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Generating Scale-free Networks with AdjustableClustering Coefficient Via Random Walks

Carlos Herrera / Pedro J. ZufiriaIEEE Network Science Workshop

West Point, NY - June 2011

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WHAT MAKES A

NETWORK TO BECOMECOMPLEX ?

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Properties of complex networks

Similar non-trivial characteristics in very different networks● Short diameter (“six degrees of separation”)→ diameter of the

networks grows as log of N (node number)

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Properties of complex networks (II)

● Scale-free networks → Power-law degree distribution

● Power law → There are hubs (Google, Miami) 

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Properties of complex networks (III)

● High clustering coefficient

 – Fraction of triangles in the network

 – Probability of having the red link, if the network has theblue ones.

A

C

B

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SummarizingSome characteristics of complex networks:

 – Short diameter 

 – High number of triangles

 – Power-law degree distribution

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Properties of complex networks (III)

● High clustering coefficient

Networks Node number Edge number   Clusteringcoefficient

Actors imdb 449 913 25 516 482 0,75

WWW (nd.edu) 269 504 1 497 135 0,29

Word co-occurrence 460 902 17 000 000 0,44

Internet (AS level) 10 697 31 992 0,39P2P Network 880 1296 0,011

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Network models●

How could these non-trivial phenomena happen inself-constructed networks?

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● Erdös-Rényi random graph●

Every connection between the N nodes has thesame probability ( p)

Network models → Erdös

Diameter Clustering Scale-free

M d l S ll W ld

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Watts – Strogatz models● High clustering coefficient & small-world● Rewire a regular lattice

Models → Small-World 

Diameter Clustering Scale-free

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● Barabási – Albert model

●

Based on growing and preferential attachment● A new node links an existing one with probability

proportional to the degree● It generates power-law degree distributions

Models → BA

 Π  k i=

k i

 Σ  j

k  j

Diameter Clustering Scale-free

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● Comments on BA model●

There were many suggestions to improve on it● Does not address the control of the clustering

feature

Models → BA

Ratio to other network models

Network C Erdös-Renyi Barabási-Albert

Flickr 0,313 0,0212 0,0397

LiveJournal 0,33 0,0084 0,0562

Orkut 0,171 0,1381 0,1898

Youtube 0,136 0,0271 0,0144

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● Criticism to BA

●

Needs global information

Models → BA

 j j

i

i k 

k 

k  Σ=Π )(

Does a blogger know thedegree distribution of thewhole WWW when helinks a webpage?

A. Vázquez (2003)

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There are plenty of models.....

Model Diameter  Adjustable Clustering

Scale-Free Local info

Ërdos (1960)

Watts (1998)

Barabási (1999)

Newman (2001)

Holme (2002)

Vázquez (2003)

Evans (2005)Toivonen (2006)

Newman (2011)

OUR GOAL

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● A random walk of length l endsmore likely in a highlyconnected node

● Already used by Vázquez[04],Evans[05] and Sarämaki[06]

● Rigorous analytical proof needed (working on it)

● According to simulations thelonger the walk the better the fit

to PA

Preferential attachment (PA) using local information:Random Walks

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Preferential attachment (PA) using local information:Random Walks

0 1 2 3 4 5 6 7 8 90

100

200

300

400

500

600

700

800

900

L=1L=10

Degree

   V   i  s   i   t  s

● A random walk of length l endsmore likely in a highlyconnected node

● Already used by Vázquez[04],Evans[05] and Sarämaki[06]

● Rigorous analytical proof needed (working on it)

● According to simulations thelonger the walk the better the fit

to PA

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First approach● If we use walks of length l=1, then we are

forcing triangles. Otherwise we start thenext walk from a different point.

● By controlling the probability CC of these

1-step walks we are controlling clustering.

l=1

Towards clustering coefficient control

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First approach results:

l=1

Towards clustering coefficient control

Model does not workproperly for low CCvalues

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Use either l=1 or l=2

l=1

Solutions

l=2

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Use either l=1 or l=2

l=1

Solutions

l=2

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Use either l=1 or l=2 walks

Solution

0 10 20 30 40 50 60 70 80 90 100

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

cc

   C

l=1

l=2

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Towards a self-organized model

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Towards a self-organized model

● It has been proved clustering coefficient involvesgenetic factors

● The are people who like their friends to meet eachother, there are people who do not.

● We use this fact for the clustering control mechanism

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● When a node “is born”, it is assigned a probability of introducing

friends to others.● Only binomial probability distribution has been tested → future

work

l=1 l=2

Genetic factor 

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Algorithm

● Start with a “seed” connected network.

●

[loop1-(n times)] Choose a random vertex● Take a walk length >=7 (ensure PA)

● [loop2-(m times)] Mark destination node

● If destination node is “friendly”, give walk length 1. Go to [loop2].

● If destination node is not “friendly”, give walk length 2. Go to [loop2].● Add vertex to network and link it to marked nodes. Unmark all nodes.

Go to [loop1]

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Simulations

Si l ti

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Simulations

R lt

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Result

Scale free networks where one canindependently select:

●

N nodes● Average degree 2m● Tunable clustering coefficient→ via genetic distribution

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Work in progress:Average degree influence in clustering

●

As average degree grows,maximum reproducibleclustering coefficientdecreases

0 2 4 6 8 10 12 14

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

Average degree

   M  a  x   C

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Work in progress:Average degree influence in clustering

●

As average degree grows,maximum reproducibleclustering coefficientdecreases

● Problem: controlling the hubs(they cannot have highclustering)

● New clustering measureshave been proposed for 

scale-free networks, avoidingdegree bias (Soffer 2005)

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 10

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

k<=77<k<= 20k>20

cc

   C

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Future steps

●

Community structure analysis● Using different distributions from binomial for genetic

assignation

● “Solve” clustering problem

● Does random walk produces preferential attachment?Analytical rigorous proof.

● Validation against real data

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THANKS!

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TRASPAEXTRA:

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Facts on the “seed” network

●

Seed network must be● Connected

● Degree (k) equal for allnodes

● K not to small to avoidwinner takes all effects

● K not too big to avoidproblems in degreedistribution

● Solution: a ring