4. PREFERENTIAL ATTACHMENT

1. 4. PREFERENTIAL ATTACHMENT The rich gets richer

2. 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

3. 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

4. Topology is a result of the dynamics • But only a random growth? • In this case the distribution is exponential!

5. 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

6. 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

7. 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 Linear attachment (more general models) Sum over all existing nodes

8. Numerical simulations • Power-law P(k)k- SF=3 • The exponent does not depend on m (the only parameter of the model)

9. =3. different m’s. P(k) changes.  not

10. Degree distribution • Handwritten notes

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

13. Average shortest-path <k>=k SF model just a fit

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

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

16. Scaling relations

17. Spectrum exponential decay around 0 power law decay for large ||

18. Nonlinear preferantial attachment • Sublinear: stretch exponential P(k) • Superlinear: winner-takes-all

19. Nonlinear growth rates • Empirical observation: the number of links increases faster than the number of nodes • Accelerated growth • Crossover with two power-laws

20. 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

21. Competition • Nodes compete for links • Power-law with a logarithmic correction

22. 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, ...)

23. 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]

24. 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

25. Does not imply preferential attachment • Classes versus actual nodes • No topology

26. 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!)

27. Removal of nodes Squares: random Circles: preferential