Scale Free Networks

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 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 1959 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.

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• 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

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Mathematical background and notation

• Degree
• Degree distribution

9
Degree distributions

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

where
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Poisson distribution
Characteristic scale. Typical average node.

• Degree distribution

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Power law distributionfor 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.
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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.

13

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Random/Exponential vs.Scale free Networks
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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

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WWW

• Slightly modified power-law distribution

c
WWW home pages
193
2.05
Companies
2.62
1370
Computer scientists
12
2.66
The WWW as a whole
0
2.1
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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.

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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.

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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.

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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.
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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

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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

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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 N1,000,000,000 we get

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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.

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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

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Internet Vulnerability

• Scale-free networks such as Internet are
  vulnerable to attacks.
• If a malicious attack could simultaneously remove
  5-15 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.

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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.