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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 Diameter bound and extensions Tight bound on decentralized routing Abstract framework for small-world

graphs

Part III: Future research

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Our new resultsFor a k-dimensional lattice model

1. The expected diameter of Kleinberg’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 more 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 settings have 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: Malkhi, Naor, and Ratajczak secure distributed protocols

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Kleinberg’s Basic setting

Based on an n by n two dimensional grid (wraparound) Lattice distance from u=(i,j ) to v= (k,l ): d(u,v)=|k-i|+|l-j| Each node has 4 local links and q directed random links; u has a link to v with probability proportional to:

d -r(u,v) (inverse second power distribution if r=2 )

A two-dimensional grid with n=6, q=0

The contacts of a node u with q=2: v and w are the two long-range contacts

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Kleinberg’s results

A decentralized routing problem Find a short path from s to t. At any step, can only use local information, Kleinberg’s greedy algorithm and analysis:

1. When u is the current node, choose next v: the closest to t (w.r.t. lattice distance) such that (u,v) is a local or random edge.

2. Achieves expected `delivery time’ of O(log2n), i.e. the st paths have expected length O(log2n).

3. This does not work if using any other inverse rth power distribution: for r2, >0 such that the expected delivery time of any decentralized algorithm is (n).

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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 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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Links into or out of a ball: the facts

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

Given any 0< <1, any integer 1 l n, for n large enough

The probability a random link from a given node u goes to outside of Bl (u) > 1--o(1)

The probability that there is a random link to u from outside of Bl (u) > 1-e+o(1)-1 (i.e. almost 1-e-1)

For a ball with radius n.51 a random link from the center leaves the ball with probability > .48 With 4 links, expect to hit 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

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 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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Extensions to Other Models

Our approach can be easily extended to other lattice-based settings which have:

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

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