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 Diameter bound and extensions Tight bound on decentralized routing Abstract framework for small-world graphs Part III: Future research

3 Our new results For a k-dimensional lattice model 1.The expected diameter of Kleinberg’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 more 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 settings have 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: Malkhi, Naor, and Ratajczak secure distributed protocols

6 Kleinberg’s Basic setting Based on an n by n two dimensional grid (wraparound) Lattice distance from u=(i,j ) to v= (k,l ): d(u,v)=|k-i|+|l-j| Each node has 4 local links and q directed random links; u has a link to v with probability proportional to: d -r (u,v) (inverse second power distribution if r=2 ) A two-dimensional grid with n=6, q=0The contacts of a node u with q=2: v and w are the two long-range contacts

7 Kleinberg’s results A decentralized routing problem Find a short path from s to t. At any step, can only use local information, Kleinberg’s greedy algorithm and analysis: 1. When u is the current node, choose next v: the closest to t (w.r.t. lattice distance) such that (u,v) is a local or random edge. 2. Achieves expected ` delivery time ’ of O(log 2 n), i.e. the s  t paths have expected length O(log 2 n). 3. This does not work if using any other inverse r th power distribution: for r  2,  >0 such that the expected delivery time of any decentralized algorithm is  (n  ).

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 distribution and directed edges 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 For set C, node u  C, a random link from u: How likely is this link to leave C ? Links into  Given: subset C, node u  C.  How likely is a link to u from outside 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 } Given any 0<  <1, any integer 1  l  n , for n large enough The probability a random link from a given node u goes to outside of B l ( u ) > 1-  -o(1) The probability that there is a random link to u from outside of B l ( u ) > 1-e  +o(1)-1 (i.e. almost 1-e  -1 ) For a ball with radius n.51 a random link from the center leaves the ball with probability >.48 With 4 links, expect to hit 4*.48 > 1.9 new nodes.

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 the ratios =d(s,t)

22 Rate of Progress To 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 k-D 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 to Other Models Our approach can be easily extended to other lattice-based settings which have: 1. Sufficiency of random links everywhere (to form super nodes) 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