Navigation in small Navigation in small worldsworlds

Social Networks: Models and Applications

Seminar

Toronto, Fall 2007

(based on a presentation by Stratis Ioannidis)

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the small-world phenomenonthe small-world phenomenon

“most people are linked by short chains of acquaintances”

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Milgram’s experiment (1960s)Milgram’s experiment (1960s)

► people in Omaha, Nebraska, were each given a letter addressed to a target person in Boston, Massachusetts, along with demographic information (name, address, profession) on this person.

► they were asked to send the letter to the target person, by forwarding it to other people that they knew on a first-name basis, instructing them to do the same.

► median number of hops to get the letter to the target: 6-> six degrees of separation

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significance of small-world phenomenonsignificance of small-world phenomenon

► qualitatively similar results by subsequent experiments on e.g. [Dodds et al. ‘03]

► small-world phenomenon also appears in other networks: powergrid actor collaboration graph WWW neural network of C. elegans semantic networks of languages food webs …

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modeling the small-world phenomenonmodeling the small-world phenomenon

► small-world network model:1. short paths between almost all pairs of nodes2. small node degree (on average)3. locally clustered:

a node’s neighbors are likely to be neighbors of each other

► a graph selected at random from all n-node graphs where each node has degree =3, has diameter O(logn), whp

► but, does not satisfy clustering requirement

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[Watts-Strogats ‘98]’s model[Watts-Strogats ‘98]’s model

► d-dimensional lattice of nd nodes [here, d=2]► for each node u:

local edges to nodes v, s.t. dist ρ(u,v) ≤ p [p=2] long-range directed edges to q random nodes selected

independently & uniformly over all nodes [q=3]► expected diameter O(logn)

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[Kleinberg’00]: a new perspective [Kleinberg’00]: a new perspective on Milgram’s experimenton Milgram’s experiment

► “short paths not only exist, but can be found by individuals using only local information !”

► proposed a simple extension to Watts-Strogats’ model► used that to demonstrated that:

ability to route efficiently with local information ≠ network diameter

this ability is affected by the correlation between local structure and long-range connections; efficient routing is possible only when this correlation is near a critical threshold; as we move away from this threshold routing deteriorates rapidly.

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Kleinberg’s (grid-based) modelKleinberg’s (grid-based) model

extends model of [Watts-Strogats ‘98]:► d-dimensional lattice of nd nodes► for each node u:

local edges to nodes v, s.t. dist ρ(u,v) ≤ p long-range directed edges to q random nodes selected

independently & uniformly over all nodes

s.t. Pr(u->v) ~ ρ(u,v)-a

► a: concentration of long-range neighbors around u a: small connections close to uniformly random

-> a = 0 [Watts-Strogats ‘98] model a: large strong preference for close connections

-> a = ∞ long-range neighbors = local neighbors

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decentralized algorithmsdecentralized algorithms

decentralized algorithm for transmitting messages:► at each step the holder u of the message passes it to one of

its neighbors (local or long-range)► u knows only

the underlying grid structure the location of the target on the lattice the location and the long-range neighbors of all nodes that have

touched the message so far

delivery time T:► expected number of steps to forward a message from a

random source to a random target

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Kleinberg’s resultsKleinberg’s results

when d = 21. for 0 ≤ a < 2, any decentralized algorithm has T = Ω(n(2-a)/3)2. for a = 2, there is a decentralized algorithm s.t. T = O(log2n)3. for a > 2, any decentralized algorithm has T = Ω(n(a-2)/(a-1))

can be extended for d ≠ 2, with 0 ≤ a < d, a = d, and a > d, respectively

the upper bound when a = 2 is achieved by greedy algorithm:► a node forwards a message for t to its neighbor v such that

ρ(v,t) is minimum

corresponding diameter results [Martel-Nguyen ’04+’05]:

1. for a ≤ d, Θ(logn)2. for d < a < 2d, Polylog(n)3. for a = 2d, ??4. for a > 2d, Poly(n)

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outline of proof of the upper boundoutline of proof of the upper bound

► a = 2, p = q = 1► in each step, the dist. from current node u to target t is halved

with prob. ~1/logn [Ω(1/logn)]► so, the expected number of steps until from u we reach a

node u’ such that ρ(u’,t) ≤ ρ(u,t)/2 is at most ~logn[O(logn)]

► the target is reached after at most logn+1 halvings, so, in ~log2n expected steps [O(log2n)]

► crucial property of a = 2:it produces long-range neighbors approx. uniformly distributed over “distance scales”: for u’s long-range neighbor v, the probability that 2j≤ρ(u,v) ≤2j+1, is the same for all j

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outline of proof of lower boundsoutline of proof of lower bounds

► a = 0, p = q = 1► Let U: set of nodes w s.t. ρ(w,t) < n2/3

► |U|~ n4/3

► Prob(s U) ~ |U|/n2 ~ 1/n2/3 -> almost certainly s U

► if s U and no node u in the path to t has a long-range neighbor in U, then the number of steps to t are ≥ n2/3

► for any u, Prob(u->U) = |U|/n2 = 1/n2/3

-> starting from s U, the expected number of steps to reach a node with a long-range neighbor in U is ~ 1/ Prob(u->U) = n2/3

► expected number of steps to t is ≥ n2/3

► a = ∞, p = q = 1► the random graph is the grid; expected number of steps ~n

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hierarchical model [Kleinberg 01]hierarchical model [Kleinberg 01]

ρ(u,v) = 2

vu

► natural model for categorizing occupation, web pages,…► ρ(u,v): height of lowest common ancestor of u,v► polylog(n) long-range neighbors from distr ~ b-aρ(u,v);

efficient routing only for a = 1► [Watts et al. 02]: many indep. trees; ρ the min of dist. in any tree

b = 3

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group-based model [Kleinberg 01]group-based model [Kleinberg 01]

► set of groups {S1,S2,…}

“bounded growth”: if Sj,Sk,… have sizes < g and all contain u, their union’s size is O(g)

► ρ(u,v): size of smallest group containing u,v► polylog(n) long-range neighbors from distr ~ ρ(u,v)-a;

efficient routing only for a = 1 (and a > 1 in some cases)

vuρ(u,v) = 6

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rank-based model [Liben-Nowel 05]rank-based model [Liben-Nowel 05]

► based on data from LiveJournal► variation of grid-based model to handle non-uniformity► each lattice point has ≥1 people associated with it► local edges to one of people in each neighboring lattice point► long-range edges to random nodes selected from distr.

~1/ranku(v)

ranku(v): rank of v when nodes sorted in increasing dist. from u

► delivery time (to lattice point) O(log3n)

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related resultsrelated results

► decentralized search with additional information:a node may “consult” a small number of nearby nodes for free [Lebhar-Schabanel ‘04]: paths of O(logn(loglogn)2) steps with

O(log2n) nodes consulted [Fraigniaud et al.’05], [Martel-Nguyen ’04]: paths of O(log3/2n) steps

by consulting neighborhood of size log(n) of current node [Manku et al.’04]: neighbor of neighbor approach: optimal for some

settings

► alternative distributions for choosing long-range neighbors:can we improve routing by choosing long-range neighbors from a distribution other than ~1/ρa

allowing variation in node degrees allowing dependence between long-range neighbors of same node?

- (almost certainly) no: [Aspnes et al. ‘02], [Flamini et al.’05], [Giakkoupis-Hadzilacos ‘07], [Woelfel ’08?]

- what if we make edges to long-range neighbors bidirectional?

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related resultsrelated results

► small-world networks on arbitrary underlying graphs: is it possible to augment any graph such that greedy routing is efficient ? (greedy: wrt initial graph) [Fraigniaud ‘05]: yes, for graphs of bounded tree-width [Duchon et al.’06]: yes, for bounded growth rate [Slivkin ‘05]: yes, for doubling dimension O(loglonn) [Fraigniaud ‘05]: no, for doubling dimension >> loglonn

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applicationsapplications

► peer-to-peer networks file sharing

► searching the web focused web crawling

► sensor networks► on-line communities

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Thank you!