Analyzing Kleinberg’s (and other)
Small-world Models
Chip Martel and Van Nguyen
Computer Science Department;
University of California at Davis
1
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
2
Our new results
For 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)
3
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”
4
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
5
Kleinberg’s Basic setting
6
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:
7
Our Main results
For Kleinberg’s small-world setting we



Analyze the Diameter for 0  r  2
Give a tight analysis of greedy routing
Suggest better routing algorithms
A framework for graphs of low diameter.
8
O(log n) Expected Diameter
Proof for simple setting:
2D grid with wraparound
4 random links per node, with r=2
Extend to:
 K-D grids, 1 random link,
 No wraparound
0r k
9
The diameter bound:
Intuition
We construct neighbor trees from s and to t:
S 0 is the nodes within logn of s in the grid
S i is nodes at distance i (random links) from S
0
s
S0
10
T-Tree
T 0 is the nodes within logn of t in the grid
Ti is nodes at distance i (random links) to
t
T0
T0
11
Subset chains
After O(logn) Growth steps S j and Ti 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
12
Proving Exponential Growth
Growth rate depends on set size and shape
We analyze using an artificial experiment
13
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)
14
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
15
nodes from u.
S-Tree growth
By making the initial set S 0 larger than
clogn, a growth step is exponential with
probability:  1  n  m
By choosing c large enough, we can
make m large enough so our sets
almost surely grow exponentially to size
nlogn
16
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)
17
Diameter Results
Thus, for a K-D grid with added link(s)
r
from u to v proportional to d ( u , v )

The expected diameter is (log n) for
0r k
18
New Diameter Results
Thus, for a K-D grid with added link(s) from
u to v proportional to
d
r
(u , v )
The expected diameter is (log n) for
0r k
 New paper: polylog expected diameter for
k  r  2k
 Expected diameter is Polynomial for
r  2k
19
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, and
Suggested by Barriere et. al 1-D result.
20
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
21
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.
22
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
m
L
t
23
Extensions
Our approach can be easily extended for other
lattice-based settings which have:
1.
2.
3.
Sufficiency of random links everywhere
(to form super node)
Rich enough in local links (to form
initial S0 and T0 with size (logn))
“Links into or out of a ball” property
24
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
25
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
26