随机网络

The Erdös–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.

无标度网络

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.

分层网络

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 Eötvös 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.

看不懂英文的看中文的，写的还可以

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