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

Analyzing Kleinberg’s (and other)Small-world Models Chip Martel and Van Nguyen Computer Science Department; University of California at Davis

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

Our new results • For the general k-dimensional lattice model • The expected diameter of Kleinbeg’s graph is (log n) • The expected length of Kleinberg’s 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(log1+1/k n)

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

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

Kleinberg’s Basic setting

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:

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.

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

We construct neighbor trees from s and to t: is the nodes within logn of sin the grid is nodes at distance i (random links) from The diameter bound:Intuition s

T-Tree is the nodes within logn of tin the grid is nodes at distance i (random links) to t

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

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

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)

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

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

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)

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 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

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.

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

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.

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 Lfrom 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

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

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

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

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