1

Complex Networks:Connectivity and Functionality

Milena MihailGeorgia Tech.

2

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.

3

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

4

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

5

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

7

Some evolutionary random graph models may capture clustering

One vertex at a time

New vertex attaches to existing vertices

Growth & Preferential Attachment

8

Open Problem: characterize clustering as a function of model parameter

ie, indicate which parameter ranges are important in simulations

?

plots as number of nodes increases

M,Saberi,Papadimitriou

Flaxman,Frieze,Vera

9

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

10

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

11

Algorithms optimizing connectivity

How do you maintain aP2P network with good search performance ?

12

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

?

Must maintain well connected topology, e.g. a random graph, an expander

Chawathe&alGkantsidis&al

Lv&al

Jerrum-Sinclair

Broder-Karlin

13

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

14

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

15

Question: How do we add new nodes with low network overhead?

Question: How do we delete nodes with low network overhead?

?

??

Gkantsidis,M,Saberi

Padurangan,Raghavan,UpfalLaw,Siu

Ajtai,Komlos,SzemerediImpagliazzo,Zuckerman

16

Link Criticality

Algorithms developing topology awareness

Boyd,Diaconis,Xiao

17

Generalized Search:

7$

3$

2S

1$local information

local in

formatio

n

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

18

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

19

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?

20

How does Capacity/Throughput/Delay Scale?

Mobility Increases Capacity, Grossgaluser & 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?