Topology of Complex Networks

1
Topology of Complex Networks
Bruno Miguel Tavares Gonçalves 2004 Universidade
de Aveiro
2
Topology of Complex Networks

• Motivation
• Fundamental Concepts
• Models
• Correlations
• Results
• Conclusion

3
Topology of Complex Networks

• Motivation
• Fundamental Concepts
• Models
• Correlations
• Results
• Conclusion

4
Importância das redes complexas

• The most efficient and robust form of
  organization
• Efficient All points are mutually accessible.
• Robust Not sensitive to random failures.

5
CN In Nature
Yeasts protein interaction network
6
Artificial Complex Networks (1)
Colaboration network between authors of
scientific papers
7
Artificial Complex Networks (2)
Visual Representation of the Internet
Infrastructure http//www.caida.org/analysis/topo
logy/as_core_network/AS_Network.xml
8
Topology of Complex Networks

• Motivation
• Fundamental Concepts
• Models
• Correlations
• Results
• Conclusion

9
Networks and Graphs

• A graph G(V,E) consists of two sets a finite
  set V of elements called vertices and a finite
  set E of elements called edges. Each edge is
  identified with a set of vertices
• Thulasitaman (1992)

10
Adjacency
1. Adjacency Matrix
2. Adjacency List
11
Adjacency Matrix

• tr(A) 0
• tr(A2) 2m (m edges)
• tr(A3) 6t (t triangles)
• (An)ij paths of size n between nodes i and j

12
Autonomous System Level
Decentralized Routing !
Topology data from BGP routing tables, collected
by NLANR, looking glass U. Oregon
13
Os Sistemas Autonomos (1)
14K nodes in 2002 ( 2K nodes in 1997) 30K
links in 2002
14
Autonomous Systems
15
Clustering Coefficient
The networks clustering coefficient is the
average of this quantity over all nodes.
16
Connectivity and Centrality
The number of connections of a node is called
connectivity. Directed networks have both in and
out connectivities
The number of shortest paths that contain a given
node determine its centrality.
17
Connectivity and Centrality (2)
Goh et al (PRL 87278701) seem to indicate a
value of ?2.2(1) for various types of networks.
18
Types of networks

• Equilibrium
• N and L are constant
• P(k) completely characterizes the network.
• No correlations
• Non Equilibrium
• N and L grow in time
• Linear growth ltkgt constant
• Accelerated growth ltkgtta
• Correlations are important
• P(k) only contains part of the information

19
Topology of Complex Networks

• Motivation
• Fundamental concepts
• Models
• Correlations
• Results
• Conclusion

20
Erdös-Rényi Model

• At t0 theres N nodes e 0 edges
• New edges are created with probability p

The connectivity distribution is
N??
21
WattsStrogatz Model (1)

• At t0 we have a ring where each node is
  connected to K neighbors.
• - At each step, a connection is modified with
  probability p.

22
Watts-Strogatz Model (2)
The clustering coefficient is In the limit
K??, C?3\4. For tN, we have modified pNK\2
connections. This implies a decrease in both C as
the diameter L(p).
23
Watts-Strogatz Model (3)
24
Barabási-Albert Model (1)
Dorogovstev, Mendes e Samukin PRL 85,4633 (2000)
25
Barabási-Albert Model (2)
Dorogovstev, Mendes e Samukin PRL 85,4633 (2000)
26
Acelerated Growth (1)
27
Accelerated Growth (2)
At each new step, a new node is added with
connections to the NN of randomly selected node
with probability p.
28
Accelerated Growth (3)
29
Accelerated Growth (4)
Logk
30
Topology of Complex Networks

• Motivation
• Fundamental Concepts
• Models
• Correlations
• Results
• Conclusion

31
Correlations
32
Shells
Shell d is defined by all the nodes at a
distânce d from the root node

• Shells can be used to study the spatial
  dependence of correlations.

33
The importance of shells

• We have defined several properties of individual
  node such as connectivity and clustering. Shells
  can help us understand
• How these properties vary with distance

2. If power-laws are universal 3. What are
the relevant exponents in each shell 4. What
are the structural properties of the network
Using these results we can learn how to build
networks that are more robust and efficient or
that are particularly well adapted to a
particular situation.
34
Topology of Complex Networks

• Motivation
• Fundamental Concepts
• Models
• Correlations
• Results
• Conclusions

35
Correlation Function
36
Shell Structure
37
Second Shell
38
Outer Shells
39
Summary
40
Topology of the Internet
41
Time evolution
42
Time Exponents (1)
43
Time Exponents (2)
44
Time Exponents (3)
45
Topology of Complex Networks

• Motivation
• Fundamental Concepts
• Models
• Correlations
• Results
• Conclusion

46
Conclusion

• You can extract a great deal of information from
  the correlation functions
• The idea of shells is useful to study the
  topology of Networks
• Power-laws can be found on all shells of the
  Internet
• The growth of the Internet is compatible with the
  accelarated growth model with p0.58
• The results we have obtained have contributed to
  a better understanding of the underlying
  structure of the Internet

47
Further Possibilities of Reasearch

• Redefine other quantities,such as the clustering
  coefficient, kin,kout, etc, using the idea of
  shell
• Apply the same methodology to other networks,
  such as protein networks, gene, words, WWW,
  etc...
• Study the correlation function applied to the
  idea of shells to try and extract more
  information from this quantity
• Apply what we have learned about the topology of
  the Internet using this method to develop a new
  generation of network generators that are more
  realistic.

48
The End