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Complex Networks Third Lecture TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA

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I. A few examples of Complex Networks II. Basic concepts of graph theory and network theory III. Models IV. Communities Program

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Three main goals: 1)Identify the “universality classes” of graphs. 2)Identify the “microsopic rules” which generate a particular class of 3)Predict the behaviour of the system when one changes the boundary conditions.

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Universality Two main classes with different behaviours of the connectivity: exponential graphs and power law graphs random graphs are exponential Almost all the biological networks are instead of the power law type

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Topological heterogeneity Statistical analysis of centrality measures: P(k)=N k /N=probability that a randomly chosen node has degree k also: P(b), P(w)…. Two broad classes homogeneous networks: light tails heterogeneous networks: skewed, heavy tails

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Topological heterogeneity Statistical analysis of centrality measures Broad degree distributions Power-law tails P(k) ~ k - typically 2< <3

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Topological heterogeneity Statistical analysis of centrality measures: Poisson vs. Power-law log-scale linear scale

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Exp. vs. Scale-Free Poisson distribution Exponential Network Power-law distribution Scale-free Network

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Consequences Power-law tails P(k) ~ k - Average= = k P(k)dk Fluctuations = k 2 P(k) dk ~ k c 3- k c =cut-off due to finite-size N 1 => diverging degree fluctuations for < 3 Level of heterogeneity:

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1)Random graph 2)Small world 3)Preferential attachment 4)Copying model Models

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Usual random graphs: Erdös-Renyi model (1960) N points, links with probability p: static random graphs Average number of edges: = pN(N-1)/2 Average degree: = p(N-1) p= c /N to have finite average degree

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Erdös-Renyi model (1960) < 1: many small subgraphs > 1: giant component + small subgraphs

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Erdös-Renyi model (1960) Probability to have a node of degree k connected to k vertices, not connected to the other N-k-1 P(k)= C k N-1 p k (1-p) N-k-1 Large N, fixed pN= : Poisson distribution Exponential decay at large k

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Erdös-Renyi model (1960) Small clustering: =p = /N Short distances l=log(N)/log( ) (number of neighbors at distance d: d ) Poisson degree distribution

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Generalized random graphs Desired degree distribution: P(k) Extract a sequence k i of degrees taken from P(k) Assign them to the nodes i=1,…,N Connect randomly the nodes together, according to their given degree

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Small-world networks Watts & Strogatz, Nature 393, 440 (1998) N = 1000 Large clustering coeff. Short typical path N nodes forms a regular lattice. With probability p, each edge is rewired randomly =>Shortcuts

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Statistical physics approach Microscopic processes of the many component units Macroscopic statistical and dynamical properties of the system Cooperative phenomena Complex topology Natural outcome of the dynamical evolution Find microscopic mechanisms

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Microscopic mechanism: An example (1) The number of nodes (N) is NOT fixed. Networks continuously expand by the addition of new nodes Examples: WWW: addition of new documents Citation: publication of new papers (2) The attachment is NOT uniform. A node is linked with higher probability to a node that already has a large number of links. Examples : WWW : new documents link to well known sites (CNN, YAHOO, NewYork Times, etc) Citation : well cited papers are more likely to be cited again

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(1) GROWTH : A t every timestep we add a new node with m edges (connected to the nodes already present in the system). (2) PREFERENTIAL ATTACHMENT : The probability Π that a new node will be connected to node i depends on the connectivity k i of that node A.-L.Barabási, R. Albert, Science 286, 509 (1999) Microscopic mechanism: An example

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Microscopic mechanism: An example

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Microscopic mechanism: An example

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BA network Connectivity distribution

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Problem with directed graphs Natural extension: What happens if k i in = 0? Nodes with zero indegree will never receive links! Bad!

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Linear preferential attachment Microscopic mechanism: S. N. Dorogovtev, J. F. F. Mendes, A. N. Samukhin, Phys. Rev. Lett. 85, 4633 (2000) (1) GROWTH : A t every timestep we add a new node with m edges (connected to the nodes already present in the system). (2) PREFERENTIAL ATTACHMENT : The probability Π that a new node will be connected to node i depends on the connectivity k i of that node and a constant k 0 (attractivity), with -m < k 0 < ∞

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Degree distribution: Extension to directed graphs: Problem of nodes with zero indegree solved!

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Microscopic mechanism: P.L. Krapivsky, S. Redner, F. Leyraz, Phys. Rev. Lett. 85, 4629 (2000) Non-linear preferential attachment (1)α <1 : P(k) has exponential decay! (2)α >1: one or more nodes is attached to a macroscopic fraction of nodes (condensation); the degree distribution of the other nodes is exponential (3)α =1: P(k) ~ k -3

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Copying model Microscopic mechanism: a. Selection of a vertex b. Introduction of a new vertex c. The new vertex copies m links of the selected one d. Each new link is kept with proba , rewired at random with proba 1- Growing network: J. M. Kleinberg, S. R. Kumar, P. Raghavan, S. Rajagopalan, A. Tomkins, Proc. Int. Conf. Combinatorics & Computing, LNCS 1627, 1 (1999)

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Copying model Microscopic mechanism: Probability for a vertex to receive a new link at time t: Due to random rewiring: (1- )/t Because it is neighbour of the selected vertex: k in /(mt) effective preferential attachment, without a priori knowledge of degrees!

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Copying model Microscopic mechanism: Degree distribution: => model for WWW and evolution of genetic networks => Heavy-tails

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