Complex Networks Third Lecture
Jan 03, 2016
Complex Networks
Third Lecture
I. A few examples of Complex Networks
II. Basic concepts of graph theory and network theory
III. Models
IV. Communities
Program
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.
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
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
Topological heterogeneityStatistical analysis of centrality measures
Broad degree distributions
Power-law tailsP(k) ~ k-typically 2< <3
Topological heterogeneityStatistical analysis of centrality measures:
Poisson vs.Power-law
log-scale
linear scale
Exp. vs. Scale-FreePoisson distribution
Exponential Network
Power-law distribution
Scale-free Network
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:
1) Random graph2) Small world3) Preferential attachment4) Copying model
Models
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
Erdös-Renyi model (1960)
<k> < 1: many small subgraphs
< k > > 1: giant component + small subgraphs
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
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
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
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
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
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
(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
BA network
Connectivity distribution
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!
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 < ∞
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 )
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
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)
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!
Copying model
Microscopic mechanism:
Degree distribution:
=> model for WWW and evolution of genetic networks
=> Heavy-tails