1 Algorithmic Performance in Complex Networks Milena Mihail Georgia Tech

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Algorithmic Performance in Complex Networks

Milena MihailGeorgia Tech.

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Outline

Metrics relevant to network function:

Expansion, Routing, Conductance, Searching Spectrum, in communication networks

Global Connectivity

Efficient maintenance of expansion

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Complex Networks

WWW 500K-3B

Internet Routing ASes: 900-15K Routers: 500-200K

P2P tens Ks-4M

Ad-hoc (wireless, mobile, sensor)

Gene-Protein Interaction

Scaling

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How does Algorithmic Performance Scale with Number of Nodes in a Complex Communication Network?

Route

Mechanism design

Efficient maintenance of metrics supporting the above

Search

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In general, between and

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 ?

How does Cover Time Scale?What algorithmic primitives can improve scaling?

Important in WWW Crawling.Important in Searching P2P.

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Demand: , uniform. What is load of max congested link, in optimal routing ?

star

expander

in general

How does Routing Congestion Scale on the Internet ?

Sparse power-law graphs ?

Important in economics.Networks with externalities.

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Demand: , uniform. What is load of max congested link, in optimal routing ?

star

expander

in general

How does Routing Congestion Scale on the AS Internet ?

Sparse scale-free graphs ?

Important in economics.Networks with externalities.

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Edge congestion under shortest path routingon the Internet graph.

Edge congestion under shortest path routingon a non blocking network(regular expander).

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How does Capacity/Throughput/Delay Scale on an Ad-Hoc Wireless Network?

Capacity of Wireless Networks, Gupta & Kumar, 2000Mobility Increases Capacity, Grossgaluser & Tse, 20001Capacity, Delay and Mobility in Wireless Networks, Bansal & Liu 2003Throughput-delay Trade-off in Wireless Networks, El Gamal, Mammen, Prabhakar & Shah 2004

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Outline



Global Connectivity


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Conductance

Sparse graphs,Demand ~ degrees

S S

Conductance and Congestion by Leighton-Rao 95

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Macroscopic Models for Scale-Free Graphs

One vertex at a time

New vertex attaches to existing vertices

EVOLUTIONARY : Growth & Preferential Attachment

Simon 55,Barabasi-Albert 99, Kumar et al 00, Bollobas-Riordan 01, Bollobas-Riordan-Spencer-Tusnady 01.

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STRUCTURAL , aka CONFIGURATIONAL MODEL

Given

Choose random perfect matching over

minivertices

“Random” graph with “power law” degree sequence.

Bollobas 80s, Molloy&Reed 90s, Aiello-Chung-Lu 00s, Sigcomm/Infocom 00s

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STRUCTURAL MODEL

Given


minivertices

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Given


STRUCTURAL MODEL

minivertices

edge multiplicity O(log n) , a.s. connected, a.s.

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Theorem [MM, Papadimitriou, Saberi 03]: For a random graph grown with preferential attachment with , , a.s.

Theorem [Gkantsidis, MM, Saberi 03]: For a random graph in the structural model arising from degree sequence ,

, a.s.

Bounds on Conductance

Previously: Cooper & Frieze 02

Independent: Chung,Lu,Vu 03

Technique: Probabilistic Counting Arguments & Combinatorics.Difficulty: Non homogeneity in state-space, Dependencies.

for a different structural random graph model

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Worst case is when all vertices have degree 3.

Structural Model, Proof Idea: Difficulty: Non homogeneity in state-space

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Growth with Preferential Connectivity Model, Proof Idea:

Difficulty:Arrival Time Dependencies

Shifting Argument

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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.

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.

Each vertex with degree in the network coreserves customers from the network periphery.

Note: Why is demand ?

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Edge congestion under shortest path routingon the Internet graph.

Edge congestion under shortest path routingon a non blocking network(regular expander).

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Conductance and Spectrum

Theorem: Eigenvalue separation

for stochastic normalization of adjacency matrix

follows by [Jerrum-Sinclair 88]

Recall: Stochastic normalizations of adjacency matrices of undirected graphs, P has n real eigenvalue-eigenvector pairs:

related to “bad cuts”

[Alon 86]

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AS

Gkantsidis, MM,Saberi ‘03

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[Gkantsidis, MM,Saberi ’03]

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[Gkantsidis, MM,Saberi ’03]

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Spectrum, Mixing and Cover Times

Rapid Mixing of Random Walk

“mixing” in

Cover Time[Broder Karlin 88]

for any constant

Simpler, by mixing and coupon collection

for

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can discover vertices

in steps.

Cover Time with Look-Ahead One In the structural model

withTheorem [MM,Saberi,Tetali 04]:

Proof

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Proof

In the structural model

with

Cover Time with Look-Ahead OneTheorem [MM,Saberi,Tetali 04]:

can discover vertices

in steps.

Adamic et al ’02 Chawathe et al 03Gkanstidis, MM, Saberi 05,Sarshar et al 05

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HYBRID SEARCH SCHEMES: Take Advantage of Local Information to Improve Global Performance

Flooding

Random Walk

Edge Criticality

Hybrid Search Schemes

Gkantsidis, MM, Saberi 04Boyd, Diaconis, Xiao 04

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Outline



Global Connectivity


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P2P Network Topology Problem: A distributed resource efficient algorithm to dynamically maintain an expander.

?

?

?

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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.

?

?

?

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?

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Theorem[Gkanstidis,MM,Saberi 04]: Construct a graph on n vertices with constant overhead per node additionwhere, for some constants a and b, every set of at least bn vertices has expansion aand where sets of size O( log n) also have constant expansion.

Proof Technique: Taking continious samples from a Markov chain achievesChernoff-like bounds [Ajtai,Komlos,Szemeredi 88, Zuckerman & Impagliazzo 89, Gillman 95]

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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.

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Complex Networks

WWW 500K-3B

Internet Routing ASes: 900-15K Routers: 500-200K

P2P tens Ks-4M

Ad-hoc (wireless, mobile, sensor)

Gene-Protein Interaction

Scaling

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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 indicationof evolutionary fitness.

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For biologists, p is an indicationof evolutionary fitness.

as a function of p, in experiment, MM & Zia ‘05

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Summary



Global Connectivity


Reverse engineering in bioinformatics

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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.

Documents

1 Algorithmic Performance in Complex Networks Milena Mihail Georgia Tech