Transcript 3. SMALL WORLDS

4. PREFERENTIAL ATTACHMENT
The rich gets richer
Empirical evidences
Many large networks are scale free
The degree distribution has a power-law
behavior for large k (far from a Poisson
distribution)
Random graph theory and the Watts-Strogatz
model cannor reproduce this feature
We can construct power-law networks by hand
Which is the mechanism that makes scale-free
networks to emerge as they grow?
Emphasis: network dynamics rather to construct
a graph with given topological features
Topology is a result of the dynamics
But only a random growth?
In this case the distribution is exponential!
Barabasi-Albert model (1999)
Two generic mechanisms common in many real
networks
– Growth (www, research literature, ...)
– Preferential attachment (idem): attractiveness of
popularity
The two are necessary
Growth
t=0, m0 nodes
Each time step we add a new node with m (m0)
edges that link the new node to m different
nodes already present in the system
Preferential attachment
When choosing the nodes to which the new
connects, the probability  that a new node will
be connected to node i depends on the degree ki
of node i
ki
 (ki ) 
kj
j
Linear attachment (more general models)
Sum over all existing nodes
Numerical simulations
Power-law P(k)k-
SF=3
The exponent does not depend on m (the only
parameter of the model)
=3. different m’s. P(k) changes.  not
Degree distribution
Handwritten notes
2m(m  1)
P(k ) 
k (k  1)(k  2)
Preferential attachment but no
growth
t=0, N nodes, no links
ki
 (ki ) 
kj
j
Power-laws at early times
P(k) not stationary, all nodes get connected
ki(t)=2t/N
Average shortest-path
<k>=k SF model
just a fit
l  A ln( N  B)  C
No theoretical stimations up to now
The growth introduces nontrivial corrections
Whereas random graphs with a power-law
degree distribution are uncorrelated
Clustering coefficient
5 times larger
CSF
N 0.75
CRG  k  N
1
SW: C is independent
of N
NO analytical prediction for the SF model
Scaling relations
Spectrum
exponential decay around 0
power law decay for large ||
1
N
1/ 4
Nonlinear preferantial attachment
Sublinear: stretch exponential P(k)
Superlinear: winner-takes-all
Nonlinear growth rates
Empirical observation: the number of links
increases faster than the number of nodes
Accelerated growth
Crossover with two power-laws
Growth constraints
Power-laws followed by exponential cutoffs
Model: when a node
– reaches a certain age (aging)
– has more than a critical number of links (capacity)
– Explains the behavior
Competition
Nodes compete for links
Power-law with a logarithmic correction
The Simon model
H.A. Simon (1955) : a class of models to account
empirical distributions following a power-law
(words, publications, city populations, incomes,
firm sizes, ...)
Algorithm
Book that is being written up to N words
fN(i) number of different words that each occurred
exactly i times in the text
Continue adding words
With probability p we add a new word
With probability 1-p the word is already written
The probability that the (n+1)th word has already
appeared i times is proportional to i fN(i) [the total
number of words that have occurred i times]
Mapping into a network model
With p a new node is added
With 1-p a directed link is added. The starting
point is randomly selected. The endpoint is
selected such that the probability that a node
belonging to the Nk nodes with k incoming links
will be chosen is
(class)  kNk
Does not imply preferential attachment
Classes versus actual nodes
No topology
Error and attack tolerance
High degree of tolerance against error
Topological aspects of robustness, caused by
edge and/or link removal
Two types of node removal:
– Randomly selected nodes (errors!)
– Most highly connected nodes are removed at each
step (this is an attack!)
Removal of nodes
Squares: random
Circles: preferential