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4. PREFERENTIAL ATTACHMENT The rich gets richer

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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

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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

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

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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

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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

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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

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

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=3. different m’s. P(k) changes. not

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Degree distribution Handwritten notes

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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

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Average shortest-path just a fit =k SF model

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

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

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Scaling relations

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

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

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

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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

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

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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,...)

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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]

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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

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Does not imply preferential attachment Classes versus actual nodes No topology

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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!)

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Removal of nodes Squares: random Circles: preferential

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