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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.

Performance analysis/evaluation in networking:

measure parameters hopefully predictive of performance.

Which are criticalnetwork parameters/metrics

that determinealgorithmic performance?

Important in network simulation and design.

Predictive of routing and searching performance

is conductance, expansion, spectral gap.

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?

Current Models for Internet Routing Topologies

focus on large degree-variance

Erdos-Renyi-like, Configurational :

Chung&Lu

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

Fabrikant,Koutsoupias,Papadimitriou

Barabasi&Albert

Bollobas&Riordan

The case of modeling the internet routing topology

Nodes are routers or Autonomous Systems

Two nodes connected by a link if they

are involved in direct exchange of traffic

Sparse small-world graphs

with large degree-variance

But are degrees the “right” parameter to measure?

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,

mixing rate, cover time.

Leighton-Rao

Jerrum-Sinclair

Broder-Karlin

How does the second eigenvalue

(more generally the principal eigenvalues)

scale as the size of the network increases?

This is also another point of view

of the small-world phenomenon

This also says that congestion

under link capacities scales smoothly

Second eigenvalue of internet is larger than that of random graphs

but spectral gap remains constant as number of nodes increases.

Gkantsidis,M,Zegura

M,Papadimitriou,Saberi

random graph

configurational model

Gkantsidis,M,Saberi

Open problem: Erdos-Renyi like, configurational models

which include spectral gap parameter?

Some evolutionary random graph models

may capture clustering

Growth & Preferential Attachment

One vertex at a time

New vertex attaches to existing vertices

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

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

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

Algorithms optimizing connectivity

How do you maintain a

P2P network with good

search performance ?

Lv&al

Chawathe&al

Gkantsidis&al

The case of Peer-to-Peer Networks

Distributed, decentralized

nnodes,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

Must maintain well connected topology,

e.g. a random graph, an expander

Jerrum-Sinclair

Broder-Karlin

P2P Network Topology Maintenance by Constant Randomization

Gnutella: constantly drops existing connections

and replaces them with new connections

random graph, expander

Theorem[Cooper, Frieze & Greenhill 04]:

The Markov chain corresponding to a 2-link switch on d-regular graphs is rapidly mixing.

Theorem[Feder, Guetz, M, Saberi 06]:

The Markov chain on d-regular graphs is rapidly mixing,

even under local 2-link switches or flips.

Question: How does the network “pick” a random 2-link switch?

In reality, the links involved in a switch are within constant distance.

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

?

?

?

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?

Algorithms developing topology awareness

Link Criticality

Boyd,Diaconis,Xiao

3$

local information

1$

7$

local information

local

information

2S

Generalized Search:

Link Criticality

A node has

a query and a budget

Subtract 1

from budget

Arbitrarily partition

the remaining budget

and forward

the parts to the neighbors

Boyd,Diaconis,Xiao

Gkantsidis,M,Saberi

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.

s.t.

Fastest Mixing Markov Chain

Boyd,Diaconis,Xiao

SDP formalization

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?

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