routing in poisson small-world networks a. j. ganesh microsoft research, cambridge joint work with...

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Routing in Poisson Routing in Poisson small-world small-world networks networks A. J. Ganesh A. J. Ganesh Microsoft Research, Cambridge Microsoft Research, Cambridge Joint work with Moez Draief Joint work with Moez Draief

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Page 1: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Routing in Poisson Routing in Poisson small-world networkssmall-world networks

A. J. GaneshA. J. Ganesh

Microsoft Research, CambridgeMicrosoft Research, Cambridge

Joint work with Moez DraiefJoint work with Moez Draief

Page 2: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

What is a small world network?What is a small world network?

Milgram (1967)Milgram (1967)• Sent letters to various people in the US Sent letters to various people in the US

addressed to targets in Bostonaddressed to targets in Boston• Demographic information about target Demographic information about target

provided: name, address, occupationprovided: name, address, occupation• Letters had to be forwarded to target via Letters had to be forwarded to target via

contacts known on first-name basiscontacts known on first-name basis• Average length of chain for successful Average length of chain for successful

delivery was found to be six.delivery was found to be six.

Page 3: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Modelling social networksModelling social networks

Random graphs have small diameterRandom graphs have small diameter• Diameter of Erdos-Renyi random graph Diameter of Erdos-Renyi random graph

on n nodes is O(log n) at connectivity on n nodes is O(log n) at connectivity thresholdthreshold

• Similar results for power-law random Similar results for power-law random graphsgraphs

But not a good model of social But not a good model of social networksnetworks• Social networks are more clusteredSocial networks are more clustered

Page 4: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Small-world network modelsSmall-world network models

Watts & StrogatzWatts & Strogatz• Lattice plus random shortcuts (uniform)Lattice plus random shortcuts (uniform)

Benjamini & BergerBenjamini & Berger• d-dimensional lattice, edge (x,y) present d-dimensional lattice, edge (x,y) present

with probability |x-y|with probability |x-y|--rr

Coppersmith et al.Coppersmith et al.• Diameter is O(log n) if r=dDiameter is O(log n) if r=d• Polynomial in n if r>2dPolynomial in n if r>2d

Page 5: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Outline of talkOutline of talk

Background: can short paths be Background: can short paths be found using local information?found using local information?• Kleinberg, Franceschetti & MeesterKleinberg, Franceschetti & Meester

Model: Poisson point process with Model: Poisson point process with local links and shortcutslocal links and shortcuts

ResultsResults Open problemsOpen problems

Page 6: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Kleinberg: can short paths be Kleinberg: can short paths be found?found?

2-dimensional lattice on N2-dimensional lattice on N22 points points Each node has q shortcutsEach node has q shortcuts Each shortcut from x is incident on y Each shortcut from x is incident on y

with probability proportional to with probability proportional to d(x,y)d(x,y)-r-r, independent of others, independent of others

Need to route a message from Need to route a message from source s to destination t using a source s to destination t using a decentralised algorithmdecentralised algorithm

Page 7: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

What is a decentralised algorithm?What is a decentralised algorithm?

Each node knows co-ordinates of its Each node knows co-ordinates of its shortcut contacts & of destination tshortcut contacts & of destination t

Suppose algorithm has currently Suppose algorithm has currently visited nodes xvisited nodes x00,…,x,…,xkk

Next node to visit must be chosen Next node to visit must be chosen from among contacts (lattice or from among contacts (lattice or shortcut) of these nodes.shortcut) of these nodes.

Page 8: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

r=2: Greedy routingr=2: Greedy routing

At each node visited by algorithmAt each node visited by algorithm• Choose shortcut contact if it is closer to Choose shortcut contact if it is closer to

destinationdestination• Else choose lattice contact which is Else choose lattice contact which is

closercloser Theorem: Number of hops to Theorem: Number of hops to

destination is O(logdestination is O(log22N)N) Key idea: Find good shortcut that Key idea: Find good shortcut that

halves distance every O(log N) stepshalves distance every O(log N) steps

Page 9: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

r r ≠ 2: Impossibility result≠ 2: Impossibility result

r≠2: No decentralised algorithm can r≠2: No decentralised algorithm can find path shorter than a polynomial find path shorter than a polynomial in Nin N

Reason:Reason:• Short-cuts lack range if r>2Short-cuts lack range if r>2• Short-cuts spread too uniformly if r<2, Short-cuts spread too uniformly if r<2,

can’t close in on target (using only local can’t close in on target (using only local information)information)

Page 10: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Continuum modelContinuum model

Franceschetti and MeesterFranceschetti and Meester Each point in plane has shortcut to Each point in plane has shortcut to

other points according to an other points according to an inhomogeneous Poisson processinhomogeneous Poisson process

Intensity at distance x proportional to Intensity at distance x proportional to xx-r-r (not integrable) (not integrable)

Objective is to deliver message from Objective is to deliver message from s to s to neighbourhood of t neighbourhood of t

Page 11: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

ResultsResults

If r=2, greedy algorithm has If r=2, greedy algorithm has expected expected delivery time delivery time

O(log d(s,t) + log(1/O(log d(s,t) + log(1/)))) If r>2, any decentralised algorithm If r>2, any decentralised algorithm

needs at least d(s,t)needs at least d(s,t)ββ steps steps If r<2, any decentralised algorithm If r<2, any decentralised algorithm

needs at least (1/needs at least (1/))ββ steps steps

Page 12: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Poisson small-world modelPoisson small-world model

Nodes:Nodes: located at points of unit rate located at points of unit rate Poisson point process on square of Poisson point process on square of area n.area n.

Local links:Local links: to all other nodes within to all other nodes within distance distance √c log(n)√c log(n)

Shortcuts:Shortcuts: qq per node, in expectation per node, in expectation Probability of shortcut to node at Probability of shortcut to node at

distance d: distance d: c(q,n) dc(q,n) d-r-r

Page 13: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

RemarksRemarks

For sufficiently large c>0, graph For sufficiently large c>0, graph formed by local links alone is formed by local links alone is connected.connected.

Hence, message can always be Hence, message can always be routed in routed in O(√n/log(n))O(√n/log(n)) hops. hops.

Do shortcuts help us do better?Do shortcuts help us do better? Can we route in Can we route in polylog(n)polylog(n) hops? hops?

Page 14: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Results: r=2Results: r=2

r=2:r=2: there is a decentralised there is a decentralised algorithm that can route a message algorithm that can route a message between any pair of nodes in between any pair of nodes in O(logO(log22n)n) hops, with high probability, hops, with high probability, for sufficiently large c>0 and any for sufficiently large c>0 and any q>0.q>0.

Page 15: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Results: rResults: r≠2≠2

r<2:r<2: Any decentralised routing Any decentralised routing algorithm needs more than algorithm needs more than nn hops hops on average, for any on average, for any < (2-r)/6 < (2-r)/6

r>2: r>2: Any decentralised routing Any decentralised routing algorithm needs more than algorithm needs more than nn hops hops on average, for any on average, for any < (r-2)/2(r-1) < (r-2)/2(r-1)

Page 16: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Algorithm for r=2Algorithm for r=2

At each hop, algorithm maintains a At each hop, algorithm maintains a `radius’ `radius’ , initialised to d(s,t)., initialised to d(s,t).

If current node x has shortcut to aIf current node x has shortcut to a node in circle of radius node in circle of radius /2 centred at /2 centred at t, message is delivered to this node.t, message is delivered to this node.

Else, it is delivered to one of the local Else, it is delivered to one of the local contacts of x which is closer to tcontacts of x which is closer to t

If d(x,t)<If d(x,t)</2, /2, is updated to is updated to /2/2

Page 17: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

In pictures + modificationIn pictures + modification

Page 18: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Sketch proof: r=2Sketch proof: r=2

If c large enough, every node x has a local If c large enough, every node x has a local contact which is closer to tcontact which is closer to t

P(good shortcut) depends on number of P(good shortcut) depends on number of nodes in annulusnodes in annulus• It is O(1/log n) if number of nodes is large It is O(1/log n) if number of nodes is large

enoughenough• Probability that number of nodes is small is Probability that number of nodes is small is

negligiblenegligible Hence, good shortcut found after Hence, good shortcut found after

geometric O(log n) stepsgeometric O(log n) steps

Page 19: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Sketch proof: r<2Sketch proof: r<2

Suppose algorithm finds a path with Suppose algorithm finds a path with fewer than nfewer than n hops hops

There has to be at least one shortcut There has to be at least one shortcut which takes path into circle of radius which takes path into circle of radius nn++ centred at t centred at t

P(shortcut between u and v) is small P(shortcut between u and v) is small for any u,v.for any u,v.

Very unlikely to find shortcut into this Very unlikely to find shortcut into this circle, by union bound.circle, by union bound.

Page 20: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Sketch proof: r>2Sketch proof: r>2

Long-range contacts penalisedLong-range contacts penalised P(shortcut has length > nP(shortcut has length > n0.5-0.5---) is too ) is too

smallsmall With high probability, there is no With high probability, there is no

such shortcut within first nsuch shortcut within first n nodes nodes seen by routing algorithmseen by routing algorithm

Page 21: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

RemarksRemarks

Results hold for inhomogeneous Poisson Results hold for inhomogeneous Poisson processes with intensity bounded away processes with intensity bounded away from zero and infinityfrom zero and infinity

Impossibility results continue to hold withImpossibility results continue to hold with• 1-step look-ahead: each node knows locations 1-step look-ahead: each node knows locations

not only of its local and long-range contacts not only of its local and long-range contacts but also their contactsbut also their contacts

• or k-step look-ahead, fixed kor k-step look-ahead, fixed k• Mean number of shortcuts per node being Mean number of shortcuts per node being

polylog(n) instead of constant polylog(n) instead of constant

Page 22: Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

Open problemsOpen problems

Other density functions for shortcutsOther density functions for shortcuts r=2: variants of proposed algorithm r=2: variants of proposed algorithm

should also work, but hard to proveshould also work, but hard to prove r=2: what if there are no local r=2: what if there are no local

contacts but c log(n) shortcuts? Is contacts but c log(n) shortcuts? Is graph connected? Is efficient routing graph connected? Is efficient routing possible?possible?

Still doesn’t explain Milgram’s Still doesn’t explain Milgram’s experiments!experiments!