Class 9: Barabasi-Albert Model Network Science: Evolving Network Models February 2015 Prof. Boleslaw...

Class 9: Barabasi-Albert Model

Network Science: Evolving Network Models February 2015

Prof. Boleslaw Szymanski

Prof. Albert-László BarabásiDr. Baruch Barzel, Dr. Mauro Martino

klog

Nloglrand

Empirical findings for real networks

N

kCrand P(k) ~ k-

Small World:

distances scale

logarithmically with the

network size

Clustered:

clustering coefficient does

not depend on network

size.

Scale-free:

The degrees follow a

power-laws distribution.


2/1NLl

The average path-length varies as

Constant degree

P(k)=δ(k-kd)

Constant clustering coefficient

C=Cd

Two-dimensional lattice:

D-dimensional lattice:

Average path-length:

Degree distribution: P(k)=δ(k-6)

Clustering coefficient:

BENCHMARK 1: Regular Lattices


Erdös-Rényi Model- Publ. Math. Debrecen 6, 290 (1959)

• fixed node number N• connecting pairs of nodes with

probability p


Path length: klog

Nloglrand

N

kpCrand

k1Nkk1Nrand )p1(pC)k(P

Degree distribution:

BENCHMARK 2: Random Network Model


Watts-Strogatz algorithm – Nature 2008

• For fixed node number N, first connect them

into even number, k, degree ring in which k/2

nearest neighbors on each side of each node

are connected to it• Then, with probability p re-wire ring edges of

each node to nodes not currently connected

to and different from it


Path length:klog

Nloglrand

Degree distribution: Exponential

BENCHMARK 3: Small World Model


Missing Hubs

Hubs represent the most striking difference between a random and a scale-free network. Their emergence in many real systems raises several fundamental questions:

• Why does the random network model of Erdős and Rényi fail to reproduce the hubs and the power laws observed in many real networks?

• Why do so different systems as the WWW or the cell converge to a similar scale-free architecture?

P(k) ~ k-

Regular network

Erdos-Renyi

Watts-Strogatz

Pathlenght Clustering Degree Distr.

klog

Nloglrand

klog

Nloglrand

N

kpCrand

P(k)=δ(k-kd)

Exponential

EMPIRICAL DATA FOR REAL NETWORKS


SCALE-FREE MODEL(BA model)


Real networks continuously expand by the addition of new nodes

Barabási & Albert, Science 286, 509 (1999)

BA MODEL: Growth

ER, WS models: the number of nodes, N, is fixed (static models)


networks expand through the addition of new nodes


BA MODEL: Growth

ER model: the number of nodes, N, is fixed (static models)

WWW


BA MODEL: Growth (www/Pubs)

Scientific Publications

http://website101.com/define-ecommerce-web-terms-definitions/ http://www.kk.org/thetechnium/archives/2008/10/the_expansion_o.php


(1) Networks continuously expand by the addition of new nodes

Add a new node with m links


BA MODEL: Growth



jj

ii k

kk

)(

PREFERENTIAL ATTACHMENT:

the probability that a node connects to a node with k links is proportional to k. New nodes prefer to link to highly

connected nodes (www, citations, IMDB).

BA MODEL: Preferential Attachment

Where will the new node link to?ER, WS models: choose randomly.


Barabási & Albert, Science 286, 509 (1999) Network Science: Evolving Network Models

Growth and Preferential Sttachment

The random network model differs from real networks in two important characteristics:

Growth: While the random network model assumes that the number of nodes is fixed (time invariant), real networks are the result of a growth process that continuously increases.

Preferential Attachment: While nodes in random networks randomly choose their interaction partner, in real networks new nodes prefer to link to the more connected nodes.


P(k) ~k-3

(1) Networks continuously expand by the addition of new nodes

WWW : addition of new documents

GROWTH:

add a new node with m links

PREFERENTIAL ATTACHMENT:

the probability that a node connects to a node with k links is proportional to k.

(2) New nodes prefer to link to highly connected nodes.

WWW : linking to well known sites

Origin of SF networks: Growth and preferential attachment

jj

ii k

kk

)(


A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)

All nodes follow the same growth law

Use: During a unit time (time step): Δk=m A=m

β: dynamical exponent


SF model: k(t)~t ½ (first mover advantage)

Fitness Model: Can Latecomers Make It?

time

Deg

ree

(k)


γ = 3

)(1)(1)())((

02

2

2

2

2

2

tmk

tm

k

tmtP

k

tmtPktkP ititi


Degree distribution

A node i can come with equal probability any time between ti=m0 and t, hence:


γ = 3


Degree distribution

(i) The degree exponent is independent of m.

(ii) As the power-law describes systems of rather different ages and sizes, it is expected that a correct model should provide a time-independent degree distribution. Indeed, asymptotically the degree distribution of the BA model is independent of time (and of the system size N) the network reaches a stationary scale-free state.

(iii) The coefficient of the power-law distribution is proportional to m2.


Stationarity: P(k) independent of N

m=1,3,5,7 N=100,000;150,000;200,000

Insert: degree dynamics

m-dependence

NUMERICAL SIMULATION OF THE BA MODEL


The mean field theory offers the correct scaling, BUT it provides the wrong coefficient of the degree distribution.

So assymptotically it is correct (k ∞), but not correct in details (particularly for small k).

To fix it, we need to calculate P(k) exactly, which we will do next using a rate equation based approach.



Number of nodes with degree k at time t.

Nr. of degree k-1 nodes that acquire a new link, becoming degree k Preferential

attachment

Since at each timestep we add one node, we have N=t (total number of nodes =number of timesteps)

2m: each node adds m links, but each link contributed to the degree of 2 nodes

Number of links added to degree k nodes after the arrival of a new node:

Total number of k-nodes

New node adds m new links to other nodes

Nr. of degree k nodes that acquire a new link, becoming degree k+1

# k-nodes at time t+1 # k-nodes at time t

Gain of k-nodes via

k-1 k

Loss of k-nodes via

k k+1

MFT - Degree Distribution: Rate Equation

# m-nodes at time t+1 # m-nodes at

time t

Add one m-degeree

node

Loss of an m-node via

m m+1

We do not have k=0,1,...,m-1 nodes in the network (each node arrives with degree m) We need a separate equation for degree m modes

# k-nodes at time t+1 # k-nodes at time t

Gain of k-nodes via

k-1 k

Loss of k-nodes via

k k+1



k>m

We assume that there is a stationary state in the N=t∞ limit, when P(k,∞)=P(k)

k>m



...m+3 k

Krapivsky, Redner, Leyvraz, PRL 2000Dorogovtsev, Mendes, Samukhin, PRL 2000 Bollobas et al, Random Struc. Alg. 2001

for large k



Its solution is:

Start from eq.

Dorogovtsev and Mendes, 2003

MFT - Degree Distribution: A Pretty Caveat


Do we need both growth and preferential

attachment?


growth preferential attachment

Π(ki) : uniform

MODEL A


tN

CttNN

Ntk

Nt

k

N

N

NkA

t

k

N

N

i

ii

i

2~

)2(

)1(2)(

1

21

1)(

)1(2

growth preferential attachment

P(k) : power law (initially)

Gaussian Fully Connected

MODEL B


Do we need both growth and preferential

attachment?

YEP.Network Science: Evolving Network Models February 2015

P(k) ~ k-

Regular network

Erdos-Renyi

Watts-Strogatz

klog

Nloglrand

klog

Nloglrand

N

kpCrand

P(k)=δ(k-kd)

Exponential

Barabasi-Albert

P(k) ~ k-




Distances in scale-free networks

Size of the biggest hub is of order O(N). Most nodes can be connected within two layers of it, thus the average path length will be independent of the system size.

The average path length increases slower than logarithmically. In a random network all nodes have comparable degree, thus most paths will have comparable length. In a scale-free network the vast majority of the path go through the few high degree hubs, reducing the distances between nodes.

Some key models produce γ=3, so the result is of particular importance for them. This was first derived by Bollobas and collaborators for the network diameter in the context of a dynamical model, but it holds for the average path length as well.

The second moment of the distribution is finite, thus in many ways the network behaves as a random network. Hence the average path length follows the result that we derived for the random network model earlier.

Cohen, Havlin Phys. Rev. Lett. 90, 58701(2003); Cohen, Havlin and ben-Avraham, in Handbook of Graphs and Networks, Eds. Bornholdt and Shuster (Willy-VCH, NY, 2002) Chap. 4; Confirmed also by: Dorogovtsev et al (2002), Chung and Lu (2002); (Bollobas, Riordan, 2002; Bollobas, 1985; Newman, 2001

Ultra Small World

Small World

DISTANCES IN SCALE-FREE NETWORKS

N

Nl

lnln

ln

Bollobas, Riordan, 2002

PATH LENGTHS IN THE BA MODEL


P(k) ~ k-

klog

Nloglrand

klog

Nloglrand

N

kpCrand

P(k)=δ(k-kd)

Exponential

P(k) ~ k-

N

Nl

lnln

ln



Regular network

Erdos-Renyi

Watts-Strogatz

Barabasi-Albert Network Science: Evolving Network Models February 2015

The numerical results indicate a slightly slower decay.

What is the functional form of C(N)?

CLUSTERING COEFFICIENT OF THE BA MODEL

Reminder: for a random graph we have:

Konstantin Klemm, Victor M. Eguiluz,Growing scale-free networks with small-world behavior,Phys. Rev. E 65, 057102 (2002), cond-mat/0107607


1

2

Denote the probability to have a link between node i and j with P(i,j)The probability that three nodes i,j,l form a triangle is P(i,j)P(i,l)P(j,l)

The expected number of triangles in which a node l with degree kl participates is thus:

We need to calculate P(i,j).



Calculate P(i,j).

Node j arrives at time tj=j and the probability that it will link to node i with degree ki already in the network is determined by preferential attachment:

Where we used that the arrival time of node j is tj=j and the arrival time of node is ti=i

Let us approximate:Which is the degree of node l at current time, at time t=N

There is a factor of two difference... Where does it come from?




Konstantin Klemm, Victor M. Eguiluz,Phys. Rev. E 65, 057102 (2002)


P(k) ~ k-



klog

Nloglrand

klog

Nloglrand

N

kpCrand

P(k)=δ(k-kd)

Exponential

P(k) ~ k-

N

Nl

lnln

ln

Regular network

Erdos-Renyi

Watts-Strogatz


The origins of preferential attachment.


t

kk

t

k ii

i

~)(

Plot the change in the degree k during a fixed time t for nodes with degree k, and you get (k)

(Jeong, Neda, A.-L. B, Europhys Letter 2003; cond-mat/0104131)

No pref. attach: κ~k

Linear pref. attach: κ~k2

kK

)K()k(

To reduce noise, plot the integral of Π(k) over k:

CAN WE MEASURE PREFERENTIAL ATTACHMENT?


neurosci collab

actor collab.

citation network

1 ,)( kAk

kK

)K()k(

Plots shows the integral of Π(k) over k:Internet

CAN WE MEASURE PREFERENTIAL ATTACHMENT?

No pref. attach: κ~k

Linear pref. attach: κ~k2


1. Copying mechanismdirected networkselect a node and an edge of this nodeattach to the endpoint of this edge

2. Walking on a networkdirected networkthe new node connects to a node, then to everyfirst, second, … neighbor of this node

3. Attaching to edgesselect an edgeattach to both endpoints of this edge

4. Node duplicationduplicate a node with all its edgesrandomly prune edges of new node

MECHANISMS RESPONSIBLE FOR PREFERENTIAL ATTACHMENT


Copying Mechanism


Proteins with more interactions are more likely to obtain new links:Π(k)~k (preferential attachment)

Wagner 2001; Vazquez et al. 2003; Sole et al. 2001; Rzhetsky & Gomez 2001; Qian et al. 2001; Bhan et al. 2002.

ORIGIN OF THE SCALE-FREE TOPOLOGY IN THE CELL:Gene Duplication


k vs. k : increase in the No. of links in a unit time

No PA: k is independent of k

PA: k ~k

t

kk

t

k ii

i

~)(

Eisenberg E, Levanon EY, Phys. Rev. Lett. 2003

Jeong, Neda, A.-L.B, Europhys. Lett. 2003

PREFERENTIAL ATTACHMENT IN PROTEIN INTERACTION NETWORKS


• Nr. of nodes:

• Nr. of links:

• Average degree:

• Degree dynamics

• Degree distribution:

• Average Path

Length:

• Clustering

Coefficient:The network grows, but the degree distribution is stationary.

β: dynamical exponent

γ: degree exponent

N

Nl

lnln

ln

SUMMARY: PROPERTIES OF THE BA MODEL


γ=1 γ=2 γ=3

<k2> diverges <k2> finite

γwin γw

out

γintern

γactor

γcollab

γmetab

γcita

γsynonyms

γsex

BA model

Can we change the degree exponent?

DEGREE EXPONENTS


Evolving network models


The BA model is only a minimal model.

Makes the simplest assumptions:

• linear growth

• linear preferential attachment

Does not capture variations in the shape of the degree distribution

variations in the degree exponentthe size-independent clustering coefficient

Hypothesis: The BA model can be adapted to describe most features of real networks.

We need to incorporate mechanisms that are known to take place in real networks: addition of links without new nodes, link rewiring, link removal; node removal, constraints or optimization

m2k

ii kk )(

EVOLVING NETWORK MODELS


(the simplest way to change the degree exponent)

2in k~)k(P

Undirected BA network:

Directed BA network:

β=1: dynamical exponent γin=2: degree exponent; P(kout)=δ(kout-m)

Undirected BA: β=1/2; γ=3

BA ALGORITHM WITH DIRECTED EDGES


Extended Model

• prob. p : internal links• prob. q : link deletion• prob. 1-p-q : add node

EXTENDED MODEL: Other ways to change the exponent

P(k) ~ (k+(p,q,m))-(p,q,m)

[1,)


P(k) ~ (k+(p,q,m))-(p,q,m) [1,) Extended Model

p=0.937

m=1

= 31.68

= 3.07

Actor network

• prob. p : internal links• prob. q : link deletion• prob. 1-p-q : add node

Predicts a small-k cutoffa correct model should predict all aspects of the

degree distribution, not only the degree exponent.Degree exponent is a continuous function of p,q, m

EXTENDED MODEL: Small-k cutoff


• Non-linear preferential attachment:

P(k) does not follow a power law for 1

<1 : stretch-exponential

>1 : no-scaling (>2 : “gelation”)

iik

kk

)(

P. Krapivsky, S. Redner, F. Leyvraz, Phys. Rev. Lett. 85, 4629 (2000)

)kk(exp)k(P 0

NONLINEAR PREFERENTIAL ATTACHMENT


Initial attractiveness shifts the degree exponent:

A - initial attractiveness

m

A2in

1 ,)( kAk

Dorogovtsev, Mendes, Samukhin, Phys. Rev. Lett. 85, 4633 (2000)

BA model: k=0 nodes cannot aquire links, as Π(k=0)=0(the probability that a new node will attach to it is zero)

Note: the parameter A can be measured from real data, being the rate at which k=0 nodes acquire links, i.e. Π(k=0)=A

INITIAL ATTRACTIVENESS


)()( iii ttkk

• Finite lifetime to acquire new edges

• Gradual aging:

withincreases

S. N. Dorogovtsev and J. F. F. Mendes, Phys. Rev. E 62, 1842 (2000)

L. A. N. Amaral et al., PNAS 97, 11149 (2000)

GROWTH CONSTRAINTS AND AGING CAUSE CUTOFFS


P(k) ~ k-


klog

Nloglrand

klog

Nloglrand

N

kpCrand

P(k)=δ(k-kd)

Exponential

P(k) ~ k-

N

Nl

lnln

ln

THE LAST PROBLEM: HIGH, SYSTEM-SIZE INDEPENDENT C(N)

Regular network

Erdos-Renyi

Watts-Strogatz


• Each node of the network can be either active or inactive.• There are m active nodes in the network in any moment.

1. Start with m active, completely connected nodes.

2. Each timestep add a new node (active) that connects to m active nodes.

3. Deactivate one active node with probability:

K. Klemm and V. Eguiluz, Phys. Rev. E 65, 036123 (2002)

1)()( jid kakP

2am

10am

makkP /2)(

kak )(

C C* when N∞

A MODEL WITH HIGH CLUSTERING COEFFICIENT


Fitness Model


SF model: k(t)~t ½ (first mover advantage)

Fitness model: fitness (h ) k(h,t)~t ( )b h

( )b h = /h C

Fitness Model: Can Latecomers Make It?

time

Deg

ree

(k)

Bianconi & Barabási, Physical Review Letters 2001; Europhys. Lett. 2001.

G. Bianconi and A.-L. Barabási, Physical Review Letters 2001; cond-mat/0011029

jjj

iii k

k

Network

)(ink

)(

Bose gas

e)(n

)(g

Fitness η Energy level ε

New node with fitness η New energy level ε

Link pointing to node η Particle at level ε

Network quantum gas

MAPPING TO A QUANTUM GAS


)(

),,(if

iii t

tmttk

.11

1)(),(

)(

epdI

1

1)(

)( e

n

f(e)=e- ( - ) b e m .

The dynamic exponent f(e) depends on m, determined by the self-consistent equation:

1)()( ngd

BOSE-EINSTEIN CONDENSATION


time

Deg

ree

(k)

Bianconi & Barabási, Physical Review Letters 2001; Europhys. Lett. 2001.

Bose-Einstein Condensation


Bianconi & Barabási, Physical Review Letters 2001; Europhys. Lett. 2001. Network Science: Evolving Network Models February 14, 2011

Bose-Einstein Condensation

Bose-Einstein condensation

Fit-gets-rich

FITNESS MODEL: Can Latecomers Make It?

1. There is no universal exponent characterizing all networks.

2. Growth and preferential attachment are responsible for the emergence of the scale-free property.

3. The origins of the preferential attachment is system-dependent.4. Modeling real networks:

• identify the microscopic processes that take place in the system

• measure their frequency from real data• develop dynamical models that capture these processes.

5. If the model is correct, it should correctly predict not only the degree exponent, but both small and large k-cutoffs.

LESSONS LEARNED: evolving network models


Philosophical change in network modeling:

ER, WS models are static models – the role of the network modeler it to cleverly place the links between a fixed number of nodes to that the network topology mimic the networks seen in real systems.

BA and evolving network models are dynamical models: they aim to reproduce how the network was built and evolved.

Thus their goal is to capture the network dynamics, not the structure. as a byproduct, you get the topology correctly

LESSONS LEARNED: evolving network models


The end


Date post:	13-Jan-2016
Category:	Documents
Upload:	colin-horn
View:	217 times
Download:	0 times

Class 9: Barabasi-Albert Model Network Science: Evolving Network Models February 2015 Prof. Boleslaw...

Documents