3. SMALL WORLDS
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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