Complex Networks Third Lecture
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Complex Networks
Third Lecture
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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 heterogeneityStatistical analysis of centrality measures:
P(k)=Nk/N=probability that a randomly chosen node has degree kalso: P(b), P(w)….
Two broad classes•homogeneous networks: light tails•heterogeneous networks: skewed, heavy tails
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Topological heterogeneityStatistical analysis of centrality measures
Broad degree distributions
Power-law tailsP(k) ~ k-typically 2< <3
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Topological heterogeneityStatistical analysis of centrality measures:
Poisson vs.Power-law
log-scale
linear scale
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Exp. vs. Scale-FreePoisson distribution
Exponential Network
Power-law distribution
Scale-free Network
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ConsequencesPower-law tailsP(k) ~ k-
Average=< k> = k P(k)dkFluctuations< k2 > = k2 P(k) dk ~ kc
3-
kc=cut-off due to finite-sizeN 1 => diverging degree fluctuations for < 3
Level of heterogeneity:
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1) Random graph2) Small world3) Preferential attachment4) 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: <E > = pN(N-1)/2
Average degree: < k > = p(N-1)
p=c/N to havefinite average degree
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Erdös-Renyi model (1960)
<k> < 1: many small subgraphs
< k > > 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)= CkN-1 pk (1-p)N-k-1
Large N, fixed pN=< k > : Poisson distribution
Exponential decay at large k
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Erdös-Renyi model (1960)
Small clustering: < C > =p =< k > /N
Short distances l=log(N)/log(< k >)(number of neighbors at distance d: < k >d )
Poisson degree distribution
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Generalized random graphs
Desired degree distribution: P(k)
• Extract a sequence ki 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 approachMicroscopic processes of the
many component units
Macroscopic statistical and dynamical properties of the system
Cooperative phenomenaComplex 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 nodesExamples: 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 : At 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 ki of that node
A.-L.Barabási, R. Albert, Science 286, 509 (1999)
jj
ii k
kk
)(
Microscopic mechanism: An example
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BA network
Connectivity distribution
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Problem with directed graphs
Natural extension:
(kiin )
kiin
jk jin
What happens if kiin = 0?
(kiin ) 0!
Nodes with zero indegree will never receivelinks! 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)
(ki) ki k0
j (k j k0)
(1) GROWTH : At 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 ki of that node and a constant k0 (attractivity), with -m < k0 < ∞
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P(k) ~ k (3k0 /m )
Degree distribution:
Extension to directed graphs:
Problem of nodes with zero indegree solved!
(kiin )
kiin k0
j (k jin k0)
P(k in ) ~ (k in ) (2k0 /m )
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Microscopic mechanism:
P.L. Krapivsky, S. Redner, F. Leyraz, Phys. Rev. Lett. 85, 4629 (2000)
(ki) ki
jki
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 vertexb. Introduction of a new vertexc. The new vertex copies m linksof the selected oned. Each new link is kept with proba , rewiredat 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: kin/(mt)
effective preferential attachment, withouta priori knowledge of degrees!
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Copying model
Microscopic mechanism:
Degree distribution:
=> model for WWW and evolution of genetic networks
=> Heavy-tails