1 Analyzing Kleinberg’s Small-world Model Chip Martel and Van Nguyen Computer Science Department; University of California at Davis.

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Analyzing Kleinberg’s Small-world Model

Chip Martel and Van NguyenComputer Science Department; University of California at Davis

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Contents

Small-world phenomenon & Models

Kleinberg’s Model Greedy routing The diameter of Kleinberg’s grid

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Small-world phenomenon

Nebraska

BostonTwo strangers meet and discover they are connected by a short chain of acquaintances

Milgram’s pioneering work (1967): “six degrees of separation between any two Americans”

Source person in Nebraska, target person in Boston People forward to someone they know on a first-name

basis

Paths were typically quite short

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Milgram’s result shows that not only do short chains exist but they can be found

using only local knowledge of links (people sent letters only knowing their own friends).

People also knew about the general geography of of the “network”: e.g. New York is close to Boston.

How can we model such networks?

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Small World Properties

Small diameter: short path between all pairs (or almost all) Efficient Greedy routing: short paths can be found with local knowledgeClustering: If there is a link (u,v) and (u,w) then more likely to be a link (v,w)

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Modeling Small-Worlds

Many networks are Small-Worlds (e.g. WWW, Social Networks, Physical systems) Motivated models of small-worlds:

(Watts-Strogatz, Kleinberg) New Analysis and Algorithms

Applications peer-to-peer systems gossip protocols secure distributed protocols

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Kleinberg’s Model

Based on an n by n, 2-D grid, where each node has 4 local undirected links

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Kleinberg’s Model

Based on an n by n, 2-D grid, where each node has 4 local undirected links

q=2

Add q directed random links per node

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Kleinberg’s Model

Based on an n by n, 2-D grid, where each node has 4 local undirected links

Add q directed random links per each node whereDefine d(u,v): lattice

distance between u and v

u

v

d(u,v)=2+5=7

Now, u has a link to v with probability proportional to d -r(u,v). Parameter r determines crucial behaviors of the model.

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Increasng r favors near nodes

r=0, Link to each other node equally likely

r=1, inverse of distance If a node is twice as far away, 1/2 as

likely

r=2, inverse squared If a node is twice as far away, 1/4 as

likely

d -r(u,v) =1 , Uniform Distribution

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Normalization Constant

For a fixed r and u sum the probabilities to each other node to get the normalization constant C.

Thus Pr[u->v] = 1/C * d -r(u,v)

r = 0, C = n2 –1 So, Pr[u->v] = 1/(n2 –1 ) For all v.

r = 2, C = O(logn)

Pr[u->v] = (1/logn) * d -2(u,v)

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Kleinberg’s SW networkis Greedy Routable iff r=2

Greedy routing algorithmusing local information only,

find a short path from s to t

When u is the current node, choose next v: the closest to t (use lattice distance) with (u,v) a local or random edge.

s

u

t

v

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Kleinberg’s SW networkis Greedy Routable iff r=2

A greedy routing algorithmusing local information only,

find a short path from s to tu

t

v

s

This greedy routing achieves expected `delivery time’ of O(log2n),

i.e. the st paths have expected length O(log2n).

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Kleinberg’s SW networkis Greedy Routable iff r=2

A greedy routing algorithmusing local information only,

find a short path from s to tu

t

v

s

This greedy routing achieves expected `delivery time’ of O(log2n), i.e. the st paths have

expected length O(log2n). This does not work unless r=2 : for r2, >0 such that the

expected delivery time of any decentralized algorithm is (n).

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Greedy Routing Analysis

We say that the algorithm is in phase i If the current node u has 2i d(u,t) < 2i+1

The initial phase has i logn since d(s,t) < 2n for any pair. If we are in phase i, how likely to jump to state i-1 ?Pretty easy to show

Pr[u->v| d(v,t) < 2i ] > 1/logn

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Greedy Routing

The initial phase has i lognWe jump to the next lower phase with probability about 1/lognSo, expected O(logn) hops/ phase After at most logn phases done. total expected hops =

logn * O(logn)= O(log2n).

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Our Results An analysis of the expected diameter of Kleinberg's setting. For a 2D-grid, One random link/node (q=1) If 0 r 2: diameter=(logn) –

PODC’04

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Our Results An analysis of the expected diameter of Kleinberg's setting. For a 2D-grid, One random link/node (q=1) If 0 r 2: diameter=(logn) – PODC’04 If 2< r <4: diameter < logcn SODA’05 If 4< r: diameter > nc for 0<c<1

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Our Results An analysis of the expected diameter of Kleinberg's setting. For a 2D-grid, One random link/node (q=1) If 0 r 2: diameter=(logn) – PODC’04 If 2< r <4: diameter <logcn for c>1 If 4< r: diameter> nc for 0<c<1

Can be generalized for k-D grid, say if k< r <2k: diameter < logcn for c>1

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Our new results: RoutingFor a k-dimensional lattice model

1. The expected length of Kleinberg’s greedy paths is (log2 n). Also, they are this long with constant probability.

2. With more local knowledge we can improve the path length to O(log1+1/k n)

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Prior work on similar (diameter) problems

Diameter of a cycle plus a random matching: Bollobas & Chung, 88 Can be seen as a special case of

Kleinberg’s grid setting where: 1-D lattice, undirected graph, r=0 (random links are uniform)

Diameter of long-range percolation graphs Benjamini & Berger, 2001 Coppersmith et al., 2002 Biskup, 2004: similar to our approach

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O(log n) Expected Diameter

Proof for simple setting: 2D grid with wraparound4 random links per node, with

r=2 Extend to: K-D grids, 1 random link, No wraparound

kr ≤≤0

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The diameter bound:Intuition

We construct neighbor trees from s and to t:

is the nodes within logn of s in the grid

is nodes at distance i (random links) from

0S

iS0S

s 0S

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T-Tree

is the nodes within logn of t in the grid

is nodes at distance i (random links) to

0T

iT 0T

t 0T

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Small-worlds: Finding a short path from s to t

S2

Sj

S1

Ti

T2

T1

t

S0

T0

s

t

Si = nodes at distance i from S0, an Initial neighbor set of sufficient sizeTi is nodes at distance i (random links) to Initial set T0

We want: the {Si} and {Ti} to grow exponentially until big enough, so the two subset chains intersect with high probability.

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After O(logn) Growth steps and are almost surely of size nlogn

Thus the trees almost surely connect

Similar to Bollobas-Chung approach for a ring + random matching. But new complications since non-uniform distribution and directed edges

Subset chains

iTjS

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Proving Exponential Growth

• Growth rate depends on set size and shape

We analyze using an artificial experiment

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Links into or out of a ball

Motivation Links to outsideFor set C , node u C, a random link from u:

How likely is this link to leave C ?

Links into Given: subset C , node u C.How likely is a link to u from outside C ?

Worst shape for C: A ball (with same size)

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Exponential Growth

u

C

Neighbor sets should have exponential growth If a node u is surrounded

by a moderate size set of vertices C, a random link from u is likely to “escape” from C.

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Links into or out of a ball: the facts

BL(u) ={nodes within distance L from u }

For any 0< <1, any integer 1 L n, for n large

The Pr[ u->v | v outside of BL (u) ] > 1--o(1) Similar for a random link to u from outside of BL

(u) Note that BL (u) has about L2 nodes.

For a ball with radius n.51 a random link from the center leaves the ball with probability > .48 With 4 links, expect 4*.48 > 1.9 new nodes.

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S-Tree growth

By making the initial set larger than clogn, a growth step is exponential with probability: By choosing c large enough, we can make m large enough so our sets almost surely grow exponentially to size nlogn

0S

mn−−>1

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The t-Tree

Ball experiment for t-tree needs some extra care (links are conditioned) Still can show exponential growth Easy to show two (nlogn) size sets of `fresh’ nodes intersect or a link from s-set hits t-set More care on constants leads to a diameter bound of 3logn + o(logn)

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Reducing the Random Links

To change from 1/node to 4/node: Collect nodes into super-nodes Each 2x2 square contracted to a super-

node

New graph has 4 random links /node and diameter differs from old by a constant factor

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Diameter Results

Thus, for a K-D grid with added link(s) from u to v proportional to The expected diameter is (log

n) for

Now look at r > k.

),( vud r−

kr ≤≤0

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The diameter of Kleinberg’s SW setting

For simplicity, use a 1-D setting

Define C(r,n) as an n-node cycle. Each node has 2 local links and One directed random-link: i is

connected to j i with

Pr[ij] ~ |i-j|-r

For 0 r 1, we showed the diameter is (logn) Now consider r>1.

0 12

n-1.

..

...

i

j

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Upper bound for the diameter of C(r,n) when 1<r<2

We use a probabilistic recurrence approach Our approach is similar to Karp's

(STOC’91) We establish a (probabilistic) relation

between the diameter of a segment and that of a smaller one.

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Upper bound for the diameter of C(r,n) when 1<r<2

We use a (probabilistic) relation between the diameter of a segment and a sub-segment. We relate D(x) , the diameter of a segment of length x, to D(y) , where y=xa for some a(0,1).

Intuitively, w.h.p, D(x) is bounded by a constant multiple of D(y).

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Upper bound for the diameter of C(r,n) when 1<r<2

Iterating the relation, starting with x=n, standard recurrence techniques bound D(n) - the graph's expected diameter - based on D(x0) for some x0 small enough (a poly-log function of n).

D(n)D(na)

D(na2)

D(x0)

…

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Partitioning:A segment of length x is divided into multiplesub-segments of length

y=x a for a(0,1).

Partitioning Hierarchy

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A partition is complete when every pair of sub-segments has two random directed edges connecting one to the other.

Partitioning A segment of length x is

divided into multiple sub-segments of length y=x a for some a(0,1).

Partitioning Hierarchy

A B

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We iterate this partitioning from x=n to some small x0

(for fixed a). We need to specify y (or a) s.t. Small enough # iterations

is order loglog (n) Not too small Almost

surely, each phase’s partition is complete

Partitioning HierarchyD(n)D(na)

D(na2)

D(x0)

…

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Supporting Facts

Fact 1: For a fixed a s.t. r/2< a <1 and for x large enough, almost surely, all partitions of length x segments are complete Note: 0<r<1 and y=x a

Implies that all sub-segments are large enough so can get to another by one link.

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Fact 2: If a partition of a segment of length x is complete, then almost surely D(x) is at most twice the maximum diameter of a subsegment, plus 1. Basically, any shortest path st can be upper bounded by two shortest paths within a

sub-segment plus 1

length(st) length(sv)+length(wt)+1for (v,w) 2 max D(y) +1

Supporting Facts

A

u u+x-1s tv w* *

B

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Fact 2 still true if we redefine D(x) as the maximum value of the diameters of all segments of length x

Supporting Facts

Fact 2: If a partition of a segment of length x is complete, then almost surely D(x) is at most twice the maximum diameter of a sub-segment, plus 1.

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Poly-log diameter for 1r2

Consider the sequence of maximum diameter values in our partitioning hierarchy

D(n), D(na), … ,D(x0)Where almost surely, D(x) 2D(x a)+1

The # of terms is (loglog n) D(x0) x0, bounded by a poly-log(n) So, D(n)= O(logcn)

for c>0 depending on r only

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The diameter of C(r,n)

For r>2, C(r,n) is a ‘large’ world expected diameter (nc), c=r-1/r

Random links tend to go to close nodes Few long links

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Higher dimensions

We generalize to k-dimensional grids If 0 r k: diameter=(logn) If k< r <2k: diameter < logcn , c>1

If 2k< r: diameter> nc for 0<c<1

The case r=2k is still open.

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Analyzing Greedy Routing

For r=k (so r=2 for 2D grid), Kleinberg shows greedy routing is O((log2n) .

We show this bound is tight, and: With probability greater than1/2, Kleinberg’s algorithm uses at least clog2n steps.

Fraigniaud et. al also show tight bound, andSuggested by Barriere et. al 1-D result.

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Proof of the tight bound (ideas)

How fast does a step reduce the remaining distance to the destination?We measure the ratio between the distance to t before and after each random trial:

We reach t when the product of the ratios =d(s,t)

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Rate of Progress To avoid a product of ratios, we transform to Zv , log of the ratio: d(v,t)/d(v’,t) where v’ is the next vertex.

Done when sum of Zv totals log(d(s,t))

Show E[Zv] = O(1/logn), so need (log2 n) steps to total log(d(s,t))= logn.

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An important technical issue: Links to a k-D surface

What is the probability to get to a given distance from t ?

Let B = {nodes within distance L from t } and SB - its surface

Given node v outside B and a random link from v, what is the chance for this link to get to SB?

v

t

m

L

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Part III: Future work

The diameter for r=2k (poly-log or polynomial)?Improved algorithms for decentralized routing A routing decision would depend on:

the distance from the new node to the destination

neighborhood information.

Better models for small-world graphs