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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
Metrics relevant to network function:
Expansion, Routing, Conductance, Searching Spectrum, in communication networks
Global Connectivity
Efficient maintenance of expansion
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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
Choose random perfect matching over
minivertices
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Given
Choose random perfect matching over
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
Metrics relevant to network function:
Expansion, Routing, Conductance, Searching Spectrum, in communication networks
Global Connectivity
Efficient maintenance of expansion
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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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P2P Network Topology Construction by Random Walk
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P2P Network Topology Construction by Random Walk
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P2P Network Topology Construction by Random Walk
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P2P Network Topology Construction by Random Walk
?
?
?
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
40
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
Metrics relevant to network function:
Expansion, Routing, Conductance, Searching Spectrum, in communication networks
Global Connectivity
Efficient maintenance of expansion
Reverse engineering in bioinformatics
43
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.