1. Scale Free Networks

2. Intro
Very large real networks (millions or billions of nodes and edges)
Occurring in nature, society, economy and technology
Evolving (growing) in time rather than designed.
Examples: Internet and WWW

3. Many networks in nature, ecology, economy, human relations & technology (Internet and WWW) have the same topological structure.
They are scale-free networks
with the same mathematical structure and behavioral properties.

4. Research objectives and some questions
Research motivation
Research objectives and some questions
Can Internet function well if hundreds of routers are out of order or damaged on purpose?
Which parts of the Internet are most vulnerable to hostile damage?
How to design efficient search engines for WWW? ( This is an algorithmic issue related to the WWW topology ).

5. Research motivation
How to prevent the current fast propagation of viruses in the Internet?
What can cause and how to prevent cascading collapse of large networks functionality ( e. g. power grids)?
How to deliver on demand computing power, huge amount of data and media functions ? ( This issue is related to Computing Grids .)

6. Random Networks
Created and researched by Paul Erdos and Alfred Renyi in and 1960.
Basic assumptions:
A fixed number of nodes.
Connected by random edges.
Nodes were “democratic” i.e. most nodes have approximately equal number of attached edges.

7. (a) growth in time by adding nodes and edges
Recent advances
Barabasi and his collaborators introduce the concept of Scale-free networks (1999). Evolving and self-organized.
Two key rules:
(a) growth in time by adding nodes and edges
(b)preferential node attachment

8. Mathematical background and notation
Degree
Degree distribution

9. Degree distributions where The probability that a vertex has k edges.
This is the Poisson distribution for the
random Erdos-Renyi
networks.
where

10. Poisson distribution Degree distribution ln P(k) ln k
Characteristic scale.
Typical average node.
Degree distribution
ln P(k)
ln k

11. Power – law distribution for evolving self-organized networks proposed by Barabasi and collaborators
Typical range
These networks have no natural average number of edges and are called scale-free.

12. Random vs.Scale Free Nets
Examples:
The network of land roads in US
is approximately a random network
with a bell shaped connectivity distribution
In contrast the airports in US
form a scale free network with several hubs connecting large number of airports.

14. Random/Exponential vs.Scale –free Networks

15. Scale free real networks
Examples:
Communication networks: The Internet , WWW
Biological : Pairwise interactions between proteins in human body.
Ecological interrelations and food webs,
Social webs, scientific citations

16. WWW c Slightly modified power-law distribution WWW home pages 2.05 193
Companies
2.62
1370
Computer scientists
2.66 12
The WWW as a whole
2.1

17. Scale-free networks
Scale-free networks are very common and a very important category of real networks.
They have strongly connected vertices (hubs) which play a key role in the network properties.
Scale-free networks are the direct result of self-organization.
Special type of growth called the preferential linking or preferential attachment.

18. Scale-free networks
Scale-free networks are dynamic , they evolve in time from small sizes to larger.
The growth follows principle of the preferential attachment.
While the network grows its new vertex becomes preferentially attached to vertices with a high number of connections. E.g. “rich gets richer”.
As a result HUBS are created.

19. Scale-free networks
A preference in the process of growth may take various forms.
The most natural linear type of preference results in scale-free networks.
Examples of a preferential attachment include WWW where more popular pages get new links.
Popularity is attractive.

20. Preferentially chosen vertex
Scale-free networks
Old network
Linear Preferential rule
Preferentially chosen vertex
New vertex
The probability that a new edge becomes attached to some vertex of degree k is proportional to k.
This leads to a scale-free network with
More general preferential attachment rules are possible.

21. Scale-free networks
The shortest path between two vertices
The average shortest path length is of the order of the LOGARITHM of the size of a network (the number of vertices)
This is also called the network DIAMETER.
Diameter of a scale-free network is short and slow growing with the size of the network.
Leads to small world networks

22. Navigating the Web Find a path from page A to page B
Given the sizes of components (the number of pages) we can estimate the probability of reaching B from A .
It is approximately 24%.
The average shortest path length of the entire WWW is 19 clicks (hyperlinks).
The 100-fold growth would add two links

23. W W W Shortest paths in the Web
For any two pages there is only 24% probability that a direct path exists from A to B.
Average shortest directed path in the Web is 19 ( the number of clicks). Undirected 6.8.
The formula for the directed path is
For N=1,000,000,000 we get

24. Long shortest paths.
According to the existing data there are pairs of pages which are separated by
a shortest directed path of length about 1,000 clicks long.

25. Internet Resilience
At any given time hundreds of routers are down but the performance is not impacted.
The Internet is robust in the presence of random failures.
This is called the topological robustness.
It will function even if we remove randomly 80% of the nodes.
Theoretical and experimental investigations show that scale-free networks are topologically robust [1] IF

26. Internet Vulnerability
Scale-free networks such as Internet are vulnerable to attacks.
If a malicious attack could simultaneously remove % of hubs (the highly connected nodes) the network would disintegrate .
A research question
Can Internet suffer from cascading failures as in power systems, economy and ecology .
We do not know.

27. More Bad News Scale-free networks are vulnerable to spreading viruses
Hubs are passing them massively to the connected multiple nodes.
This suggests immunizing hubs.