Analyzing Kleinbergs and other Smallworld Models

1
Analyzing Kleinbergs (and other)Small-world
Models

• Chip Martel and Van Nguyen
• Computer Science Department
• University of California at Davis

2
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

3
Our new results

• For the general k-dimensional lattice model
• The expected diameter of Kleinbegs graph is
  ?(log n)
• The expected length of Kleinbergs greedy paths
  is ?(log2 n). Also, they are this long with
  constant probability.
• With some extra local knowledge we can improve
  the path length to O(log11/k n)

4
BackgroundSmall-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
• Milgrams pioneering work (1967) six degrees of
  separation between any two Americans

5
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

6
Kleinbergs Basic setting
7
Kleinbergs 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

8
Our Main results

• For Kleinbergs 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.

9
O(log n) Expected Diameter

• Proof for simple setting
• 2D grid with wraparound
• 4 random links per node, with r2
• Extend to
• K-D grids, 1 random link,
• No wraparound

10
The diameter boundIntuition

• 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

s
11
T-Tree

• is the nodes within logn of t in the
  grid
• is nodes at distance i (random links) to

t
12
Subset chains

• 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

13
Proving Exponential Growth

• Growth rate depends on set size and shape
• We analyze using an artificial experiment

14
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)

15
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

16
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

17
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)

18
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

19
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

20
Analyzing Greedy Routing

• For rk (so r2 for 2D grid), Kleinberg shows
  greedy routing is O((log2n) .
• We show this bound is tight, and
• With probability greater than ½, Kleinbergs
  algorithm uses at least clog2n steps.
• Fraigniaud et. al also show tight bound, and
• Suggested by Barriere et. al 1-D result.

21
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

22
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 EZv O(1/logn), so need ?(log2 n)
• steps to total log(d(s,t)) logn.

23
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
t
L
24
Extensions

• Our approach can be easily extended for other
  lattice-based settings which have
• 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

25
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

26
Part III Future work

• The diameter for r2k (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