4. PREFERENTIAL ATTACHMENT

The rich gets richer

Empirical evidences

Many large networks are scale freeThe 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 handWhich 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 dynamicsBut 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 twotwo 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 choosingchoosing 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

( ) ii

jj

kk

k

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

2 ( 1)( )

( 1)( 2)

m mP k

k k k

Preferential attachment but no growth

t=0, N nodes, no links

Power-laws at early timesP(k) not stationary, all nodes get connectedki(t)=2t/N

( ) ii

jj

kk

k

Average shortest-path

just a fit

ln( )l A N B C

<k>=k SF model

No theoretical stimations up to nowThe growth introduces nontrivial correctionsWhereas random graphs with a power-law degree distribution are uncorrelated

Clustering coefficient

NO analytical prediction for the SF model

5 times larger

0.75SF

1RG

C N

C k N

SW: C is independent of N

Scaling relations

Spectrum

exponential decay around 0

power law decay for large ||

1/ 41 N

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 nodesAccelerated growthCrossover with two power-laws

Growth constraints

Power-laws followed by exponential cutoffsModel: when a node– reaches a certain age (aging)– has more than a critical number of links (capacity)

– Explains the behavior

Competition

Nodes compete for linksPower-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 wordsfN(i) number of different words that each occurred exactly i times in the textContinue adding wordsWith probability p we add a new wordWith probability 1-p the word is already writtenThe 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 addedWith 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) kkN

Does not imply preferential attachmentClasses versus actual nodesNo topology

Error and attack tolerance

High degree of tolerance against error Topological aspects of robustness, caused by edge and/or link removalTwo 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