Self-Similarity of Complex Networks

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Self-Similarity of Complex Networks

• Maksim Kitsak
• Advisor
• H. Eugene Stanley

Collaborators Shlomo Havlin Gerald Paul Zhenhua
Wu Yiping Chen Guanliang Li
Kitsak, Havlin, Paul, Pammolli, Stanley,
(submitted), Phys. Rev. E. (2006)
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Motivation and Objectives

• Many real networks are fractals.
• Fractal real networks are shown to have a
  topology distinct from non-fractal networks.
• Do fractal and non-fractal networks have
  different properties? (Transport properties)
• What are the possible applications of these
  properties?

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Networks Definitions
1) Network is a set of nodes (objects) connected
with edges (relations).
2) Degree (k) of a node is a number of edges
connected to it.
3) Degree Distribution P(k) is the probability
that a randomly chosen node has degree k
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Networks Properties

1. Small World property. Despite the large size,
   the shortest path between any two nodes is small.
   (WWW, Internet, Biological)Number of nodes
   accessible from a random node (seed) grows
   exponentially with the distance measured from the
   seed.

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Networks Properties
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Networks Properties
3) Self-SimilaritySelf-similar network is
approximately similar to a part of itself and is
fractal. Fractal typically has fractional
dimension and doesntpossess translational
symmetry.
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Networks Properties
It turns out that many real networks possess all
three properties (Small World, Scale-Free,
Fractal)!!!
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Dimension Calculation Box Covering Algorithm
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Fractal analysis with box-covering algorithm
Song, Havlin, Makse, 2005
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Origin of fractals in scale-free networks
Repulsion between hubs
In fractal networks large degree nodes (hubs)
tend to connect to small degree nodes and not to
each other!
Song, Havlin, Makse, 2005
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Transport on networks Betweenness Centrality
Most of the transport on the network flows along
the shortest paths. Central nodes are critical
if they are blocked transport becomes
inefficient
Number of shortest paths between nodes and
that pass node .
Sociology - L.C. Freeman, 1979
Total number of shortest paths between nodes
and .
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Transport on networks Betweenness Centrality
How do we identify nodes with high Centrality? Is
it true that high centrality nodes also have
large degree?
Centrality is weakly correlated with degree in
fractal scale-free networks!
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Transport on networks Betweenness Centrality
Why is centrality weakly correlated with degree
in fractal scale-free networks?
Due to repulsion between hubs small degree
nodes appear at all parts of the fractal network.
Thus, their centralities can have both small and
large values.
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Centrality-degree correlation in real networks
One cant compare centralities of networks
directly due to uniqueness of real networks.
The network can be compared to its random
counterpart !
Rewired network has degree distribution identical
to the original network. Repulsion between hubs
is broken by random rewiring. The random network
is always non-fractal.
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Centrality-degree correlation.
Kitsak, Havlin, Paul, Pammolli, Stanley, 2006
Centrality degree correlation in non-fractal
scale-free networks is much stronger than that in
SF fractal networks.
Average centrality of small degree nodes in
scale-free fractal networks is significantly
larger due to repulsion between hubs.
Fractal networks should be more stable to
conventional degree attacks. Immunization/Attack
strategies should be optimized for fractal
networks.
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What is the overall Centrality distribution in
scale-free networks?
Kitsak, Havlin, Paul, Pammolli, Stanley, 2006
Nodes of fractal networks generally have larger
centrality than nodes of non-fractal networks
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Transition from Fractal to Non-Fractal Behavior.
Real networks are neither pure fractals nor
non-fractals due to statistical effects. What
happens if we add random edges to a scale-free
fractal network?
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Summary and Conclusions

• 1. Centrality-degree correlation is much weaker
  in fractal networks than in non-fractal.
• Fractal networks should be more stable to
  conventional degree attacks.
• Immunization/Attack strategies should be
  optimized for fractal networks.
• 2. Power-law centrality distribution
• Centralities of nodes are larger in fractal
  scale-free networks.
• fractal networks have different transport
  properties.
• 3. Transition from fractal to non-fractal
  networks.
• A crossover is observed from fractal to
  non-fractal networks.
• Relatively small percent of edges is needed to
  turn fractal network into non-fractal.
• Findings of present work have been submitted to
  Phys. Rev. E.

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Centrality-degree correlation.
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Transition from Fractal to Non-Fractal Behavior
Analytical Consideration
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Centrality distribution Analytical Consideration
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Contents

• 1. Introduction.
• (Definitions, Properties)
• 2. Self-Similarity and Fractality.
• (Fractal Networks General Results)
• 3. Betweenness Centrality in Fractal and
  Non-Fractal Networks.
• (Centrality distribution, Centrality-Degree
  correlation)
• 4. Transition from Fractal to Non-Fractal
  networks.
• (Crossover phenomenon, Scaling)
• 5. Summary and Conclusions

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Betweenness Centrality
Betweenness Centrality is a measure of Importance
of a node in the network.
L.C. Freeman, 1979
Is it true that larger degree nodes generally
have larger centrality?
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Transition from Fractal to Non-Fractal Behavior.
What happens when random edges are added to a
fractal network?
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Networks Properties
It turns out that many real networks possess all
three properties (Small World, Scale-Free,
Fractal)!!!