Algorithmic Performance
in Complex Networks
Milena Mihail
Georgia Tech.
1
Outline
Metrics relevant to network function:
Expansion,
Conductance,
Spectrum,
Routing,
Searching
in communication networks
Global Connectivity
Efficient maintenance of expansion
2
Complex Networks
Scaling
WWW
500K-3B
Internet Routing
ASes: 900-15K
Routers: 500-200K
P2P
tens Ks-4M
Ad-hoc (wireless, mobile, sensor)
Gene-Protein Interaction
3
How does Algorithmic Performance
Scale with Number of Nodes
in a Complex Communication Network?
Search
Route
Mechanism design
Efficient maintenance of metrics supporting the above
4
How does Cover Time Scale?
What algorithmic primitives can improve scaling?
Random walk on
nodes.
What is the expected time
to visit all the nodes ?
What is the expected time
to visit a constant fraction of the
nodes ?
Important in WWW Crawling.
Important in Searching P2P.
In general, between
and
5
How does Routing Congestion Scale on
the Internet ?
Demand:
, uniform.
What is load of max
congested link, in
optimal routing ?
star
expander
Sparse power-law graphs ?
Important in economics.
Networks with externalities.
in general
6
How does Routing Congestion Scale on
the AS Internet ?
Demand:
, uniform.
What is load of max
congested link, in
optimal routing ?
star
expander
Sparse scale-free graphs ?
Important in economics.
Networks with externalities.
in general
7
Edge congestion under
shortest path routing
on the Internet graph.
Edge congestion under
shortest path routing
on a non blocking network
(regular expander).
8
How does Capacity/Throughput/Delay Scale
on an Ad-Hoc Wireless Network?
Capacity of Wireless Networks, Gupta & Kumar, 2000
Mobility Increases Capacity, Grossgaluser & Tse, 20001
Capacity, Delay and Mobility in Wireless Networks, Bansal & Liu 2003
Throughput-delay Trade-off in Wireless Networks,
El Gamal, Mammen, Prabhakar & Shah 2004
9
Outline
Metrics relevant to network function:
Expansion,
Conductance,
Spectrum,
Routing,
Searching
in communication networks
Global Connectivity
Efficient maintenance of expansion
10
Conductance
S
Conductance and Congestion
S
by Leighton-Rao 95
Sparse graphs,
Demand ~ degrees
11
Macroscopic Models for Scale-Free Graphs
EVOLUTIONARY : Growth & Preferential Attachment
One vertex at a time
New vertex attaches to
existing vertices
Simon 55,Barabasi-Albert 99, Kumar et al 00,
Bollobas-Riordan 01,
Bollobas-Riordan-Spencer-Tusnady 01.
12
STRUCTURAL , aka CONFIGURATIONAL MODEL
“Random” graph with “power law” degree sequence.
Given
Choose random perfect matching over
minivertices
Bollobas 80s, Molloy&Reed 90s, Aiello-Chung-Lu 00s, Sigcomm/Infocom 00s 13
STRUCTURAL MODEL
Given
Choose random perfect matching over
minivertices
14
STRUCTURAL MODEL
Given
edge multiplicity
O(log n) , a.s.
Choose random perfect matching over
connected, a.s.
minivertices
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Bounds on Conductance
Technique: Probabilistic Counting Arguments
& Combinatorics.
Difficulty: Non homogeneity in state-space,
Dependencies.
Theorem [MM, Papadimitriou, Saberi 03]: For a random graph grown with
preferential attachment with
,
, a.s.
Previously:
Cooper & Frieze 02
Theorem [Gkantsidis, MM, Saberi 03]: For a random graph in the structural
model arising from degree sequence
,
, a.s.
Independent:
Chung,Lu,Vu 03
for a different structural random graph model16
Structural Model, Proof Idea:
Difficulty: Non homogeneity in state-space
Worst case is when all vertices have degree 3.
17
Growth with Preferential Connectivity Model, Proof Idea:
Difficulty:
Arrival Time Dependencies
Shifting Argument
18
Theorem [MM, Papadimitriou, Saberi 03]: For a random graph grown with
preferential attachment with
there is a poly time computable flow
that routes demand
between all vertices i and j with max link
congestion
, a.s.
Theorem [Gkantsidis,MM, Saberi 03]: For a random graph in the structural
model arising from degree sequence
there is a poly time computable flow that routes demand
between all
vertices i and j with max link congestion
a.s.
Note: Why is demand
?
Each vertex with degree
in the network core
serves
customers from the network periphery.
19
Edge congestion under
shortest path routing
on the Internet graph.
Edge congestion under
shortest path routing
on a non blocking network
(regular expander).
20
Conductance and Spectrum
Theorem:
Eigenvalue separation
for stochastic normalization of adjacency matrix
follows by
[Alon 86]
[Jerrum-Sinclair 88]
Recall: Stochastic normalizations of adjacency matrices of undirected graphs,
P has n real eigenvalue-eigenvector pairs:
related to “bad cuts”
21
AS
Gkantsidis, MM,
Saberi ‘03
22
23
[Gkantsidis, MM,
Saberi ’03]
24
[Gkantsidis, MM,
Saberi ’03]
25
Spectrum, Mixing and Cover Times
Rapid Mixing of Random Walk
for
“mixing” in
Cover Time
[Broder Karlin 88]
for any constant
Simpler, by mixing and coupon collection
26
Cover Time with Look-Ahead One
Theorem [MM,Saberi,Tetali 04]:
In the structural model
with
can discover
Proof
in
vertices
steps.
27
Cover Time with Look-Ahead One
Theorem [MM,Saberi,Tetali 04]:
In the structural model
with
can discover
vertices
in
steps.
Proof
Adamic et al ’02
Chawathe et al 03
Gkanstidis, MM, Saberi 05,
Sarshar et al 05
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HYBRID SEARCH SCHEMES: Take Advantage of Local Information
to Improve Global Performance
Random Walk
Edge Criticality
Hybrid Search Schemes
Flooding
Gkantsidis, MM, Saberi 04
Boyd, Diaconis, Xiao 04
29
Outline
Metrics relevant to network function:
Expansion,
Conductance,
Spectrum,
Routing,
Searching
in communication networks
Global Connectivity
Efficient maintenance of expansion
30
P2P Network Topology Problem: A distributed resource
efficient algorithm to dynamically maintain an expander.
?
?
?
31
P2P Network Topology Construction by Random Walk
?
?
?
Theorem [Law & Siu ‘03]: Construct a constant expander on n vertices
with overhead O( log n) per node addition.
32
P2P Network Topology Construction by Random Walk
33
P2P Network Topology Construction by Random Walk
34
P2P Network Topology Construction by Random Walk
35
P2P Network Topology Construction by Random Walk
?
?
?
Theorem[Gkanstidis,MM,Saberi 04]: Construct a graph on n vertices
with constant overhead per node addition
where, for some constants a and b, every set of at least bn vertices has expansion a
and where sets of size O( log n) also have constant expansion.
Proof Technique: Taking continious samples from a Markov chain achieves
Chernoff-like bounds [Ajtai,Komlos,Szemeredi 88, Zuckerman & Impagliazzo 89,
36
Gillman 95]
37
P2P Network Topology Maintenance by 2-Link Switches
Theorem [Cooper, Frieze & Greenhill 04]:
The corresponding random walk on d-regular graphs is rapidly mixing.
Question: How does the network “pick” a random 2-Link Switch?
In reality, the links involved in a switch are within constant distance.
38
Complex Networks
Scaling
WWW
500K-3B
Internet Routing
ASes: 900-15K
Routers: 500-200K
P2P
tens Ks-4M
Ad-hoc (wireless, mobile, sensor)
Gene-Protein Interaction
39
Gene-Protein Interaction Networks
Copying Random Graph Model:
a new node v attaches with d links
as follows:
(1)Picks a random node u
(2) For i:=1 to d
with probability p,
v copies the ith link of u
with probability 1-p , v attaches
to a uniformly random node.
The exponent of the resulting
Power-law graph is a function of p.
[Kumar et al 01, Chung & Lu 04]
For biologists, p is an indication
of evolutionary fitness.
40
as a function of p,
in experiment, MM & Zia ‘05
For biologists, p is an indication
of evolutionary fitness.
41
Summary
Metrics relevant to network function:
Expansion,
Conductance,
Spectrum,
Routing,
Searching
in communication networks
Global Connectivity
Efficient maintenance of expansion
Reverse engineering in bioinformatics
42
References
On the Eigenvalue Powerlaw, M. Mihail and C. Papadimitriou, RANDOM 02.
Spectral Analysis of Internet Topologies, C. Gkantsidis, M. Mihail
and E. Zegura, INFOCOM 03.
Conductance and Congestion in Powerlaw Graphs, C. Gkantsidis, M. Mihail
and A. Saberi, SIGMETRICS 03.
On Certain Connectivity Properties of the Internet Topology, M. Mihail,
C. Papadimitriou and A. Saberi, FOCS 03.
On the Random Walk Method for P2P Networks, C. Gkantsidis, M. Mihail
and A. Saberi, INFOCOM 05.
Hybrid Search Schemes in P2P Networks, C. Gkantsidis, M. Mihail
and A. Saberi, INFOCOM 05.
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