lyon lecture iib
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Deep learning tutorialTRANSCRIPT
Statistical physics of complexnetworks
Marc BarthelemyCEA, FranceEHESS-CAMS, France
IXXI Lyon July 2008
Outline I. Introduction: Complex networks
1. Complex systems and networks
2. Graph theory and characterization of large networks: tools
3. Characterization of large networks: results
4. Models
II. Dynamical processes
1. Resilience and vulnerability
2. Epidemiology
III. Advanced topics
1. Global disease spread
2. Community detection
3. Evolution and formation of the urban street network in cities
I. 4 Models
Models: static vs. dynamic
• Static: N nodes from the beginning, connection rules- Erdos-Renyi- Generalized random graphs- Watts-Strogatz model- Fitness models (hidden variables)
• Dynamic: the network is growing
- Barabasi-Albert- Copy model- Weighted network model
Simplest model of random graphs:Erdös-Renyi (1959)
N nodes, connected with probability p
Paul Erdős and Alfréd Rényi(1959)"On Random Graphs I" Publ.Math. Debrecen 6, 290–297.
Simplest model of random graphs:Erdös-Renyi (1959)
Some properties:
- Average number of edges
- Average degree
Finite average degree
Erdös-Renyi model: degree distribution
Proba to have a node of degree k=• connected to k vertices, • not connected to the other N-k-1
Large N, fixed : Poisson distribution
Exponential decay at large k
• N points, links with proba p:
Erdös-Renyi model: clustering and averageshortest path
• Neglecting loops, N(l) nodes at distance l:
For
: many small subgraphs
: giant component + small subgraphs
Erdös-Renyi model: components
Erdös-Renyi model: summary
- Small clustering
- Small world
- Poisson degree distribution
Generalized random graphs
Desired degree distribution: P(k)
Extract a sequence ki of degrees taken fromP(k)
Assign them to the nodes i=1,…,N
Connect randomly the nodes together,according to their given degree
Generalized random graphs
Average clustering coefficient
Average shortest path
Small-world and randomness
Watts-Strogatz (1998)
Lattice Random graphReal-Worldnetworks*
Watts & Strogatz, Nature 393, 440 (1998)
* Power grid, actors, C. Elegans
Watts-Strogatz (1998)
N nodes forms a regular lattice.With probability p,each edge is rewired randomly =>Shortcuts
• Large clustering coeff.• Short typical path
MB and Amaral, PRL 1999
Barrat and Weight EPJB 2000
N = 1000
Watts-Strogatz (1998)
Fitness model (hidden variables)
Erdos-Renyi: p independent from the nodes
• For each node, a fitness
Soderberg 2002Caldarelli et al 2002
• Connect (i,j) with probability
• Erdos-Renyi: f=const
Fitness model (hidden variables)
• Degree
• Degree distribution
Fitness model (hidden variables)
• If power law -> scale free network
• If and
Generates a SF network !
Barabasi-Albert (1999) Everything’s fine ?
Small-world network
Large clustering
Poisson-like degree distribution
Except that for
Internet, Web
Biological networks
…
Power-law distribution:
Diverging fluctuations !
Internet growth
Moreover - dynamics !
Barabasi-Albert (1999)
(1) The number of nodes (N) is NOT fixed.Networks continuously expand by theaddition of new nodes Examples:
WWW : addition of new documentsCitation : publication of new papers
(2) The attachment is NOT uniform.A node is linked with higher probability to a node thatalready has a large number of links: ʻʼRich get richerʼʼ
Examples :WWW : new documents link to wellknown sites (google, CNN, etc)Citation : well cited papers aremore likely to be cited again
Barabasi-Albert (1999)
(1) GROWTH : At every time step we add a new node with medges (connected to the nodes already present in thesystem).(2) PREFERENTIAL ATTACHMENT :The probability Π that a new node will be connected to node idepends on the connectivity ki of that node
A.-L.Barabási, R. Albert, Science 286, 509 (1999)
jj
ii
k
kk
!=" )(
P(k) ~k-3
Barabási & Albert, Science 286, 509 (1999)
jj
ii
k
kk
!=" )(
Barabasi-Albert (1999)
Barabasi-Albert (1999)
Barabasi-Albert (1999)
Clustering coefficient
Average shortest path
Copy model
a. Selection of a vertex
b. Introduction of a new vertex
c. The new vertex copies m linksof the selected one
d. Each new link is kept with proba 1-α, rewiredat random with proba α
1−α
α
Growing network:
Copy model
Probability for a vertex to receive a new link at time t (N=t):
• Due to random rewiring: α/t
• Because it is neighbour of the selected vertex: kin/(mt)
effective preferential attachment, withouta priori knowledge of degrees!
Copy model
Degree distribution:
model for WWW and evolution of genetic networks
=> Heavy-tails
Preferential attachment: generalization
Rank known but not the absolute value
Who is richer ?
Fortunato et al, PRL (2006)
Preferential attachment: generalization
Rank known but not the absolute value
Scale free network even in the absence of the value of the nodesʼ attributes
Fortunato et al, PRL (2006)
Weighted networks
• Topology and weights uncorrelated
• (2) Model with correlations ?
Weighted growing network
• Growth: at each time step a new node is added with m links tobe connected with previous nodes
• Preferential attachment: the probability that a new link isconnected to a given node is proportional to the nodeʼs strength
Barrat, Barthelemy, Vespignani, PRL 2004
Redistribution of weights
New node: n, attached to iNew weight wni=w0=1Weights between i and its other neighbours:
The new traffic n-i increases the traffic i-j
Onlyparameter
n i
j
Evolution equations
Evolution equations
Correlations topology/weights:
Numerical results: P(w), P(s)
(N=105)
Another mechanism:Heuristically Optimized Trade-offs (HOT)
Papadimitriou et al. (2002)
New vertex i connects to vertex j by minimizing the function Y(i,j) = a d(i,j) + V(j)d= euclidean distanceV(j)= measure of centrality
Optimization of conflicting objectives
Analytical results
Correlations topology/weights: wij ~ min(ki,kj)a , a=2d/(2d+1)
•power law growth of s
•k proportional to s