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Analyzing Kleinberg’s (and other)Small-world Models

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

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Contents

Part I: An introduction Background and our initial results

Part II: Our new results The tight bound on decentralized

routing The diameter bound and extensions An abstract framework for small-world

graphs

Part III: Future research

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Our new resultsFor the general k-dimensional lattice

model

1. The expected diameter of Kleinbeg’s graph is (log n)

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

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

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Background

Small-world phenomenon

From a popular situation where two completely unacquainted people meet and discover that they are two ends of a very short chain of acquaintances

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

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Modeling Small-Worlds

Many real settings exhibit small-world properties Motivated models of small-worlds:

(Watts-Strogatz, Kleinberg) New Analysis and Algorithms

Applications: gossip protocols: Kemper, Kleinberg, and Demers peer-to-peer systems: Malki, Naor, and Ratajczak secure distributed protocols

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Kleinberg’s Basic setting

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Kleinberg’s results

A decentralized routing problem For nodes s, t with known lattice coordinates,

find a short path from s to t. At any step, can only use local information, Kleinberg suggests a simple greedy algorithm

and analyzes it:

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Our Main results

For Kleinberg’s small-world setting we Analyze the Diameter for Give a tight analysis of greedy routing Suggest better routing algorithms

A framework for graphs of low diameter.

20 r

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O(log n) Expected Diameter

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

r=2Extend 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

iS 0S

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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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 distiribution

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 outside

Given: subset C , node u, a random link from u. What is the chance for this link to get out of C ?

Links into Given: subset C , node u C. What is the chance to have a link to u from

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

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Links into or out of a ball: the facts

Bl (u) ={nodes within distance l from u }

For a ball with radius n.51 a random link from the center leaves the ball with probability at least .48

With 4 links, expected to hit 4*.48 > 1.9 new nodes from u.

For the general K(n,p,q) with wraparound or not

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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

mn1

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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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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

),( vud r

kr 0

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New 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

New paper: polylog expected diameter for

Expected diameter is Polynomial for

kr 0

),( vud r

kr 2

krk 2

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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 than ½, 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 these ratios is 1

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Rate of Progress

To avoid 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 spherical 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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ExtensionsOur approach can be easily extended for other lattice-based settings which have:

1. Sufficiency of random links everywhere (to form super node)

2. Rich enough in local links (to form initial S0 and T0 with size (logn))

3. “Links into or out of a ball” property

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An abstract framework Motivation: capture the characteristics of KSW model formalize more general classes of SW graphs In the abstract: a base graph, add new

random links under a specific distribution Abstract characteristics which result in

small diameter and fast greedy routing

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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