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Properties common to many large-scale networks, independently of their origin and function:
- The degree and betweenness distribution are decreasing functions, usually power-laws.
- The distances scale logarithmically with the network size
- The clustering coefficient does not seem to depend on the network size, and is larger than the clustering coefficient of comparable random graphs
There are two model families proposed to explain these properties: Small world network models and scale-free network models.
Network models
log k
logN l ≈
scale - free
small world
Benchmark 1: regular lattices
One-dimensional lattice:
1 / 2
l ≈ L ≈ N
= =const.forinside nodes 15
C
k = 6 =const.forinside nodes
The average path-length varies as Constant degree (coordination number), constant clustering coefficient.
l ≈ N , k = const , C = const
l ≈ N^1 /^ D
Two-dimensional lattice:
D-dimensional lattice:
Benchmark 2: random graph theory
Erdös-Rényi algorithm - Publ. Math. Debrecen 6, 290 (1959)
- fixed node number N
- connecting pairs of nodes with probability p
Expected clustering coefficient:
Expected path length: (^) logk
logN l (^) rand ≈
N
k C (^) rand = p =
k k N 1 k Prand (k) CN 1 p( 1 p) −− Expected degree distribution: ≅ − −
Path length and order in real networks
log k
logN l (^) rand = N
k C (^) rand =
100 102 104 106 108 1010 N
0
5
10
15
l^ log
food webs neural networkpower grid collaboration networks WWW metabolic networksInternet
10 0 102 104 106 108 N
10 -
10 -
10 -
10 -
100
C/
(^) food webs neural network metabolic networkspower grid collaboration networksWWW
Real networks have short distances like random graphs yet show signs of local order.
Small-world networks
Watts-Strogatz model - D. Watts, S. Strogatz, Nature 393, 440 (1998)
- lattice with K neighbors
- rewire edges with probability p
2 K
N
l =
Real networks resemble both regular lattices and random graphs – perhaps they are in between.
4 (K 1 )
3 (K 2 )
C
logK
logN l ≈ N
K
C ≈
Is there a regime with small l and large C?
Transition from a lattice to a small world
lattice small world random
There is a broad interval of p over which (^) C ( p ) ≅ C ( 0 ) but l ( p ) ≅ l ( 1 )
The onset of the small-world behavior
depends on the system size
f(pKN ) K
N
l(N,p)
1 /d ≈
C( p) = C( 0 )( 1 − p)^3
d is the dimension of the lattice
These results cannot be directly compared to most real networks because the rewiring probability p is not known.
f (u) =
const if u << 1 ln u/u if u >> 1
lattice - like random graph - like
The transition point depends on the rewiring probability, the size of the network and the average degree.
Degree distribution of a small-world
network
P(k) depends on the rewiring parameter p, but is always centered around .
k = K
Rewiring does not change the average degree, but modifies the degree distribution.
Degree distribution similar to that of a random graph, with exponentially small probability for very highly connected nodes_._
General properties of the network
- nr. of nodes:
- nr. of edges:
- average degree:
- degree distribution:
2 m
N
2 E
k = →
10 0 10 1 10 2 10 3 k
10 -
10 -
10 -
10 -
10 0
P(k)
N = m 0 + t
m t
m(m 1 )
E 0 0 +
3
P( k) t Ak
−
Although the network grows, the degree distribution becomes stationary.
The degree of “old” nodes increases by acquiring new edges. The probability of an old node with degree k (^) i receiving a new edge is
Degree increase:
Choices: follow the increase in the number of nodes with degree k (^) i (rate equation approach) follow the increase in time in k (^) i (continuum theory)
Analytical determination of P(k)
2 t
k
k
k
m (k) m i
j j
i
i = ≈
∑
otherwise
withprob. 0
2 t
k 1 k
i ∆ i
Change in average number of nodes with degree k
Plug in:
Rate equation approach
k, m k
k
k k 1 k kN(t)
(k 1 )N (t) kN(t) m dt
dN
∑
−
P( k) N(t)N limNk(t) t k (^) t →∞
k-1 k
first node
number of edges normalization of new node
k, m k
kP(k)t
(k 1 )P(k 1 )t kP(k)t P(k) m + δ
k k+
The rate equation leads to a recursive relationship between P(k) and P(k-1)
Leads to
Degree distribution
3 ( 1 )( 2 )
= k kk k
mm Pk
P. Krapivsky, S. Redner, F. Leyvraz, Phys. Rev. Lett. 85, 4629 (2000)
⎪
⎪ ⎩
⎪⎪ ⎨
⎧
=
− >
−
= for k m m
Pk for k m k
k
Pk
2
2
( 1 ) 2
1
()
Stationary power law with an exponent γ= 3
k, m 2
(k 1 )P(k 1 ) kP(k) P(k) + δ
Ex. 1 Start from a seed of two nodes connected by an edge. At each step, add a new node, and connect it by a single edge to a preexisting node.
Let the probability of selection be directly proportional with the degree of the “old” node. (Is there an easy way to do this?)
Continue growing the graph until you reach 15 nodes. Describe the graph (average degree, degree distribution, clustering coefficient, connectivity, maximum distance).
Ex. 2 How will the properties of the graph change if at each step a new node and two new edges are added?
m
k
exp(
m
e
P(k)
m t 1
m
A (k)
dt
dk
0
i
i
Model A
growth preferential attachment
Π( k i ) : uniform
m = 7
m = 1
m = 3 m = 5
Characteristic degree scale: m
N
2 mt k
N
2 t
k N 1
N
N
A (k) dt
dk (^) i i
i
Model B
growth preferential attachment
P( k ) : power law (initially) ⇒ Gaussian
t = N
t = 5 N
t = 40 N
Fixed N, edges connect a randomly selected node with a preferentially selected node
BA algorithm with directed edges
New edges are directed from the new to the old nodes
k m
in
0
out
ki = m fori > m
kiin varies
t
k k
k m (k ) m dt
dk (^) iin
j
in j
in in i i
in i (^) = = = ∑
Pin (k)~k
−
The degree exponent of the directed scale-free network is 2!
- Copying mechanism directed network select a node and an edge of this node attach to the endpoint of this edge
- Walking on a network directed network the new node connects to a node, then to every first, second, … neighbor of this node
- Attaching to edges select an edge attach to both endpoints of this edge
- Node duplication duplicate a node with all its edges randomly prune edges of new node
Mechanisms for preferential attachment Growth constraints and aging cause
cutoffs
∏ ( )∝ (− )−^ ν ki kit ti
- Finite lifetime to acquire new edges
γ increaseswith ν
S. N. Dorogovtsev and J. F. F. Mendes, Phys. Rev. E 62, 1842 (2000)
L. A. N. Amaral et al., PNAS 97, 11149 (2000)
Additional processes also change the
degree exponent
S. N. Dorogovtsev, J. F. F. Mendes, Europhys. Lett. 52, 33 (2000)
− γ
- mp new edges P ( k )≈ ( k + k 0 ) k^0 ,^ γ= f ( p , q , m )
- mq edges rewired
R. Albert, A.-L. Barabási, Phys. Rev. Lett 85, 5234 (2000)
1 2 c
in 2
A model with high clustering coefficient
- Each node of the network can be either active or inactive.
- There are only m active nodes in the network at any instance.
- Start with m active, completely connected nodes
- At each timestep add a new node (active) that connects to m active nodes.
- Deactivate one active node
K. Klemm and V. Eguiluz, Phys. Rev. E 65, 036123 (2002)
( ) ∝ ( + ) −^1 Pd ki a kj
m = a = 2
m = a = 10
am P k k 2 / ( ) −− ≈
Π ( k ) ≈ a + k
- Start with a completely connected graph with five nodes (one “central” , four peripheral
- Make four copies of the graph, keep the original in the center. Connect the four peripheral nodes of each copy to the central node of the original.
- Make four copies of the graph, again connect peripheral nodes to the central node.
A deterministic scale-free model
5-clique
A deterministic scale-free model
E. Ravasz, A.-L. Barabasi, Phys Rev E 67, 026112 (2003)
5-clique
connect peripheries to central node
Ex. 1 How does the number of nodes increase as a function of time steps?
Ex. 2 How does the degree of the central node increase in time?
Ex. 3 How does the number of edges increase as a function of time steps?
Ex. 4 Can you identify the highest degree nodes?
Properties of the model
100 10 1 10 2 103 104 k
10 -
10 -
10 -
10 -
10 -
10 -
10 -
10 -
100
P(k)
100 10 1 10 2 103 104 k
10 -
10 -
10 -
10 -
10 -
10 -
10 -
10 -
100
P(k)
(a)
102 103 104 105 N
10 -
10 -
10 -
10 -
100
C(N)
(c)
Degree distribution
Clustering coefficient independent of network size
1 ln 4 /ln 3 P (k) k −− ∝
C ≈ 0****. 6
