) and J is the index denoting the sum over
network nodes. The network that is generated by this growth process has a power-law degree distribution that is characterized by the degree
exponent γ = 3. Such distributions are seen as a straight line on a log–log plot (see figure, part Bb). The network that is created by the
Barabási–Albert model does not have an inherent modularity, so C(k) is independent of k (see figure, part Bc). Scale-free networks with degree
exponents 2<γ<3, a range that is observed in most biological and non-biological networks, are ultra-small34,35, with the average path length
following ~ log log N, which is significantly shorter than log N that characterizes random small-world networks.

分層網絡

To account for the coexistence of modularity, local clustering and scale-free topology in many real systems it has to be assumed that clusters
combine in an iterative manner, generating a hierarchical network47,53 (see figure, part C). The starting point of this construction is a small cluster
of four densely linked nodes (see the four central nodes in figure, part Ca).Next, three replicas of this module are generated and the three external
nodes of the replicated clusters
connected to the central node of
the old cluster, which produces a
large 16-node module. Three
replicas of this 16-node module
are then generated and the 16
peripheral nodes connected to
the central node of the old
module, which produces a new
module of 64 nodes. The
hierarchical network model
seamlessly integrates a scale-free
topology with an inherent
modular structure by generating
a network that has a power-law
degree distribution with degree
exponent γ = 1 + n4/n3 = 2.26
(see figure, part Cb) and a large,
system-size independent average
clustering coefficient ~ 0.6.
The most important signature of
hierarchical modularity is the
scaling of the clustering
coefficient, which follows
C(k) ~ k –1 a straight line of slope
–1 on a log–log plot (see figure,
part Cc). A hierarchical
architecture implies that sparsely
connected nodes are part of
highly clustered areas, with
communication between the
different highly clustered
neighbourhoods being
maintained by a few hubs
(see figure, part Ca).

From Network Biology - Understanding the Cell's Functional Organization. Albert Laszlo Barabasi, Zoltan Oltvai 2004.

About the Authors
Albert-László Barabási is the Emil T. Hofman Professor of physics
at the University of Notre Dame, USA. His research group introduced
the concept of scale-free networks and studied their relevance
to biological and communication systems. He obtained his
M.Sc. degree in physics in 1991 from E?tv?s Loránd University,
Budapest, Hungary, and his Ph.D. in 1994 from Boston University,
USA. After a year as a postdoctoral fellow at IBM Thomas J.
Watson Research Center, USA, he joined the University of Notre
Dame in 1995. He is a fellow of the American Physical Society, and
the author of the general audience book Linked: The New Science of
Networks.
Zoltán Nagy Oltvai is an assistant professor of pathology at
Northwestern University’s Feinberg School of Medicine, USA. His
clinical interest is molecular pathology and he is the director of
diagnostic molecular pathology at the medical school and
Northwestern Memorial Hospital. His research group’s interest is
the theoretical and experimental study of intracellular molecular
interaction networks. He received his M.D. degree from
Semmelweiss Medical University, Budapest, Hungary, and did his
clinical pathology/molecular biology research residency at
Washington University/Barnes Hospital in St. Louis, USA.

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看不懂英文的看中文的，寫的還可以

參考：http://www.cnblogs.com/peon/archive/2009/08/23/1552472.html

[筆記+整理]隨機網絡和無標度網絡

傳統的隨機網絡（如ER模型），盡管連接是隨機設置的，但大部分節點的連接數目會大致相同，即節點的分布方式遵循鐘形的泊松分布，有一個特征性的“平均數”。連接數目比平均數高許多或低許多的節點都極少，隨著連接數的增大，其概率呈指數式迅速遞減。故隨機網絡亦稱指數網絡。

節點連接數的泊松分布：

一個隨機網絡：

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現實世界的網絡大部分都不是隨機網絡，少數的節點往往擁有大量的連接，而大部分節點卻很少，一般而言他們符合zipf定律，(也就是80/20馬太定律)。人們給具有這種性質的網絡起了一個特別的名字——無標度網絡。這里的無標度是指網絡缺乏一個特征度值（或平均度值），即節點度值的波動范圍相當大。

節點連接數的zipf分布：

符合zipf分布的無標度網絡：

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現實中的交通網，電話網和Internet都是無標度網絡，在這種網絡中，存在擁有大量連接的集散節點，比如交通樞紐就是這樣的節點。下面是Internet的連接模型：

分布滿足冪律的無標度網絡還有一個奇特的性質——“小世界”特性[49]，雖然WWW中的頁面數已超過80億，但平均來說，在WWW上只需點擊19次超鏈接，就可從一個網頁到達任一其它頁面。“小世界”現象在社會學上也稱為“六度分離”。

Barabási與Albert針對復雜網絡中普遍存在的冪律分布現象，提出了網絡動態演化的BA模型[42, 59]，他們解釋，成長性和優先連接性是無標度網絡度分布呈現冪律的兩個最根本的原因。所謂成長性是指網絡節點數的增加，像Internet中自治系統或路由器的添加，以及WWW中網站或網頁的增加等，優先連接性是指新加入的節點總是優先選擇與度值較高的節點相連，比如，新網站總是優先選擇人們經常訪問的網站作為超鏈接。隨著時間的演進，網絡會逐漸呈現出一種“富者愈富，貧者愈貧”的現象。社會學家所說的“馬太效應”[72]，《新約》圣經所說的“凡有的，還要加給他，叫他有余”，同優先連接也有某種相通之處。

引用：

冪律分布研究簡史

無標度網絡及其系統科學意義

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下面是我的其他博客：
博客園，寫一些工作和學習的筆記：?http://www.cnblogs.com/peon/
博客堂，開發方面的一些文章：http://blog.joycode.com/peon/
流媒體博客，流媒體方面的一些文章：http://blog.lmtw.com/b/peon/

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