class 12: barabasi-albert model-part ii prof. albert-lászló barabási dr. baruch barzel, dr. mauro...

Class 12: Barabasi-Albert Model-Part II

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

Network Science: Evolving Network Models February 2015

Prof. Boleslaw Szymanski

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

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

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

# 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

MFT - Degree Distribution: Rate Equation

Network Science: Evolving Network Models February 2015

k>m

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

k>m

MFT - Degree Distribution: Rate Equation

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...m+3 k

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

for large k

MFT - Degree Distribution: Rate Equation

Network Science: Evolving Network Models February 2015

Its solution is:

Start from eq.

Dorogovtsev and Mendes, 2003

MFT - Degree Distribution: A Pretty Caveat

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γ = 3

Network Science: Evolving Network Models

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.

for large k

NUMERICAL SIMULATION OF THE BA MODEL

Absence of growth or preferential attachment

Section 6

growth preferential attachment

Π(ki) : uniform

MODEL A

Network Science: Evolving Network Models February 2015

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

Network Science: Evolving Network Models February 2015

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-

EMPIRICAL DATA FOR REAL NETWORKS

Pathlenght Clustering Degree Distr.

Network Science: Evolving Network Models February 2015

Diameter and clustering coefficient

Section 10

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

Section 10 Diameter

Bollobas, Riordan, 2002

P(k) ~ k-

klog

Nloglrand

klog

Nloglrand

N

kpCrand

P(k)=δ(k-kd)

Exponential

P(k) ~ k-

N

Nl

lnln

ln

EMPIRICAL DATA FOR REAL NETWORKS

Pathlenght Clustering Degree Distr.

Regular network

Erdos-Renyi

Watts-Strogatz

Barabasi-Albert Network Science: Evolving Network Models February 2015

Section 10 Clustering coefficient

What is the functional form of C(N)?

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).

CLUSTERING COEFFICIENT OF THE BA MODEL

Network Science: Evolving Network Models February 2015

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?

CLUSTERING COEFFICIENT OF THE BA MODEL

Network Science: Evolving Network Models February 2015

CLUSTERING COEFFICIENT OF THE BA MODEL

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

Network Science: Evolving Network Models February 2015

P(k) ~ k-

EMPIRICAL DATA FOR REAL NETWORKS

Pathlenght Clustering Degree Distr.

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

Nonlinear preferential attachment

Section 8

Section 8 Nonlinear preferential attachment

α=0: Reduces to Model A discussed in Section 5.4. The degree distribution follows the simple exponential function.

α=1: Barabási-Albert model, a scale-free network with degree exponent 3.

0<α<1: Sublinear preferential attachment. New nodes favor the more connected nodes over the less connected nodes. Yet, for the bias is not sufficient to generate a scale-free degree distribution. Instead, in this regime the degrees follow the stretched exponential distribution:

Section 8 Nonlinear preferential attachment

α=0: Reduces to Model A discussed in Section 5.4. The degree distribution follows the simple exponential function.

α=1: Barabási-Albert model, a scale-free network with degree exponent 3.

α>1: Superlinear preferential attachment. The tendency to link to highly connected nodes is enhanced, accelerating the “rich-gets-richer” process. The consequence of this is most obvious for , when the model predicts a winner-takes-all phenomenon: almost all nodes connect to a single or a few super-hubs.

Section 8 Nonlinear preferential attachment

The growth of the hubs. The nature of preferential attachment affects the degree of the largest node. While in a scale-free network the biggest hub grows as (green curve), for sublinear preferential attachment this dependence becomes logarithmic (red curve). For superlinear preferential attachment the biggest hub grows linearly with time, always grabbing a finite fraction of all links (blue curve)). The symbols are provided by a numerical simulation; the dotted lines represent the analytical predictions.

The origins of preferential attachment

Section 9

Section 9 Link selection model Link selection model -- perhaps the simplest example of a local or random mechanism capable of generating preferential attachment.

Growth: at each time step we add a new node to the network.

Link selection: we select a link at random and connect the new node to one of nodes at the two ends of the selected link.

To show that this simple mechanism generates linear preferential attachment, we write the probability that the node at the end of a randomly chosen link has degree k as

Section 9 Originators of preferential attachments

Measuring preferential attachment

Section 7

Section 7 Measuring preferential attachment

t

kk

t

k ii

i

~)(

Plot the change in the degree Δk during

a fixed time Δt for nodes with degree 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:

Network Science: Evolving Network Models

neurosci collab

actor collab.

citation network

1 ,)( kAk

kK

)K()k(

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

Network Science: Evolving Network Models

Section 7 Measuring 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

Network Science: Evolving Network Models February 2015

Section 9 Copying model

(a) Random Connection: with probability p the new node links to u.(b) Copying: with probability we randomly choose an outgoing link of node u and connect the new node to the selected link's target. Hence the new node “copies” one of the links of an earlier node

(a) the probability of selecting a node is 1/N. (b) is equivalent with selecting a node linked to a randomly selected link. The probability of selecting a degree-k node through the copying process of step (b) is k/2L for undirected networks. The likelihood that the new node will connect to a degree-k node follows preferential attachment

Social networks: Copy your friend’s friends.Citation Networks: Copy references from papers we read.Protein interaction networks: gene duplication,

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

Network Science: Evolving Network Models February 2015

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

Network Science: Evolving Network Models February 2015

• 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

Network Science: Evolving Network Models February 2015

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

Network Science: Evolving Network Models February 2015

Evolving network models

Network Science: Evolving Network Models February 2015

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

Network Science: Evolving Network Models February 2015

(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

Network Science: Evolving Network Models February 2015

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,)

Network Science: Evolving Network Models February 2015

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

Network Science: Evolving Network Models February 2015

• 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: MORE MODELS

Network Science: Evolving Network Models February 2015

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

Network Science: Evolving Network Models February 2015

)()( 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

Network Science: Evolving Network Models February 2015

P(k) ~ k-

Pathlenght Clustering Degree Distr.

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

Barabasi-Albert Network Science: Evolving Network Models February 2015

• 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

Network Science: Evolving Network Models February 2015

Fitness Model

Network Science: Evolving Network Models February 2015

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.

Bose-Einstein condensation

Fit-gets-rich

FITNESS MODEL: Can Latecomers Make It?

The network grows, but the degree distribution is stationary.

Section 11: Summary

The network grows, but the degree distribution is stationary.

Section 11: Summary

Section 11: Summary

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

Network Science: Evolving Network Models February 2015

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

Network Science: Evolving Network Models February 2015

The end

Network Science: Evolving Network Models February 2015