Spectral Analysis for Stochastic Models of LargeScale Complex Dynamical Networks

1
Spectral Analysis for Stochastic Models of
Large-Scale Complex Dynamical Networks

• Victor M. Preciado
• Supervisor
• Prof. George Verghese, LEES/LIDS
• Thesis Committee
• Prof. Pablo Parrilo, LIDS
• Prof. Devavrat Shah, LIDS
• May 16th, 2008

2
Introduction Complex Networks
Internet

• Generic features
• Large number of nodes
• Sparse connectivity
• Absence of symmetries

• Challenges
• Topology discovery
• Topology storage/retrieval
• Standard algorithms not an alternative

i
Ni
3
Main Objective
Stochastic Graph Modeling
Spectral Graph Theory
??

• TO Estimation of global structure and dynamics.

4
Spectral Graph Theory
1
2
4
3
Adjacency, A
5
Synchronization of Identical Oscillators

• Question Under what conditions do the
  oscillators synchronize?

6
Synchronization of Oscillators

• Master Stability Function Pecora and Carrol,
  98 Approach to decouple dynamics and topology.

Dynamics f(.)
7
Spectral Analysis of Complex Networks
Small/medium size topology
8
Outline of Thesis
3
Chapters 1. Introduction 2. Spectral Analysis of
Dynamical Processes in Networks
4
6
4
3. Stochastic Modeling of Large-Scale Complex
Networks
3
5
7
4. Spectral Analysis of Random Static Graphs
6
7
5. Estimation of Spectral Properties
6. Spectral Analysis of Evolving Networks
7. Analysis of Laplacian and Kirchhoff Matrices
2
2
8. Future Research
9
Random Models
10
Traditional Models
Erdös-Rényi Small-Worlds Preferential
Attachment (Watts and Strogatz)
(Barabasi and Albert)
11
Modeling Limitations

• Example
• High-speed backbone of an European ISP (Tiscali
  SpA)
• 161 high-speed router, 532 optical links

Not Poisson-like distribution!
12
Static Model with Prescribed Expected Degree
Distribution

• Modeling limitations
• Preferential attachment model is limited to
  power-law distributions
• Traditional static models limited to Poisson-like
  distributions

• Generalized static models Chung Lu
• Graph ensemble with prescribed expected degree
  distribution

13
Modeling ISP high-speed core
14
Asymptotic Spectral Moments

• Main Problem Determine the spectral eigenvalue
  distribution for very large (Chung-Lu) random
  networks

• Numerical Experiment Represent the histogram of
  eigenvalues for several random realizations of
  random graphs

• Limiting Spectral Density Analytical expression
  only possible for very particular cases.

15
Prior Work

• Prior work on CL model only for spectral radius
• Comments on what I achieve
• Highlights

16
Method of Moments

• Alternative Approach Characterize spectral
  distribution by its moments

• From algebraic graph theory
• where Ck denotes the set of closed walks of
  length k.

• In a random graph we have
• Each possible closed walk of length k is a
  probabilistic structure presenting a particular
  probability of existence
• The spectral moment is a random variable
  proportional to the number of existing walks

17
Spectral Moments of a Random Graph

• Analysis of expected spectral moments

• Strategy to compute the expected spectral
  moments
• Under some technical conditions, there is a very
  particular subset of closed walks, Dk in Ck, that
  dominates the probabilistic sum.

18
Polynomial Expressions for the Expected Spectral
Moments

• We codify each walk using (rooted plane) trees

• This codification allows us to count the dominant
  closed walks by counting trees

19
Symbolic Polynomials for Expected Spectral Moments

• Symbolic expressions
• Numerical verification 500 nodes random
  power-law,b2.5

20
What can we do with these moments?

• Techniques to extract spectral information
• Wigners high-order moment method
• Piecewise-linear reconstruction
• Optimal SDP bounds

21
Wigners High-Order Method

• Main Idea If we can upper-bound the rate of
  growth of the sequence of even spectral moments

we would have the following probabilistic upper
bound on the spectral support

• Numerical Example Erdös-Rényi graph
• Special case CL
• consistent with the support of Wigners
  semicircle law

22
PWL Spectrum Reconstruction

• We propose a piecewise linear reconstuction that
  preserves the truncated moment sequence

Knots xp are fixed Compute yp?

• We can fit a number of moments, with an
  appropriate number of affine pieces, by solving a
  linear system of equations

23
Stochastic Graph Models
24
Dynamically Evolving Models

• Explain the developmental process of a complex
  network.

• Barabási-Albert model Sequential addition of
  nodes with preferential attachment.

• Balls-and-bins model

25
Comments about the moments

• Apply again the method of moments
• Counting closed walks
• Order of magnitude hit by closed SAWs define
  SAWs
• TRY TO BE MORE EXPLICIT!!!

26
Mean-Field Analysis of Walks

• Our Problem Evolution of (self-avoiding) walks

• Generation of new walks

1. Initial network with n-1 nodes
27
Evolution of Walks

• Generic equation for evolution of closed SAWs

• Conjectures on weak correlations split
  expectations of products
• mean-field approximations replace quantities by
  expectations
• There is a differential equation that
  approximates the above discrete-time evolution

28
Solving the Evolution of Walks

• Exploiting recursive structure of the ODE, we
  reach

Eqn.
where the evolution of triangles (already
studied in the literature) provides us the
initial condition to this ODE.

• Solution

Eqn.

• Numerical details
• m2, n up to 104.
• qln for l3,4, and 5.

29
Further Matricial Representations
Adjacency
Static Models
Spectral Graph Analysis
Stochastic Network Modeling
Kirchhoff
Dynamic Models
Laplacian

• Tools to analyze low-order spectral moments of
  Laplacian and Kirchhoff matrices

30
L and K in words

• For L and K we have analytical expressions for
  the first three moments for any graph
• For CL model, we have the expected values for the
  first 3 moments
• EXAMPLE, triangular fitting (maybe new slide)

31
Algebraic Analysis of Kirchhoff Matrices

• Algebraic analysis using algebraic graph theory,
  we deduce the following expressions for the
  spectral moments of the Kirchhoff matrix
• Probabilistic analysis REMOVE THEOREM FORM

32
Numerical Results

• Power-law model by Chung
• Numerical and analytical moments (for a single
  realization)

where
Network description N500 Expected average
degree50 Max expected degree100 b3.0
33
Conclusion

• This thesis is devoted to the spectral analysis
  of several stochastic models of large-scale
  networks of relevance.
• We use the method of moments to analyze the
  expected eigenvalue distribution of static random
  graphs.
• We propose several methods to extract information
  about the spectral distribution from the sequence
  of spectral moments.
• We study the evolution of spectral properties in
  a dynamically evolving random graph.
• We derive closed-form expressions for the
  low-order spectral moments of the Laplacian and
  Kirchhoff matrices of random graphs.

34
Acknowledgements

• George C. Verghese
• Technical support Bill, Laura, Tushar, Faisal,
• Emotional support Zahi, Demba, Al, Ali,
• My parents, sisters, brother.
• My fiancee Farah.
• Finantial support La Caixa Foundation, DoD
  project,

35
Questions
36
Topological Metrics

• FORWARD TO THE POINT OF LAP AND KIRCHHOFF.
• Joint-degree distribution
• Graphical examples
• How to measure
• Real examples

37
Topological Metrics

• Distribution of triangles
• Graphical examples
• How to measure
• Real examples

38
Algebraic Analysis of Laplacian Matrices

• Algebraic analysis using algebraic graph theory,
  we deduce the following expressions for the
  spectral moments of the Laplacian matrix
• Probabilistic analysis

where
39
Structure of Our Approach
SWITCH