Transcript Complex Networks: Characterization and Dynamics

Complex Networks:
Structures and Dynamics
Changsong Zhou
AGNLD, Institute für Physik
Universität Potsdam
Summary
• Part - I
– Characterization of Complex Networks.
• Part - II
– Dynamics on Complex Networks.
• Part - III
– Relevance to Neurosciences.
Part-I: Network Structures
1.Introduction:
• Reductionism, Linearity and Complexity.
• Examples of networks.
2.Characterization of Networks...
•
•
•
•
Basic measures.
Common topology features in realistic networks.
Basic models (E-R, W-S, A-B)
Properties of the models.
3.... and More Realistic Properties ...
• Weighted and directed networks.
1. INTRODUCTION
Reductionism and complexity
• Linear and nonlinear systems
x
f(x)
y
f(y)
x
y
x
Linear f(x) + f(y)
System
y
result ≠
Non-Linear f(x) + f(y) !!
System
1. INTRODUCTION
Reductionism and complexity
• Complicated and complex systems
D.R. Chialvo “Critical Brain Networks” Physica A 340 (2004)
1. INTRODUCTION
Reductionism and complexity
•
Brain in ``DynamicsLand´´
1. INTRODUCTION
Reductionism and complexity
• Connection topology
Crystal Lattices
1. INTRODUCTION
Reductionism and complexity
• Connection topology
Crystal Lattices
All-to-all interactions
Internet
1. INTRODUCTION
Reductionism and complexity
• Connection topology
Crystal Lattices
Diffusion
All-to-all interactions
Mean field
Internet
?
1. INTRODUCTION
Technological Networks
World-Wide Web
Internet
Power Grid
1. INTRODUCTION
Social Networks
Citation Networks
Friendship Net
Movie Actors
Sexual Contacts
Collaboration Networks
1. INTRODUCTION
Transportation Networks
Airport Networks
Local Transportation
Road Maps
1. INTRODUCTION
Biological Networks
Protein interaction
Ecological Webs
Neural Networks
Genetic Networks
Metabolic Networks
2. NETWORKS...
A Unified Approach
towards the
Connection Topology
of various
Complex Systems
A food web
2. NETWORKS...
Networks Approach
Basic Graphs
Symmetrical Adjacency Matrix
1
2
4
3
5
6
7
8
Aij =
1
2
3
4
5
6
7
8
1
0
1
1
0
0
0
0
0
2
1
0
1
1
1
0
0
0
3
1
1
0
0
0
1
0
0
4
0
1
0
0
1
0
1
0
5
0
1
0
1
0
1
1
0
6
0
0
1
0
1
0
0
1
7
0
0
0
1
1
0
0
1
8
0
0
0
0
0
1
1
0
2. NETWORKS...
Networks Approach
Basic Graphs
Non-Symmetrical
Adjacency Matrix
DiGraphs
1
2
4
3
5
6
Aij =
7
8
1
2
3
4
5
6
7
8
1
0
1
1
0
0
0
0
0
2
0
0
1
0
1
0
0
0
3
1
0
0
0
0
0
0
0
4
0
1
0
0
1
0
0
0
5
0
0
0
0
0
1
0
0
6
0
0
1
0
1
0
0
0
7
0
0
0
1
1
0
0
1
8
0
0
0
0
0
1
1
0
2. NETWORKS...
Networks Approach
Basic Graphs
DiGraphs
Weighted Graphs
7.4
0.4
1.8
0.6
0.7
4.5
2.8
0.4
8
0.5
8
5
2
2. NETWORKS...
Characterization
• Vertex degree: k(v)
Basic Graphs
Friendship
k( )  4
2. NETWORKS...
Characterization
• Clustering Coeficient: C(v)
Basic Graphs
Friendship
2. NETWORKS...
Characterization
• Clustering Coeficient: C(v)
Basic Graphs
• Number of existing connections: 2
Friendship
2. NETWORKS...
Characterization
• Clustering Coeficient: C(v)
Simple Graphs
Friendship
• Number of existing connections: 2
• Total Number of possible connections:
➡ ½·kv·(kv-1) = ½·(4·3) = 6
2. NETWORKS...
Characterization
• Clustering Coeficient: C(v)
Basic Graphs
Friendship
• Number of existing connections: 2
• Total Number of possible connections:
➡ ½·kv·(kv-1) = ½·(4·3) = 6
• Cv = 2 / 6 = 0.333
2. NETWORKS...
Characterization
• Clustering Coeficient: C(v)
Basic Graphs
Friendship
• Number of existing connections: 2
• Total Number of possible connections:
➡ ½·kv·(kv-1) = ½·(4·3) = 6
• Cv = 2 / 6 = 0.333
1
Cv 
k v (k v  1)
A
vj Avh A jh
j ,h
How well are the neighbours connected!
2. NETWORKS...
Characterization
• Distance (pathlength)
Basic Graphs
j
Friendship
i
2. NETWORKS...
Characterization
• Distance (pathlength)
Basic Graphs
j
Friendship
i
2. NETWORKS...
Characterization
• Distance (pathlength)
Basic Graphs
j
Friendship
i
2. NETWORKS...
Characterization
• All-to-all distance matrix:
Lij Length of the shortest paths
Lij =
1
2
3
4
5
6
7
8
9
1
2
1
1
2
2
2
3
3
4
2
1
2
1
1
1
2
2
3
4
3
1
1
2
2
2
1
3
2
3
4
2
1 2
5
2
1
6
2
2
1
2
1
2
3
2
1
2
1
1
2
3
2
1
2
1
2
2
2
0
7
3
2
3
1
1
2
2
1
2
8
3
3
2
2
2
1
1
2
1
9
4
4
3
3
3
2
2
1
2
3
2
1
3
2
1
0
1
2
2. NETWORKS...
General Features of Real Networks
• Scale-free
structure
Power-law distribution of degrees
ki :
P(k ) ~ k
k

2. NETWORKS...
General Features of Real Networks
• Small world structure
ki :
i
Small distance
L   Lij  : ln ln N ~ ln N
( N  109 , L ~ 3  20)
High clustering
2. NETWORKS...
Random Network Models
ERDOS - RÉNYI MODELL (E-R)
Connecting a pair of nodes with probability p
2. NETWORKS...
Random Network Models
ERDOS - RÉNYI MODELL (E-R)
• Degree distribution:
Poissonian!
Mean degree K=NP
2. NETWORKS...
Random Network Models
ERDOS - RÉNYI MODELL (E-R)
• Degree distribution:
Poissonian!
• Giant Component:
2. NETWORKS...
Random Network Models
WATTS - STROGATZ MODELL (W-S):
• Degree?
• Clustering?
• Pathlength?
2. NETWORKS...
Random Network Models
WATTS - STROGATZ MODELL (W-S):
Rewiring a link with probability p
2. NETWORKS...
Random Network Models
WATTS - STROGATZ MODELL (W-S):
Having shortcuts now!
2. NETWORKS...
Random Network Models
WATTS - STROGATZ MODELL (W-S):
• SMALL - WORLD NETS =
– High clustering
– Short distance
Watts, Strogatz. Nature 393/4, 1998
2. NETWORKS...
Comparison
Regular Lattice
P(k) = δ(k-Z) :
Z= number of neighbours
Small-World Net
Random Graph
Poissonian!
2. NETWORKS...
Random Network Models
EVOLVING NETWORKS, Barabási-Albert model (B-A)
• Ingredients:
– Growing AND
– Preferential attachment
pi 
ki
k
j
j
2. NETWORKS...
Random Network Models
EVOLVING NETWORKS, Barabási-Albert model (B-A)
• Ingredients:
– Growing AND
– Preferential attachment
pi 
ki
k
j
j
• Results:
– “Richer-Gets-Richer”
– k distribution: Scale Invariant!
2. NETWORKS...
Random Network Models
EVOLVING NETWORKS, Barabási-Albert model (B-A)
SCALE - FREE NETWORKS
Barabási, Albert. Science 286 (1999)
2. NETWORKS...
Properties of the models
Lattice
Small-World
Random
Pathlength
Long
>
Short
≥
Short
Clustering
Large
≥
Large
≥
Small
Scale-Free
≥
Short
Small
Large in many real scale-free networks !
2. NETWORKS...
Significant Impacts
• Network Resiliance:
– Highly robust agains RANDOM failure of node.
2. NETWORKS...
Significant Impacts
• Network Resiliance:
– Highly robust agains RANDOM failure of node.
2. NETWORKS...
Significant Impacts
• Network Resiliance:
– Highly robust agains RANDOM failure of node.
– Highly vulnerable to deliberate attack on HUBS.
2. NETWORKS...
Significant Impacts
• Network Resiliance:
– Highly robust agains RANDOM failure of node.
– Highly vulnerable to deliberate attack on HUBS.
2. NETWORKS...
Significant Impacts
• Network Resiliance:
– Highly robust agains RANDOM failure of node.
– Highly vulnerable to deliberate attack on HUBS.
• Applications:
– Inmunization in computer networks and populations
Cohen et al PRL, (2000, 2002)
2. NETWORKS...
Communities and Overlapping Nodes
Palla et al. Nature 435, 9 (2005)
Cat cortico-cortical connections
Physics collaboration network
3. ... AND MORE REALISTIC CHARACTERIZATION
Weighted and Directed Networks
Graphs
Weighted
Directed
9.3
9.3
7.0
7.2
0.3
0.3
5.1
Degree k
degree, k( ) = 4
Intensity
6.0
Si 
5.1
W
ij
j
degree, k( ) = 4
In/out-degree
In/out-intensity
out-degree, out-k( ) = 3
in- degree, in-k( ) = 2
intensity. S( ) = 21.7
out-intensity, out-S( ) =24.9
in-intensity, in-S( ) = 12.3
3. ... AND MORE REALISTIC CHARACTERIZATION
Weighted Networks
• Are weights correlated with degrees?
– NO ⇒
Scientific Collaborations (SCN)
Wij ~ const
– YES ⇒ World-Airport-Networks (WAN)

Wij ~ (ki k j )
3. ... AND MORE REALISTIC CHARACTERIZATION
Weighted Networks
• Are weights correlated with degrees?
– NO ⇒
SCN
S (k ) ~ const* k
– YES ⇒ WAN
S (k ) ~ k
1
3. ... AND MORE REALISTIC CHARACTERIZATION
Weighted Networks
• Weighted Clustering Coeficient:
1
Cv 
k v (k v  1)
(WAN)
v
v
j
CvW
1

S v (kv  1)

j ,h
A
vj Avh A jh
j ,h
Wvj  Wvh
2
Avj Avh A jh
h
Barrat et al. (2004) PNAS vol.101, 11