1 Analyzing Kleinberg’s (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 1.The expected diameter of Kleinbeg’s graph is  (log n) 2.The expected length of Kleinberg’s greedy paths is  (log 2 n). Also, they are this long with constant probability. 3.With some extra local knowledge we can improve the path length to O(log 1+1/k n)

4 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”

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 Kleinberg’s Basic setting

7 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:

8 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.

9 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

10 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 s

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

12 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

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 B l (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 r=k (so r=2 for 2D grid), Kleinberg shows greedy routing is O((log 2 n). We show this bound is tight, and: With probability greater than ½, Kleinberg’s algorithm uses at least clog 2 n 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 Z v, log of the ratio d(v,t)/d(v’,t) where v’ is the next vertex. Done when sum of Z v totals log(d(s,t)) Show E[Z v ] = O(1/logn), so need  (log 2 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 S B - its surface  Given node v outside B and a random link from v, what is the chance for this link to get to S B ? v t m L

24 Extensions Our 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 S 0 and T 0 with size  (logn)) 3. “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 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