Scale Free Networks
Mar 31, 2015
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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.