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

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What is a small world network? Milgram (1967) Milgram (1967) Sent letters to various people in the US addressed to targets in BostonSent letters to various people in the US addressed to targets in Boston Demographic information about target provided: name, address, occupationDemographic information about target provided: name, address, occupation Letters had to be forwarded to target via contacts known on first-name basisLetters had to be forwarded to target via contacts known on first-name basis Average length of chain for successful delivery was found to be six.Average length of chain for successful delivery was found to be six.

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Modelling social networks Random graphs have small diameter Random graphs have small diameter Diameter of Erdos-Renyi random graph on n nodes is O(log n) at connectivity thresholdDiameter of Erdos-Renyi random graph on n nodes is O(log n) at connectivity threshold Similar results for power-law random graphsSimilar results for power-law random graphs But not a good model of social networks But not a good model of social networks Social networks are more clusteredSocial networks are more clustered

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Small-world network models Watts & Strogatz Watts & Strogatz Lattice plus random shortcuts (uniform)Lattice plus random shortcuts (uniform) Benjamini & Berger Benjamini & Berger d-dimensional lattice, edge (x,y) present with probability |x-y| -rd-dimensional lattice, edge (x,y) present with probability |x-y| -r 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

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Outline of talk Background: can short paths be found using local information? Background: can short paths be found using local information? Kleinberg, Franceschetti & MeesterKleinberg, Franceschetti & Meester Model: Poisson point process with local links and shortcuts Model: Poisson point process with local links and shortcuts Results Results Open problems Open problems

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Kleinberg: can short paths be found? 2-dimensional lattice on N 2 points 2-dimensional lattice on N 2 points Each node has q shortcuts Each node has q shortcuts Each shortcut from x is incident on y with probability proportional to d(x,y) -r, independent of others Each shortcut from x is incident on y with probability proportional to d(x,y) -r, independent of others Need to route a message from source s to destination t using a decentralised algorithm Need to route a message from source s to destination t using a decentralised algorithm

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What is a decentralised algorithm? Each node knows co-ordinates of its shortcut contacts & of destination t Each node knows co-ordinates of its shortcut contacts & of destination t Suppose algorithm has currently visited nodes x 0,…,x k Suppose algorithm has currently visited nodes x 0,…,x k Next node to visit must be chosen from among contacts (lattice or shortcut) of these nodes. Next node to visit must be chosen from among contacts (lattice or shortcut) of these nodes.

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r=2: Greedy routing At each node visited by algorithm At each node visited by algorithm Choose shortcut contact if it is closer to destinationChoose shortcut contact if it is closer to destination Else choose lattice contact which is closerElse choose lattice contact which is closer Theorem: Number of hops to destination is O(log 2 N) Theorem: Number of hops to destination is O(log 2 N) Key idea: Find good shortcut that halves distance every O(log N) steps Key idea: Find good shortcut that halves distance every O(log N) steps

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r 2: Impossibility result r2: No decentralised algorithm can find path shorter than a polynomial in N r2: No decentralised algorithm can find path shorter than a polynomial in N Reason: Reason: Short-cuts lack range if r>2Short-cuts lack range if r>2 Short-cuts spread too uniformly if r<2, cant close in on target (using only local information)Short-cuts spread too uniformly if r<2, cant close in on target (using only local information)

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Continuum model Franceschetti and Meester Franceschetti and Meester Each point in plane has shortcut to other points according to an inhomogeneous Poisson process Each point in plane has shortcut to other points according to an inhomogeneous Poisson process Intensity at distance x proportional to x -r (not integrable) Intensity at distance x proportional to x -r (not integrable) Objective is to deliver message from s to neighbourhood of t Objective is to deliver message from s to neighbourhood of t

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Results If r=2, greedy algorithm has expected delivery time If r=2, greedy algorithm has expected delivery time O(log d(s,t) + log(1/)) If r>2, any decentralised algorithm needs at least d(s,t) β steps If r>2, any decentralised algorithm needs at least d(s,t) β steps If r<2, any decentralised algorithm needs at least (1/) β steps If r<2, any decentralised algorithm needs at least (1/) β steps

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Poisson small-world model Nodes: located at points of unit rate Poisson point process on square of area n. Nodes: located at points of unit rate Poisson point process on square of area n. Local links: to all other nodes within distance c log(n) Local links: to all other nodes within distance c log(n) Shortcuts: q per node, in expectation Shortcuts: q per node, in expectation Probability of shortcut to node at distance d: c(q,n) d -r Probability of shortcut to node at distance d: c(q,n) d -r

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Remarks For sufficiently large c>0, graph formed by local links alone is connected. For sufficiently large c>0, graph formed by local links alone is connected. Hence, message can always be routed in O(n/log(n)) hops. Hence, message can always be routed in O(n/log(n)) hops. Do shortcuts help us do better? Do shortcuts help us do better? Can we route in polylog(n) hops? Can we route in polylog(n) hops?

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

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Results: r2 r<2: Any decentralised routing algorithm needs more than n hops on average, for any < (2-r)/6 r<2: Any decentralised routing algorithm needs more than n hops on average, for any < (2-r)/6 r>2: Any decentralised routing algorithm needs more than n hops on average, for any 2: Any decentralised routing algorithm needs more than n hops on average, for any < (r-2)/2(r-1)

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Algorithm for r=2 At each hop, algorithm maintains a `radius, initialised to d(s,t). At each hop, algorithm maintains a `radius, initialised to d(s,t). If current node x has shortcut to a node in circle of radius /2 centred at t, message is delivered to this node. If current node x has shortcut to a node in circle of radius /2 centred at t, message is delivered to this node. Else, it is delivered to one of the local contacts of x which is closer to t Else, it is delivered to one of the local contacts of x which is closer to t If d(x,t)

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In pictures + modification

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Sketch proof: r=2 If c large enough, every node x has a local contact which is closer to t If c large enough, every node x has a local contact which is closer to t P(good shortcut) depends on number of nodes in annulus P(good shortcut) depends on number of nodes in annulus It is O(1/log n) if number of nodes is large enoughIt is O(1/log n) if number of nodes is large enough Probability that number of nodes is small is negligibleProbability that number of nodes is small is negligible Hence, good shortcut found after geometric O(log n) steps Hence, good shortcut found after geometric O(log n) steps

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Sketch proof: r<2 Suppose algorithm finds a path with fewer than n hops Suppose algorithm finds a path with fewer than n hops There has to be at least one shortcut which takes path into circle of radius n+ centred at t There has to be at least one shortcut which takes path into circle of radius n+ centred at t P(shortcut between u and v) is small for any u,v. P(shortcut between u and v) is small for any u,v. Very unlikely to find shortcut into this circle, by union bound. Very unlikely to find shortcut into this circle, by union bound.

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Sketch proof: r>2 Long-range contacts penalised Long-range contacts penalised P(shortcut has length > n ) is too small P(shortcut has length > n ) is too small With high probability, there is no such shortcut within first n nodes seen by routing algorithm With high probability, there is no such shortcut within first n nodes seen by routing algorithm

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Remarks Results hold for inhomogeneous Poisson processes with intensity bounded away from zero and infinity Results hold for inhomogeneous Poisson processes with intensity bounded away from zero and infinity Impossibility results continue to hold with Impossibility results continue to hold with 1-step look-ahead: each node knows locations not only of its local and long-range contacts but also their contacts1-step look-ahead: each node knows locations not only of its local and long-range contacts but also their contacts or k-step look-ahead, fixed kor k-step look-ahead, fixed k Mean number of shortcuts per node being polylog(n) instead of constantMean number of shortcuts per node being polylog(n) instead of constant

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Open problems Other density functions for shortcuts Other density functions for shortcuts r=2: variants of proposed algorithm should also work, but hard to prove r=2: variants of proposed algorithm should also work, but hard to prove r=2: what if there are no local contacts but c log(n) shortcuts? Is graph connected? Is efficient routing possible? r=2: what if there are no local contacts but c log(n) shortcuts? Is graph connected? Is efficient routing possible? Still doesnt explain Milgrams experiments! Still doesnt explain Milgrams experiments!

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