analyzing kleinberg’s (and other) small-world models
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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
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