Optimization Based Modeling of Social Network Yong-Yeol Ahn, Hawoong Jeong.

Optimization Based Modeling of Social Network

Yong-Yeol Ahn, Hawoong Jeong

Outline

• About real networks and models• Motivation• Simulation method• Result

• Conclusion

Real Complex Networks

• Social networks • Acquaintance, scientific collaboration, acto

r, bbs, etc. • Internet, WWW, e-mail, other communicatio

n networks

Real Complex Networks

• Biological networks• Metabolic network• Genetic network• Protein interaction network• Neuronal network

Basic Concepts of Network

Nodes

Links

Degree = 3

A shortest path with path length=3 (Equivalent with 3 clicks in WWW)

Clustering Coefficient

AB

C

A knows B

A knows C

The probability that B knows C is large

• Clustering coefficient for a node represent how many links are there between neighbors

• Clustering coefficient for a network is the average of all nodes’s clustering coefficient

C= # of links between neighbor

# of possible neighbor pair

Clustering Coefficient

A clique or a communityC=1

C=0

Clustering Coefficient

‘Triangle’ the building block.

Alternative definition of clustering coefficient 3 x # of triangle

# of connected triplesC=

Real Network’s Universal Characters

•Short path length •High clustering •Large inhomogeneity (power-law deg

ree distribution)

Modeling Real Networks

• Static network model• Erdös-Rényi model(random network)

Connect All pairs of nodes with probability p

Erdös-Rényi Model

• Randomness short path length

• Homogeneous model

Modeling Real Networks

• Static network model• Watts-Strogatz model (small world)

Modeling Real Networks

• Watts-Strogatz model C: clustering coefficientL: average path length

Small World Network Model

• Randomness short path length

• Regularity high clustering

• Balance between regularity and randomness

Modeling Real Networks

• Growing network model• BA model

• From the power law degree distribution of real networks

• Many models after BA model adopted the growing scheme

Network Models: BA Model

• Growing• New nodes and links are added

continuously

• Linear preferential attachment• New nodes make links with

preferential attachment rule• Rule : “Riches get richer”

Scale-free Network Model

• Scale-free network model• ‘Hub’ and power-law degree distribution

‘inhomogeneity’• “Network is growingand inhomogeneous”

P(k) ~k-3

New Scheme: Optimization

BA model says “A network is growing”

New models say “The evolution is more important than growth. Let’s ignore the growth” (Mathias et al.)

Growth and Evolution

• WWW is growing exponentially• Rewiring in WWW is faster than growth • Bacteria Human (Growth of biological

networks)• Origin of species (Numerous rewiring in

biological networks)

Growth : Addition of nodesEvolution : Rewiring of links

Evolutionary Pressure

• So, the rewiring occur randomly? No.

• Biological networks • Natural selection

• Artificial networks(electrical circuit,…)• Cost, High performances

New Design : Optimization Models

• Origin of biological networks and man-made networks

• Timescale of link dynamics vs. Timescale of node dynamics

Take a snapshot

• ‘Growth’ ‘rewiring, evolution’

Examples of Optimization

• In biosystems• Metabolic network’s path length conservatio

n• Allometric scaling

• In artificial systems• JAVA class network(A structure of computer

program)• Electric circuit

Optimization Scheme

• How to model the natural selection and optimization? Nature want to enlarge network’s ‘efficiency’ while wan

t to cut down ‘cost’

So,• High ‘efficiency’ short path length (Information flo

w)• Low ‘cost’ fewer links Energy = p L + (1-p)E (p:parameter, L:path length, E: expense, cost)

Star Network

• Trivial case: optimizing only average path length

‘Star network’

To shorten path lengthmakes a hub

Result of Optimization Model

• Power law degree distribution in some range of p (parameter)

k (Degree)

P(k)(Cancho and Sole)

Our Motivation

• Real networks have large clustering coefficient and community structures

Then, • What kind of network will we get, if we m

aximize a network’s clustering cofficient?

Method

• Greedy algorithm• Choose a link and rewire it randomly• If energy decreases, keep it• If energy increases, discard it• We calculate with or without ‘connection constraint’

A triangle is formed, we’ll keep this rewiring

Method: Supplement

This link is weak underour method

Strong link

Energy Optimization

• Maximizing clustering coefficient

• Energy= 1 - C (C: Clustering coefficient)• We try to maximize clustering coefficient

• Generalized form• Energy= p(1-C) + (1-p)d• P balances contributions from C and d • We try to maximize clustering and to minimize

normalized vertex-vertex distance

Results:Clustering Only (NotConnected)

Scale free network with exponent –2.2

(N=10000,L~20000)

Clustering coeff. : 0.83

P(k)

Degree

Results:Clustering Only (NotConnected)

Structure of the network.

N=300, L~600, Clustering coeff. ~0.9

Results: Clustering Only(connected)

P(k)

k

Exponent ~ –2.9

(N=10000,L~20000)

Clustering coeff. : 0.79

Results: Clustering Only(connected)

Structure of the network

Results: Clustering and Distance

p=0

p=0.1

p=1

We can observe large differences in topology

Only by path lengthOnly by clustering coefficient

Discussion

• Let’s see social networks• Can we define ‘cost’ in social

networks?• Can we define ‘efficiency’ in social

networks?

Social networks are different from biological and artificial networks.

Discussion

• Functional networks : Metabolic network, Electrical circuit network, ..

‘global’

• Non-functional network : Social networks, e-mail network, ..

‘Local’

Discussion

• Creation and deletion of a link in non-functional network. • Creation of link through friends• Deletion of link through ‘out of sight, out

of mind’

Simplified to ‘rewiring’

Discussion

• Two forces• Make triangles!

• Make hubs!

Discussion

• The two forces make power-law degree distribution

• If we add average path length in energy function, large hubs result.

Conclusion

• We categorize networks into two groups

• We explain the meaning of clustering-driving scheme

• With clustering optimization, we get highly clustered scale-free network