Spectra of RealWorld graphs: Beyond the semicircle law

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Networks in life Scaling properties and
eigenvalue spectra
Tamás Vicsek Dept. of Biological Physics, Eötvös
University, Hungary http//angel.elte.hu/vicsek
Collaborators A.-L. Barabási, I. Derényi, I.
Farkas, Z. Néda, Z.-N. Oltvai, E. Ravasz, and A.
Schubert
Why networks (topological features of
interactions)? The simplest (still rich)
approach to complex systems consisting of many
similar, but still specific and individually
relevant units.
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• Introduction to graph models
• Example networks
• Deterministic scale-free
• Collaboration graph of scientists
• A biochemical network
• Structural analysis of real-world graphs via
  their spectrum

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Graph models
degree of a vertex of edges
Each pair of vertices is connected with equal and
independent probability p
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Small World
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Graph models
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Graph models
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Graph models
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Evolution of the social network of scientific
collaborations
A.-L. Barabási, H.Jeong, Z.Néda, E.Ravasz, A.
Schubert, T. Vicsek (cond-mat/0104162)
The Erdos graph
co-author first in 1973
1976
L. Lovasz
1979
Data collaboration graphs in (M) Mathematics
and (NS) Neuroscience
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Collaboration network
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Collaboration network
Internal preferential attachment
Measured data shows
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Collaboration network
Modeling the Web of Science
(continuum model)
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Collaboration network
Modeling the Web of Science (contd)
degree of one vertex
degree distribution of the graph
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Collaboration network
Measured data shows diameter decreases with time
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Protein Network
Jeong et al, Nature (2001)
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A biochemical network
Description of data / 1 The current view of how
biological information is stored and used in the
cell
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Transcriptome similarity graph
data source Hughes et.al., Cell 102 109-126
(2000)
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Transcriptome similarity graph
( The structure of the transcriptional response
to perturbations. )

• vertex an experiment
• (a single-gene deletion strain)
• edge high number of partial similarities ( C
  gt 0.8 ) between the two transcrip-tomes
  (comparing the two columns of the matrix using
  small groups of rows)
• color of an edge strength

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Transcriptome similarity graph
Structural analysis of the transcriptome
similarity graph
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Spectral analysis
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Spectral analysis
Spectral densities of the graph models
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Spectral analysis
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Scale-free graph ( pN 2m const. )
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Spectral analysis
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Spectral analysis
Testing the structure of a small measured graph
Question Given a measured graph, which of the
graph models describes it well ? (a test graph
same number of edges and vertices)

• Answers
• For large graphs ( N gt 1000 ) the degree
  sequence can be informative power-law,
  exponential,
• For small graphs ( N 100 - 500 ) alternative
  structural tests can be also useful.

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Transcriptome similarity graph
Structural analysis of the transcriptome
similarity graph
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Summary

• Gene expression and collaboration networks are
  scale-free graphs
• The topology of real-world networks can be
  characterized by spectral methods
• Anomalous eigenvalue spectra
• Characteristic inverse participation ratio

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Collaboration network
Measured data shows diameter decreases with time
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Collaboration network
Continuum theory
Degree distribution
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