class 9: barabasi-albert model-part i network science: evolving network models february 2015 prof....

Class 9: Barabasi-Albert Model-Part I

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) excluding corners and boundaries Clustering coefficient:

BENCHMARK 1: Regular Lattices


6 neighbors, each with 2 edges = 12/30

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

• fixed node number N• connecting pairs of nodes with

probability p

Clustering coefficient:

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

Clustering coefficient:

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

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


∑𝑗

𝑘 𝑗=2𝑚 (𝑡−1 )+𝑚0 (𝑚0−1)

2

In limit: So:

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

Fitness Model: Can Latecomers Make It?

time

Deg

ree

(k)


Section 5.3

Degree dynamics

Section 4

γ = 3

A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)Network Science: Evolving Network Models

Degree distribution

A random node j at time t then is with equal probability 1/N=1/(m0+t-1) onr of the nodes 1, 2,…. N ( we assume the initial m0 nodes create a fully connected graph), its degree will grow with the above equation, so

for t ≥ m0+i and 0 otherwise as system size at t is N=m0+t-1

𝑃 (𝑘 𝑗(𝑡)¿¿𝑘 )=𝑃 (𝑡 𝑗>𝑚

1𝛽 𝑡

𝑘1𝛽 )=1−𝑃 (𝑡 𝑗≤𝑚

1𝛽 𝑡

𝑘1𝛽 )=1− 𝑚

1𝛽 𝑡

𝑘1𝛽 (𝑡+𝑚0−1)

For the large times t (and so large network sizes) we can replace t-1 with t above, so

γ = 3

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

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.


P (k )=2𝑚2𝑡

𝑡− 𝑡01𝑘3

𝑘−𝛾

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.


The end


class 9: barabasi-albert model-part i network science: evolving network models february 2015 prof....

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