complex networks: connectivity and functionality
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Complex Networks: Connectivity and Functionality. Milena Mihail Georgia Tech. Search and routing networks, like the WWW , the internet , P2P networks, ad-hoc ( mobile, wireless, sensor ) networks are pervasive and scale at an unprecedented rate. - PowerPoint PPT PresentationTRANSCRIPT
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Complex Networks:Connectivity and Functionality
Milena MihailGeorgia Tech.
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Search and routing networks, like the WWW, the internet, P2P networks, ad-hoc (mobile, wireless, sensor) networks are pervasive and scale at an unprecedented rate.
Performance analysis/evaluation in networking:measure parameters hopefully predictive of performance.
Important in network simulation and design.
Which are critical network parameters/metrics that determine algorithmic performance?
Predictive of routing and searching performanceis conductance, expansion, spectral gap.
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How can network models capture the parameters/metrics that are critical in network performance?
Can we design network algorithms/protocolsthat optimize these critical network parameters?
This talk: The case ofinternet routing topology
This talk: The case of P2P networks
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The case of modeling the internet routing topology
Nodes are routers or Autonomous Systems
Two nodes connected by a link if theyare involved in direct exchange of traffic
Sparse small-world graphs with large degree-variance
But are degrees the “right” parameter to measure?
Current Models for Internet Routing Topologies focus on large degree-variance
Erdos-Renyi-like, Configurational : A random graph with given degrees Evolutionary, macroscopic and microscopic :
The graph grows one vertex at a time and attaches preferentially to degrees or according to some optimization criterion
Chung&Lu
Barabasi&AlbertBollobas&Riordan
Fabrikant,Koutsoupias,Papadimitriou
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An important metric: Conductance and the second eigenvalue of the stochastic normalization of the adjacency matrix characterize routing congestion under link capacities, mixing rate, cover time.
Leighton-Rao
Jerrum-Sinclair
How does the second eigenvalue (more generally the principal eigenvalues) scale as the size of the network increases?
Broder-Karlin
computationally softMatlab does 1-2M nodesparse graphs
6Open problem: Erdos-Renyi like, configurational models which include spectral gap parameter?
This is also another point of view of the small-world phenomenon
random graphconfigurational model
Gkantsidis,M,Zegura
M,Papadimitriou,Saberi
Gkantsidis,M,Saberi
Second eigenvalue of internet is larger than that of random graphsbut spectral gap remains constant as number of nodes increases.
This also says that congestionunder link capacities scales smoothly
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Some evolutionary random graph models may capture clustering
One vertex at a time
New vertex attaches to existing vertices
Growth & Preferential Attachment
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Open Problem: characterize clustering as a function of model parameter
ie, indicate which parameter ranges are important in simulations
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plots as number of nodes increases
M,Saberi,Papadimitriou
Flaxman,Frieze,Vera
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real network
random graph,evolutionary model
random graph,configurational model
Other discrepancies of random graph models from real internet topologies:
high degree nodes away from “network core”
what do internet topologies “optimize” ?
Li, Alderson, Willinger, Doyle
high degrees mostly connected to low degrees“core” of low degrees
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Given total length l and n random points in a metric spaceconstruct a graph with total link length l that - maximizes spectral gap, conductance - minimizes congestion under node capacities
Open Problem: Research direction:Algorithms improving congestionconductance and spectral gap
Boyd&Saberi
Rao&Vazirani
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Algorithms optimizing connectivity
How do you maintain aP2P network with good search performance ?
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The case of Peer-to-Peer Networks
n nodes, d-regular graph
each node has resources O(polylogn)and knows a constant size neighborhood
Distributed, decentralized
Search for content, e.g. by flooding or random walk
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Must maintain well connected topology, e.g. a random graph, an expander
Chawathe&alGkantsidis&al
Lv&al
Jerrum-Sinclair
Broder-Karlin
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Theorem [Feder, Guetz, M, Saberi 06]: The Markov chain on d-regular graphs is rapidly mixing, even under local 2-link switches or flips.
P2P Network Topology Maintenance by Constant Randomization
Theorem [Cooper, Frieze & Greenhill 04]: The Markov chain corresponding to a 2-link switch 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.
random graph, expander Gnutella: constantly drops existing connections and replaces them with new connections
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Space of d-regular graphsgeneral 2-link switch Markov chain
Space of connected d-regular graphs local Flip Markov chain
Define a mapping from to such that
(a) (b) each edge in maps to a path of constant length in
The proof is a Markov chain comparison argument
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Question: How do we add new nodes with low network overhead?
Question: How do we delete nodes with low network overhead?
??
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Gkantsidis,M,Saberi
Padurangan,Raghavan,UpfalLaw,Siu
Ajtai,Komlos,SzemerediImpagliazzo,Zuckerman
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Link Criticality
Algorithms developing topology awareness
Boyd,Diaconis,Xiao
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Generalized Search:
7$
3$
2S
1$local information
local inform
ation
local information
A node has a query and a budget
Arbitrarily partition the remaining budgetand forward the parts to the neighbors
Subtract 1 from budget
Link Criticality
Gkantsidis,M,SaberiBoyd,Diaconis,Xiao
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Fastest Mixing Markov Chain Boyd,Diaconis,Xiao
s.t.
Let be a graph.Assign symmetric transition probabilities to links in (and self loops)so that the resulting matrix is stochasticand the second in absolute value largest eigenvalue is minimized.
SDP formalization
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Fastest Mixing Markov Chain Subgradient Algorithm
is some vector on of initial transition probabilities
is the eigenvector corresponding to second in absolute value largest eigenvalue
is a vector on with
repeat
subgradient step
projection to feasible subspace
Open Question: Is there a decentralized implementation or algorithm?
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How does Capacity/Throughput/Delay Scale?
Mobility Increases Capacity, Grossglauser & Tse, 2001Capacity, Delay and Mobility in Wireless Networks, Bansal & Liu 2003Throughput-delay Trade-off in Wireless Networks, El Gamal, Mammen, Prabhakar & Shah 2004
The Case of Ad-Hoc Wireless Networks
Capacity of Wireless Networks, Gupta & Kumar, 2000 Is there a connection with Lipton & Tarjan’s separators for planar graphs?