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
‘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
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
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 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
• The two forces make power-law degree distribution
• If we add average path length in energy function, large hubs result.