In the BA model of a scale –free network, at each time point a node with M links is added
to the network, which connects to an already existing node I with probability
 I  k I /  I k J , where kI is the degree of node I 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 r=3. Such distributions
are seen as a straight line on a log-log plot. The network that is created by the BA model
does not have an inherent modularity, so C(k) is independent of k.
ER model (random graph model, 1959, 1960, 1961)
Assume that we start with N vertices and connect each pair of vertices with probability p
In the model, the probability that vertex has k edges follows a Poisson distribution
P(k )  e  k / k! , where
 N  1 k
 p (1  p) N 1 k
  N 
k


-
(Newman, 2004)
G n ,m is the ensemble of all graphs having n vertices and m edges, each possible graph
appearing with equal probability.
Many properties of the random graph are exactly solvable in the limit of large graph size
The limit of large n is taken holding the mean degree z = p(n-1) constant, in which case
the model has a Poisson degree distribution, since the presence or absence of edges is
independent, and hence the probability of a vertex having degree k is
 n
z k e z
pk    p k (1  p) nk 
k!
k 
with the last approximate equality becoming exact in the limit of large n and fixed k
WS model (small-world model)
N vertices form a one-dimensional lattice, each vertex being connected to its two nearest
and next-nearest neighbors.
With probability p, each edge is reconnected to a vertex chosen at random.
The long-range connections generated by this process decrease the distance between the
vertices, leading to a small-world phenomenon (six degrees of separation)
For p=0, the probability distribution of the connectivities is p(k )   (k  z ) , where z is
the coordination number in the lattice; whereas for finite p, P(k) still peaks around z, but
it gets broader.
- (Newman, 2004)
The degree distribution of the small-world model does not match most real-world
network well. Each vertex has degree at least 2k, for the degree of the underlying regular
lattice, plus a binomially distributes number of shortcuts. Hence probability p_j of having
degree j is
j 2 k
L j  2 k
 L  2kp 
 2kp 

p j  

1  L 
 j  2k   L 
for j  2k and p j  0 for j  2k . The distribution has a lower cutoff at k rather than 2k,
j k n
k 
n k  n ( pk )


(
1

p
)
p
e  pk

 n
(
j

k

n
)!
n 0
 
for j  k and p j  0 for j  k .
pj 
min( j  k ,k )
A common feature of the ER and WS models is that
the probability of finding a highly connected vertex (a large k) decreases exponentially
with k; thus, vertices with large connectivity are practically absent.
In contrast, the power-law tail characterizing P(k) for the networks studied indicates that
highly connected (large k) vertices have a large chance of occurring, dominating the
connectivity.
1) Growth
We start with a fixed number (N) of vertices that are then randomly connected (ER
model), or reconnected (WS model), without modifying N.
In contrast, most real world networks are open and they form by the continuous addition
of new vertices to the system, thus N increases throughout the life time of the network.
Ex) WWW – new Web pages, research literature – the publication of new papers,
2)
The random network models assume that the probability that two vertices are connected
is random and uniform.
In contrast, most real networks exhibit preferential connectivity.
Ex) new actor – to be cast in supporting role with more established and better-known
actors, newly created web page (new manuscript) – to include links to well-known
popular documents with already-high connectivity (much cited paper)
To incorporate the growing character of the network, starting with a small number ( m0 )
of vertices, at every time step we add a new vertex with m( m0 ) edges that link the new
vertex to m different vertices already present in the system.
To incorporate the preferential attachment, we assume that the probability  that a new
vertex will be connected to vertex i depends on the connectivity ki of that vertex,
 ( ki )  ki /  j k j .
After t step, the model leads to a random network with t  m0 vertices and mt edges.
This network involve into the scale invariant state with the probability that a vertex has k
edges, following a power law with an exponent  mod el  2.9  0.1
P(k) is independent of time and system size t  m0 . – despite its continuous growth, the
system organizes it self into a scale-free stationary state.
Test
Model A (fig2B) keeps the growth, but preferential attachment is eliminated by assuming
that a new vertex is connected with equal probability to any vertex in the system
(k )  const  1/( m0  t  1)
Such a model leads to P(k ) ~ exp(  k ) , indicating that the absence of preferential
attachment eliminates the scale-free feature of the distribution.
In model B, we start with N vertices and no edges. At each time step, we random select a
vertex and connect it with probability (ki )  ki /  j k j to vertex i in the system.
Although at early times the model exhibits power-law scaling, P(k) is not stationary :
because N is constant and the number of edges increases with time, after T  N 2 time
steps the system reaches a state in which all vertices are connected.
The failure of models A and B indicates that both ingredients are needed for the
development of the stationary power-law distribution (fig1)
An initial difference in the connectivity between two vertices will increase further as the
network grows. The rate at which a vertex acquires edge is ki / t  ki / 2t , which gives
ki (t )  m(t / ti )0.5 , where ti is the tie at which vertex i was added to the system, a scaling
property that could be directly tested once time-resolved data on network connectivity
becomes available.
Thus older (with smaller t_i) vertices increase their connectivity at the expense of the
younger (with larger t_i) ones. (rich-get-richer phenomenon)
The probability that a vertex i has a connectivity smaller than k,
P[ki (t )  k ] , can be written as P(ti  m 2t / k 2 ) .
Assuming that we add the vertices to the system at equal time intervals, we obtain
P(ti  m 2t / k 2 )  1  P(ti  m 2t / k 2 )  1  m 2t / k 2 (t  m0 ) .
The probability density P(k) can be obtained from ,
P(k )  P[ki (t )  k ] / k , which over long time periods leads to the stationary solution
2m 2
k3
giving   3 , independent of m.
Although it reproduces the observed scale-free distribution, the proposed model cannot
be expected to account for all aspects of the studied networks.
For this, we need to model these systems in more detail. Ex) in the model we assumed
linear preferential attachment; that is (k ) ~ k .
However, although in general (k ) could have an arbitrary nonlinear form (k ) ~ k  ,
simulations indicate that scaling is present only for  =1.
Furthermore, the exponents obtained for the different networks are scattered between 2.1
and 4. – modified  ( p)  3  p , which is supported by numerical simulations.
- how to explain about disappearing vertices in the real network?
P(k ) 
-
Newman 2004
The probability that a new edge attaches to any of the vertices with degree k is thus
(k  1) pk
(k  1) pk
(58)

k (k  1) pk m  1
The probability that a new edge attaches to a vertex of degree k
kpk
kp
 k (65)
k kpk 2m
Mean degree of the network 2m, since there are m edges for each vertex added, and each
edge, being now undirected, contributes two ends to the degrees of network vertices.
Now the mean number of vertices of degree k that gain an edge when a single new vertex
with m edges is added is m  kpk / 2m  kpk / 2 , independent m. The number np k of
vertices with degree k thus decreases by this same amount, since the vertices that get new
edges become vertices of degree k+1.
The number of vertices of degree k also increases because of influx from vertices
previously of degree k-1 that have also just acquired a new edge, except for vertices of
degree m, which have an influx of exactly 1.
p k ,n the value of pk when the graph has n vertices, the net change in np k per vertex
added is
1
1
(n  1) pk ,n1  npk ,n  (k  1) pk 1,n  kpk ,n for k >m
2
2
1
(n  1) pm ,n1  npm ,n  1  mpm ,n for k =m
2
No vertices for k <m
Stationary solutions pk ,n1  pk ,n  pk
1
1
(k  1) pk 1  kpk for k  m
2
pk  2
1
1  mpm for k  m
2
pk  2 /( m  2) pk pk  pk 1 (k  1) /( k  2)
(k  1)( k  2)..m
2m(m  1)
pm 
(k  2)( k  1)...( m  3)
(k  2)( k  1)k
In the limit of large k this gives a power law degree distribution p k ~ k 3 , with only the
single fixed exponent γ=3.
pk 