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