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“Expansion” in Power Law
and Scale Free Graphs
Milena Mihail
Georgia Tech
with Christos Gkantsidis, Christos Papadimitriou
and Amin Saberi
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Graphs with Skewed Degree Sequences
Communication Networks
Metabolic Networks
This Talk: Algorithmic Issues,
“Expansion”, spectral gap determine
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performace of key algorithms.
How does Congestion Scale?
Demand: n2, uniform.
What is load of max
congested link, in
optimal routing ?
Sprint
ISPs:
900-14K
Routers: 500-200K
500K-3B
AT&T WWW:
P2P:
hundred Ks
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Real Internet Topologies
E[degree]~3
Degrees not
Concentrated
around mean
Not Erdos-Renyi
CAIDA http://www.caida.org
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Degree-Frequency Power Law
Models by Kumar et al 00,
x
Bollobas et al 01,
x
Fabrikant et al 02
frequency
Erdos-Renyi
sharp
concentration
1
2
3
4 5
E[d] = const., but
No sharp concentration
10
degree
100
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Power Laws
[Interdomain Routing: Faloutsos et al 99]
[WWW: Kumar et al 99, Barabasi-Albert 99]
Degree-Frequency
Rank-Degree
Eigenvalues
(Adjacency Matrix)
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Models for Power Law Graphs
EVOLUTIONARY
Macroscopic : Growth & Preferential Attachment
Simon 55, Barabasi-Albert 99, Kumar et al 00,
Bollobas-Riordan 01.
Microscopic : Growth & Multiobjective Optimization,
QoS vs Cost
Fabrikant-Koutsoupias-Papadimitriou 02.
STRUCTURAL (aka CONFIGURATIONAL)
“Random” graph with “power law” degree sequence.
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Structural Random Graph Model
Given
Choose random perfect matching over
minivertices
Molley&Reed 95-98, Aiello,Chung,Lu 00, Tagmunarunkit et al 02
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Congestion in the “Core”
Theorem [Gkantsidis,MM, Saberi 02]:
For a random graph arising from
degree sequence
O(n½ ) ≥ d1≥d2≥…≥dn ≥3
there is a flow that routes demand
di * dj between all vertices i and j
with max link congestion O(n log2n)
almost surely.
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Proof :
Step 1 : Approximation algorithms for
multicommodity flow reduce
congestion to conductance
(special case of sparsest cut).
Step 2 : Bound conductance - MAIN LEMMA.
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Proof, Step 1 : Reduce to Conductance
By Maximum multi-commodity flow, [Leighton & Rao 95]
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Proof, Step 2 :
Main Lemma [Gkantsidis,MM, Saberi 02]:
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Proof of MAIN LEMMA:
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Proof of MAIN LEMMA:
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Proof of MAIN LEMMA:
Stirling
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Proof of MAIN LEMMA:
BIG
SMALL
ignore
Stirling
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In an Evolutionary Model ?
Growth with Pref. Attachment
One vertex at a time
New vertex attaches to d
existing vertices
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Reduction to Random Matching
[Bollobas & Riordan 01]
t=5
t=1
t=2
t=3
t=4
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Reduction to Random Matching
[Bollobas & Riordan 01]
t=5
t=1
t=2
t=3
t=4
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Reduction to Random Matching
[Bollobas & Riordan 01]
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In an Evolutionary Model ?
Growth with Pref. Attachment
Theorem [MM, Saberi 02]:
For a graph grown with preferential attachment with d ≥ 3
there is a flow that routes demand di * dj between all vertices
i and j with max link congestion O(n log n) almost surely.
Main Lemma:
almost surely.
Open Question: Analyze a graph grown one vertex or edge
at a time, where with probability
a new vertex comes
and attaches preferentially and with probability
a new
edge grows preferentially between existing vertices.
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Spectral Implication
Theorem: Eigenvalue separation
for stochastic normalization of adjacency matrix [Alon 85,
Jerrum&Sinclair 88]
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Spectra of “Real” Internet
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Spectral Implications
Theorem: Eigenvalue separation
for stochastic normalization of adjacency matrix [Alon 85,
Jerrum&Sinclair 88]
On the eigenvalue Power Law
[M.M. & Papadimitriou 02]
Rank-Degree
Eigenvalues
(Adjacency Matrix)
Using matrix perturbation
[Courant-Fisher theorem]
in a sparse random graph
model
.
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Theorem :
[M.M. & Papadimitriou 02]
Wwith probability at least
Ffor large enough
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Proof : Step 1. Decomposition
LR = Vertex Disjoint Stars - LR-extra
LL
RR
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Proof: Step 2: Vertex Disjoint Stars
Degrees of each Vertex Disjoint Stars
Sharply Concentrated around its Mean d_i
Hence Principal Eigenvalue Sharply Concentrated around
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Proof: Step 3: LL, RR, LR-extra
LR-extra has max degree
LL has
edges
RR has max degree
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Proof: Step 3: LL, RR, LR-extra
LR-extra has max degree
LL has
edges
RR has max degree
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Proof: Step 4: Matrix Perturbation Theory
Vertex Disjoint Stars
have principal
eigenvalues
All other parts have max eigenvalue
QED
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Implication for Info Retrieval
Term-Norm Distribution
Problem :
Spectral filtering,
without preprocessing,
reveals only
the large degrees.
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Implication for Info Retrieval
Term-Norm Distribution
Problem :
Spectral filtering,
without preprocessing,
reveals only
the large degrees.
Local information.
No “latent semantics”.
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Implication for Information Retrieval
Term-Norm Distribution
Problem :
Application specific
preprocessing (normalization
of degrees) reveals clusters:
WWW: related to searching,
Kleinberg 97
IR, collaborative filtering, …
Internet: related to
congestion, Gkantsidis et al 02
Open : Formalize “preprocessing”.
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Further Directions:
Generalize theory of
Regular Expanders
• Routing
Integral paths?
Peleg&Upfal’88 …
Broder,Frieze&Upfal’01
Short paths?
Kleinberg&Rubinfeld’97
Reliability?
•Cover time?
(Experimental work: Gkantsidis, MM, Zegura 02.)
•Hitting time?
•Planted model?
Related to Crawling
Related to Searching
Information Retrieval
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Metabolic Networks
Statistics of fixed size subgraphs?
Related to “motifs” in metabolic networks.
Model (explain) heavy tailed statistics in
noncoding part of DNA?
Related to stages of species evolution.
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Evaluation of Synthetic Topology
Generators
Core of the Network
Entire Topology
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