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Introduction to complex networksIntroduction to complex networksPart II: ModelsPart II: Models
Ginestra Bianconi
Physics Department,Northeastern University, Boston,USA
NetSci 2010 Boston, May 10 2010
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Random graphs
Binomial Poisson distribution distribution
G(N,p) ensembleG(N,p) ensemble
Graphs with N nodesEach pair of nodes linked
with probability p
G(N,L) ensembleG(N,L) ensemble
Graphs with exactly N nodes and
L links
€
P(k) =N −1
k
⎛
⎝ ⎜
⎞
⎠ ⎟pk (1 − p)N −1−k
P(k) =1
k!c ke−c
Random graphs
Poisson distribution
G(N,L) ensembleG(N,L) ensemble
Graphs with exactly N nodes and
L links
Small clustering coefficient
Small average distance
€
C(N) ∝1
N
l ∝log(N)
log(c)
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
Regular lattices
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d dimensions
Large average distance
Significant local interactions
€
L ≈ N1/ d
Universalities:Small worldUniversalities:Small world
Ci =# of links between 1,2,…ki neighbors
ki(ki-1)/2
Networks are clustered (large average Ci,i.e.C)
but have a small characteristic path length
(small L).
Network C Crand L N
WWW 0.1078 0.00023 3.1 153127
Internet 0.18-0.3 0.001 3.7-3.763015-6209
Actor 0.79 0.00027 3.65 225226
Coauthorship 0.43 0.00018 5.9 52909
Metabolic 0.32 0.026 2.9 282
Foodweb 0.22 0.06 2.43 134
C. elegance 0.28 0.05 2.65 282
Ki
i
Watts and Strogatz (1999)
Watts and Strogatz small world model
Watts & Strogatz (1998)Variations and characterizations
Amaral & Barthélemy (1999)
Newman & Watts, (1999)
Barrat & Weigt, (2000)
There is a wide range of values of p in which high clustering coefficientcoexist with small average distance
Small world and efficiency
Small worlds are both
locally and globally efficient
Boston T is only
globally efficient€
E =1
N(N −1)
1
diji≠ j
∑
Latora & Marchiori 2001, 2002
Degree distribution of the small world model
The degree distribution of the small-world model is homogeneous
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Barrat and Weigt 2000
Universalities:Scale-free degree distribution
)exp()(~)( 00
τ
γ
kkk
kkkP+
−+ −
€
P(k)∝ k −γ γ ∈ (2,3)
k
Actor networks WWW Internet
∞→2k
finitek
k
Faloutsos et al. 1999Barabasi-Albert 1999
Scale-free networksScale-free networks • Technological networks:
– Internet, World-Wide Web
• Biological networks :Biological networks : – Metabolic networks,
– protein-interaction networks,– transcription networks
• Transportation networks:Transportation networks:
– Airport networks
• Social networks:Social networks: – Collaboration networks
– citation networks
• Economical networks:Economical networks: – Networks of shareholders
in the financial market – World Trade Web
Why this universality?
• Growing networks:Growing networks:– Preferential attachment
Barabasi & Albert 1999,Dorogovtsev Mendes 2000,Bianconi & Barabasi 2001,
etc.
• Static networks:Static networks:– Hidden variables mechanism
Chung & Lu 2002, Caldarelli et al. 2002, Park & Newman 2003
Motivation for BA modelMotivation for BA model
1) The network growNetworks continuously expand by the addition of new nodesEx. WWW : addition of new documents Citation : publication of new papers2) The attachment is not uniform
(preferential attachment).
A node is linked with higher probability to a node that already has a large number of links.
Ex: WWW : new documents link to well known sites (CNN, YAHOO, NewYork Times, etc) Citation : well cited papers are more likely to be cited again
BA modelBA model(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
Barabási et al. Science (1999)
jj
ii k
kk
Σ=Π )(
P(k) ~k-3
Result of the BA scale-free Result of the BA scale-free modelmodel
The connectivity of each node increases in time as a power-law with exponent 1/2:
The probability that a node has k links follow a power-law with exponent γ:
€
ki(t) = mt
ti
€
P(k) = 2m2 1
k 3
Initial attractiveness
The initial attractiveness can change the value of the power-law exponent γ
€
Πi ∝ ki + A
€
γ∈ (2,∞)
β(γ −1) =1
€
ki ∝t
ti
⎛
⎝ ⎜
⎞
⎠ ⎟
β
P(k) ∝ k −γ
A preferential attachment with initial attractiveness A yields
Dorogovtsev et al. 2000
Non-linear preferential attachment
<1 Absence of power-law degree distribution
=1 Power-law degree distribution>1 Gelation phenomena
The oldest node acquire most of the links
First-mover-advantage
€
Πi ∝ kiα
Krapivski et al 2000
Gene duplication modelGene duplication model
Duplication of a gene Duplication of a gene
adds a node.adds a node.
New proteins will be New proteins will be
preferentially connected preferentially connected
to high connectivity.to high connectivity.
A. Vazquez et al. (2003).
Effective Effective preferential preferential attachmentattachment
Other variationsScale-free networks with high-clustering coefficient
Dorogovtsev et al. 2001
Eguiluz & Klemm 2002
Aging of the nodes
Dorogovstev & Medes 2000
Pseudofractal scale-free network
Dorogovtsev et al 2002
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Features of the nodesFeatures of the nodesIn complex networks
nodes are generally heterogeneous and they are characterized by specific features
Social networks: age, gender, type of jobs, drinking and smoking habits, Internet: position of routers in geographical space, … Ecological networks: Trophic levels, metabolic rate, philogenetic distance Protein interaction networks: localization of the protein inside the cell, protein
concentration
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Fitness the nodes Fitness the nodes
23
1
45
6
Not all the nodes are the same!
Let assign to each node an
energy
of a node
In the limit =0 all the nodes have same fitness
TheThe fitness model fitness model
Growth:
–At each time a new node and m links are added to the network.
–To each node i we assign a energy i from a p() distribution
Generalized preferential attachment:–Each node connects to the rest of the network by m links attached preferentially to well connected, low energy nodes.
2 3
1
45
6
€
Πi ∝ e−βε i ki
Results of the model
€
P(k) ≈ k −γ 2 < γ < 3
Power-law degree distribution
Fit-get-rich mechanismFit-get-rich mechanism
€
kη (t) = mt
ti
⎛
⎝ ⎜
⎞
⎠ ⎟
η i /C
Fit-get rich mechanismFit-get rich mechanismThe nodes with higher fitness
increases the connectivity faster
satisfies the condition
€
ki =t
ti
⎛
⎝ ⎜
⎞
⎠ ⎟
f (ε i )
.1
1)(1
)(∫ −=
−e
pd€
f (ε) = e−β (ε − μ )
Mapping to a Bose gasMapping to a Bose gasWe can map the fitness model to a Bose
gas with
– density of states p( );– specific volume v=1;– temperature T=1/.
In this mapping, – each node of energy corresponds to
an energy level of the Bose gas – while each link pointing to a node of
energy , corresponds to an occupation of that energy level.
Network
Energy diagramG. Bianconi, A.-L. Barabási 2001
23
1
45
6
Bose-Einstein Bose-Einstein condensation in trees condensation in trees scale-free networksscale-free networks
In the network there is a critical temperature Tc such that
•for T>Tc the network is in the
fit-get-rich phase
•for T<Tc the network is in the
winner-takes-all
or Bose-condensate phase
Correlations in the InternetCorrelations in the Internetand the fitness modeland the fitness model
knn (k) mean value of the connectivity of neighbors sites of a node with connectivity k
C(k) average clustering coefficient of nodes with connectivity k.
Vazquez et al. 2002
Growing weighted models
With new nodes arriving at each time
Yook & Barabasi 2001
Barrat et al. 2004With weight-degree
correlations
And possible condensation of the links
G. Bianconi 2005
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Growing Cayley-treeGrowing Cayley-tree
Each node is either at the interface ni=1 or in the bulk ni=0
At each time step a node at the interface is attached to m new nodes with energies from a p( ) distribution.
High energy nodes at the interface are more likely to grow.
The probability that a node i grows is given by
G. Bianconi 2002
4
1
23
5 6
7
8
9
ii ne i∝Π
Nodes at the interface
Mapping to a Fermi gasMapping to a Fermi gas
The growing Cayley tree network can be mapped into a Fermi gas – with density of states
p();– temperature T=1/;– specific volume v=1-1/m.
In the mapping the nodes corresponds to the energy levels
the nodes at the interface to the occupied energy levels
4
1
23
5 6
7
8
9
Network Energy diagram
Why this universality?
• Growing networks:Growing networks:– Preferential attachment
Barabasi & Albert 1999,Dorogovtsev Mendes 2000, Bianconi & Barabasi 2001,
etc.
• Static networks:Static networks:– Hidden variables mechanism
Chung & Lu 2002, Caldarelli et al. 2002, Park & Newman 2003
Molloy Reed configuration model
Networks with given degree distribution
Assign to each node a degree from the given degree distributionCheck that the sum of stubs is evenLink the stubs randomlyIf tadpoles or double links are
generated repeat the construction
∏ ∑−Σ=
i jiji akGP )(
1)(
1
δ
Molloy & Reed 1995
Caldarelli et al. hidden variable model
Every nodes is associated with an hidden variable xi
The each pair of nodes are linked with probability
€
pij = f (x i, x j )
k(x) = N dy ρ(y) f (x, y)∫
Caldarelli et al. 2002Soderberg 2002Boguna & Pastor-Satorras 2003
Park & Newman Park & Newman Hidden variables modelHidden variables model
J. Park and M. E. J. Newman (2004).
€
H = θ iki = (θ i +θ j )ai, ji, j
∑i
∑
pij =eθ i +θ j
1+ eθ i +θ j
∫+
θρθ−= θ+θ 1
11
'iie
)'('d)N(k
The system is defined through an Hamiltonian
pij is the probability of a link
The “hidden variables” θi are quenched and distributed through the nodes with probability ρθ
There is a one-to-one correspondence between θ and the average connectivity of a node
Random graphs
Binomial Poisson distribution distribution
G(N,p) ensembleG(N,p) ensemble
Graphs with N nodesEach pair of nodes linked
with probability p
G(N,L) ensembleG(N,L) ensemble
Graphs with exactly N nodes and
L links
€
P(k) =N −1
k
⎛
⎝ ⎜
⎞
⎠ ⎟pk (1 − p)N −1−k
P(k) =1
k!c ke−c
Statistical mechanics and
random graphs
Microcanonical Configurations G(N,L) GraphsEnsemble with fixed energy E Ensemble with fixed # of links L
Canonical Configurations G(N,p) GraphsEnsemble with fixed average Ensemble with fixed average energy <E> # of links <L>
Statistical mechanics Random graphs
Gibbs entropy and entropy of the G(N,L) random
graph
))(log( EkS Ω= )log(ZN
1=Σ
)( EΩ
Gibbs Entropy
Statistical mechanicsMicrocanonical ensemble
Random graphsG(N,L) ensemble
Total number of microscopic configurations with energy E
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
L
NNZ
21 /)(
Entropy per node of the G(N,L)ensemble
Total number of graphs in the G(N,L) ensembles
Shannon entropy and entropy of the G(N,p) random
graph
€
S = − p(E)lnp(E)E
∑
€
S = −1
Np(aij )ln
a ij{ }
∑ p(aij )
€
p(E) =1
Ze−bE
Shannon Entropy
Statistical mechanicsCanonical ensemble
Random graphsG(N,p) ensemble
Typical number of microscopic configurations with temperature €
Σ =−c
2lnc +
N
2ln N −
(N − c)
2ln(N − c)
Entropy per node of the G(N,p)ensemble
Total number of typical graphs the G(N,p) ensembles
Hypothesis:Hypothesis: Real networks are single instances
of an ensemble of possible networks which would equally well perform the function of
the existing network
The “complexity” of a real network is a decreasing function
of the entropy of this ensemble
Complexity of a real networkComplexity of a real network
G. Bianconi EPL (2008)
The complexity of networks is indicated by their organization at
different levels
• Average degree of a network• Degree sequence• Degree correlations• Loop structure• Clique structure• Community structure• Motifs
Relevance of a network characteristics
How many networks
have the same:
Added features
Entropy of network ensembles with given features
Degree sequence
Degree correlations
Communities
The relevance of an additional feature is quantified by the entropy drop
Networks with given degree Networks with given degree sequence sequence
∏ ∑−Σ=
i jiji akGP )(
1)(
1
δ
Microcanonical ensemble Canonical ensemble
Ensemble of network with exactly M links Ensemble of networks with average number of links M
∏<
−−=ji
aij
aij
ijij ppGP 1)1()(
Molloy-Reed Hidden variables
Shannon Entropy of canonical Shannon Entropy of canonical ensemblesensembles
€
S = −1
Npijlnpij + (1− pij )ln(1 − pij )
ij
∑ ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
We can obtain canonical ensembles by maximizing this entropy conditional to given constraints
Link probabilities
Constraints Link probability
Total number of links L=cN
Degree sequence {ki}
Degree sequence {ki} and number of links in within and
in between communities {qi}
In spatial networks, degree sequence {ki} and number of links at distance d
€
pij =θ iθ jW (qi,q j )
1+θ iθ jW (qi,q j )
€
pij =θ iθ j
1+θ iθ j
€
pij =θ iθ jW (dij )
1+θ iθ jW (dij )
€
pij =c
N
Anand & Bianconi PRE 2009
Partition function randomized Partition function randomized microcanonical network microcanonical network
ensemblesensembles
Statistical mechanics on the adjacency matrix of the network
hij auxiliary fields
€
Zκ =k
∏ δ(K constraint k K )a ij{ }
∑ e i , j
∑ hij a ij
otherwisea
jtolinkedisiifa
ij
ij
0
1
=
=
The entropy of the The entropy of the randomized ensemblesrandomized ensembles
0)log(1
==Σ hZN κκ
0
)log(
=∂
∂=
hijij h
Zp κ
Gibbs Entropy per nodeof a randomized network ensemble
Probability of a link.
The link probabilitiesin microcanonical and canonical ensembles
are the same-Example Microcanonical ensemble Canonical ensemble
Regular networks Poisson networks
but
€
pij =c
N
€
pij =c
N
€
Σ < S Anand & Bianconi 2009
Two examples of given Two examples of given degree sequencedegree sequence
k=2
k=1
k=2k=2
Zero entropy Non-zero entropy
k=3
The entropy The entropy of random scale-free of random scale-free
networksnetworksγ−∝ kkP )(
The entropy decreases as decrease toward quantifying a higher order in networks with fatter tails
€
γ
Bianconi 2008
Change and Necessity:Change and Necessity:
Randomness is not all
Selection
or
non-equilibrium processes
have to play a role
in the evolution of
highly organized networks
Quantum statistics in equilibrium network models
Simple networks
Fermi-like distribution
Weighted network
Bose-like distributions
€
pij =θ iθ jW (dij )
1+θ iθ jW (dij )=
1
1+ eβε ij
βε ij = −lnθ i − lnθ j − lnW (dij )
G. Bianconi PRE 2008D. Garlaschelli, Loffredo PRL 2009
€
wij =1
eβ (xi +x j ) −1
Other related works
Ensembles of networks with clustering, acyclic
Newman PRL 2009, Karrer Newman 2009Entropy origin of disassortativity in complex
networks
Johnson et al. PRL 2010Assessing the relevance of node features for
network structure
Bianconi et al. PNAS 2009Finding instability in the community structure of
complex networks
Gfeller et al. PRE 2005
The spatial structure of the The spatial structure of the airport networkairport network
)(1
)(
ijji
ijjiij dW
dWp
θθθθ
+=Link probability
€
W (d) ≈ d−α
€
≈3G. Bianconi et al. PNAS 2009
Models in hidden hyperbolic spaces
The linking probability is taken to be
dependent on the hyperbolic distance x
between the nodes
€
€
p(x) =1
1+ eβ (x −R )
x = r + r'+2
ζlnsin
Δθ
2
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Krioukov et al. PRE 2009
Dynamical networks
At any given time the network looks disconnected
Protein complexes during the cell cycle of yeast
Social networks(phone calls, small
gathering of people)
De Lichtenberg et al.2005
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Barrat et al.2008
ConclusionsConclusions
The modeling of complex networks is a continuous search to answer well studied questions as
Why we observe the universality network structure?
How can we model a network at a given level of coarse-graining?
And new challenging questions…
What is the relation between network models and quantum statistics?
Space: What is the geometry of given complex networks?
Time: How can we model the dynamical behavior of complex social and biological networks?