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 two The two are necessary

Growth t=0, m 0 nodes Each time step we add a new node with m (  m 0 ) edges that link the new node to m different nodes already present in the system

Preferential attachment choosing 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 k i of node i 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

Preferential attachment but no growth t=0, N nodes, no links Power-laws at early times P(k) not stationary, all nodes get connected k i (t)=2t/N

Average shortest-path just a fit =k SF model

No theoretical stimations up to now The growth introduces nontrivial corrections Whereas random graphs with a power-law degree distribution are uncorrelated

Clustering coefficient NO analytical prediction for the SF model 5 times larger SW: C is independent of N

Scaling relations

Spectrum exponential decay around 0 power law decay for large | |

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 f N (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 f N (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 N k nodes with k incoming links will be chosen is

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