Introduction to complex networks Part II: Models Ginestra Bianconi Physics Department,Northeastern University, Boston,USA NetSci 2010 Boston, May 10 2010.

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



L links

Small clustering coefficient

Small average distance

€

C(N) ∝1

N

l ∝log(N)

log(c)

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

Regular lattices





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



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



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



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



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)

Citrate cycle:highly Citrate cycle:highly preservedpreserved

Homo SapiensEscherichia coli

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



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



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?

Introduction to complex networks Part II: Models Ginestra Bianconi Physics Department,Northeastern University, Boston,USA NetSci 2010 Boston, May 10 2010.

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