shi zhou university college london second-order mixing in networks shi zhou university college...
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Shi Zhou
University College London
Second-order mixingin networks
Shi Zhou
University College London
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Dr. Shi Zhou (‘Shee Joe’)
Lecturer of Department of Computer Science, University College London (UCL)
Holder of The Royal Academy of Engineering/EPSRC Research Fellowship
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Outline
Part 1. Power-law degree distribution (1st-order) assortative coefficient Rich-club coefficient
Collaborated with Dr. Raúl Mondragón, Queen Mary University of London (QMUL).
Part 2. Second-order mixing in networks Ongoing work collaborated with
Prof. Ingemar Cox, University College London (UCL) Prof. Lars K. Hansen, Danish Technical University (DTU).
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Power-law degree distribution,
assortative mixing and rich-club
Part 1
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Complex networks
Heterogeneous, irregular, evolving structure
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Degree
Degree, k Number of links a node has
Average degree <k> = 2L / N
Degree distribution, P(k) Probability of a node having degree k
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Poisson vs Power-law distributions
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Yeast protein network
Trading networks
Business networksFood web
Power law (or scale-free) networks are everywhere
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Here we study two typical networks
Scientific collaborations network in the research area of condense matter physics Nodes: 12,722 scientists Links: 39,967 coauthor relationships
The Internet at autonomous systems (AS) level Nodes: 11,174 Internet service providers (ISP) Links: 23,409 BGP peering relationship
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Degree properties
Average degree is about 6
Sparsely connected L/N(N-1)/2 <0.04%
Degree distribution Non-strict power-law
Maximal degree 97 for scientist network 2389 for Internet
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Degree-degree correlation / mixing pattern
Correlation between degrees of the two end nodes of a link Assortative coefficient r
Assortative mixing Scientific collaboration r = 0.161
Disassortative mixing Internet r = - 0.236
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Rich-club
Connections between the top 20 best-connected nodes themselves
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Rich-club coefficient
N>k is the number of nodes with degrees > k.
E>k is the number of links among the N>k nodes.
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Scientist network Internet
Degree distribution
Power-law Power-law,extra long tail
Mixing pattern Assortative Disassortative
Rich nodesSparsely
connected,no rich-club
Tightly interconnected,
rich-club
Summary
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Discussion 1
Mixing pattern and rich-club are not trivially related Mixing pattern is between TWO nodes Rich-club is among a GROUP of nodes. Each rich node has a large number of links, a small
number of which are sufficient to provide the connectivity to other rich nodes whose number is small anyway.
These two properties together provide a much fuller picture than degree distribution alone.
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Discussion 2
Why is the Internet so ‘small’ in terms of routing efficiency? Average shortest path between two nodes is only 3.12
This is because Disassortative mixing: poorly-connected, peripheral
nodes connect with well-connected rich nodes. Rich-club: rich nodes are tightly interconnected with
each other forming a core club. shortcuts redundancy
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Jellyfish model
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Discussion 3 (optional)
How are the three properties related? Degree distribution Mixing pattern Rich-club
We use the link rewiring algorithms to probe the inherent structural constraints in complex networks.
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Link-rewiring algorithm
Each node’s degree is preserved
Therefore, surrogate networks is generated with exactly the same degree distribution. Random case, maximal assortative case and
maximal disassortative
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Mixing pattern vs degree distribution
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Rich-club vs degree distribution
P(k) does not constrain rich-club. For the Internet, a minor change of the value of r is
associated with a huge change of rich-club coef.
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Observations
Networks having the same degree distribution can be vastly different in other properties.
Mixing pattern and rich-club phenomenon are not trivially related. They are two different statistical projections of the joint
degree distribution P( k, k’ ). Together they provide a fuller picture.
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Part 2
Second-order mixing in networks
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Neighbour’s degrees
Considering the link between nodes a and b
1st order mixing: correlation between degrees of the two end nodes a and b.
2nd order mixing: correlation between degrees of neighbours of nodes a and b
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1st and 2nd order assortative coefficients
Assortative coefficient
1st order
(degree)
2nd order
(Neighbours average degree)
2nd order
(Neighbours max
degree)
r Ravg Rmax
Scientist Collaborations
0.161 0.680 0.647
Internet -0.195 -0.097 0.036
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Assortative coefficients of networks
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Statistic significance of the coefficients
The Jackknife method For all networks under study, the expected standard deviation of the coefficients are very small.
Null hypothesis test The coefficients obtained after random permutation (of one of the two value sequences) are close to zero with minor deviations.
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Mixing properties of the scientist network
Frequency distribution of links as a function of1. degrees, k and k’
2. neighbours average degrees, Kavg and K’avg
of the two end nodes of a link.
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Mixing properties of the scientist network
1. Link distribution as a function of neighbours max degrees, Kmax and K’max
2. That when the network is randomly rewired preserving the degree distribution.
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Discussion (1)
Is the 2nd order assortative mixing due to increased neighbourhood? No. In all cases, the 3rd order coefficient is smaller.
Is it due to a few hub nodes?No. Removing the best connected nodes does not result in smaller values of Rmax or Ravg.
Is it due to power-law degree distribution?No. As link rewiring result shows, degree distribution has little constraint on 2nd order mixing.
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Discussion (2)
Is it due to clustering?No.
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Discussion (3)
How about triangles? There are many links where the best-connected neighbours of the two end nodes are one and the same, forming a triangle.
But there is no correlation between the amount of such links and the coefficient Rmax. There are also many links where the best-connected neighbours are not the same. And there are many links …
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Discussion (4)
Is it due to the nature of bipartite networks? No. The secure email network is not a bipartite network, but it shows very strong 2nd order assortative mixing. The metabolism network is a bipartite network, but it shows weaker 2nd order assortative mixing.
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Discussion - summary
Each of the above may play a certain role
But none of them provides an adequate explanation.
A new property? New clue for networks modelling
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Implication of 2nd order assortative mixing
It is not just how many people you know, Degree 1st order mixing
But also who you know. Neighbours average or max degrees 2nd order mixing
Collaboration is influenced less by our own prominence, but more by the prominence of who we know?
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Reference
The rich-club phenomenon in the Internet topology IEEE Comm. Lett. 8(3), p180-182, 2004.
Structural constraints in complex networks New J. of Physics, 9(173), p1-11, 2007
Second-order mixing in networks http://arxiv.org/abs/0903.0687 An updated version to appear soon.