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Scale Free Networks

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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

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• 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.

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Research motivationResearch 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 ).

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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 .)

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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

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Degree distributions

• The probability that a vertex has k edges.

• This is the Poisson distribution for the

random Erdos-Renyi

networks.€

P(k) =e− k

_

k k

k!

∑∞

=

=0

_

)(k

kkPk

where

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Poisson distribution• Degree distribution

ln P(k)

ln k

Characteristic scale.Typical average node. k

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Power – law distributionfor evolving self-organized

networks proposed by Barabasi and collaborators

€

P(k)∝ k−γ

32 <<γThese networks have no natural average number of edges and are called scale-free.

Typical range

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Random vs.Scale Free Nets

Examples:The network of land roads in US is approximately a random networkwith a bell shaped connectivity

distributionIn contrast the airports in US form a scale free network with several

hubs connecting large number of airports.

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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 γ−+∝ )()( ckkP

WWW home pages

γ c

Companies

2.05 193

2.62 1370

Computer scientists 2.66 12

The WWW as a whole 2.1 0

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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

• Linear Preferential rule

Preferentially chosen vertexNew vertex

Old network

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 3=γMore 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

Nlave 10log06.23.0 +=

For N=1,000,000,000 we get 19≈avel

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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

3≤γ

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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.