1. 1 Analyzing Kleinberg’s Small-world Model Chip Martel and Van Nguyen Computer Science Department; University of California at Davis

2. 2 Contents Small-world phenomenon & Models Kleinberg’s Model Greedy routing The diameter of Kleinberg’s grid

3. 3 Small-world phenomenon Nebraska Boston Two strangers meet and discover they are connected by a short chain of acquaintances Milgram’s pioneering work (1967): “six degrees of separation between any two Americans” Source person in Nebraska, target person in Boston People forward to someone they know on a first-name basis Paths were typically quite short

4. 4 Milgram’s result shows that not only do short chains exist but they can be found using only local knowledge of links (people sent letters only knowing their own friends). People also knew about the general geography of of the “network”: e.g. New York is close to Boston. How can we model such networks?

5. 5 Small World Properties Small diameter: short path between all pairs (or almost all) Efficient Greedy routing: short paths can be found with local knowledge Clustering: If there is a link (u,v) and (u,w) then more likely to be a link (v,w)

6. 6 Modeling Small-Worlds Many networks are Small-Worlds (e.g. WWW, Social Networks, Physical systems) Motivated models of small-worlds: (Watts-Strogatz, Kleinberg) New Analysis and Algorithms Applications peer-to-peer systems gossip protocols secure distributed protocols

7. 7 Kleinberg’s Model Based on an n by n, 2-D grid, where each node has 4 local undirected links

8. 8 Kleinberg’s Model Based on an n by n, 2-D grid, where each node has 4 local undirected links q=2 Add q directed random links per node

9. 9 Kleinberg’s Model Based on an n by n, 2-D grid, where each node has 4 local undirected links Add q directed random links per each node where  Define d(u,v): lattice distance between u and v u v d(u,v)=2+5=7  Now, u has a link to v with probability proportional to d -r (u,v). Parameter r determines crucial behaviors of the model.

10. 10 Increasng r favors near nodes r=0, Link to each other node equally likely r=1, inverse of distance If a node is twice as far away, 1/2 as likely r=2, inverse squared If a node is twice as far away, 1/4 as likely d -r (u,v) =1, Uniform Distribution

11. 11 Normalization Constant For a fixed r and u sum the probabilities to each other node to get the normalization constant C. Thus Pr[u->v] = 1/C * d -r (u,v) r = 0, C = n 2 –1 So, Pr[u->v] = 1/(n 2 –1 ) For all v. r = 2, C = O(logn) Pr[u->v] = (1/logn) * d -2 (u,v)

12. 12 Kleinberg’s SW network is Greedy Routable iff r=2 Greedy routing algorithm using local information only, find a short path from s to t  When u is the current node, choose next v: the closest to t (use lattice distance) with (u,v) a local or random edge. s u t v

13. 13 Kleinberg’s SW network is Greedy Routable iff r=2 A greedy routing algorithm using local information only, find a short path from s to t u t v s This greedy routing achieves expected ` delivery time ’ of O(log 2 n), i.e. the s  t paths have expected length O(log 2 n).

14. 14 Kleinberg’s SW network is Greedy Routable iff r=2 A greedy routing algorithm using local information only, find a short path from s to t u t v s This greedy routing achieves expected ` delivery time ’ of O(log 2 n), i.e. the s  t paths have expected length O(log 2 n). This does not work unless r=2 : for r  2,  >0 such that the expected delivery time of any decentralized algorithm is  (n  ).

15. 15 Greedy Routing Analysis We say that the algorithm is in phase i If the current node u has 2 i  d(u,t) < 2 i+1 The initial phase has i  logn since d(s,t) < 2n for any pair. If we are in phase i, how likely to jump to state i-1 ? Pretty easy to show Pr[u->v| d(v,t) 1/ logn

16. 16 Greedy Routing The initial phase has i  logn We jump to the next lower phase with probability about 1/logn So, expected O(logn) hops/ phase After at most logn phases done. total expected hops = logn * O(logn)= O(log 2 n).

17. 17 Our Results An analysis of the expected diameter of Kleinberg's setting. For a 2D-grid, One random link/node (q=1) If 0  r  2 : diameter=  (logn) – PODC’04

18. 18 Our Results An analysis of the expected diameter of Kleinberg's setting. For a 2D-grid, One random link/node (q=1) If 0  r  2: diameter=  (logn) – PODC’04 If 2< r <4: diameter < log c n SODA’05 If 4 n c for 0<c<1

19. 19 Our Results An analysis of the expected diameter of Kleinberg's setting. For a 2D-grid, One random link/node (q=1) If 0  r  2: diameter=  (logn) – PODC’04 If 2 1 If 4 n c for 0<c<1 Can be generalized for k-D grid, say if k 1

20. 20 Our new results: Routing For a k-dimensional lattice model 1.The expected length of Kleinberg’s greedy paths is  (log 2 n). Also, they are this long with constant probability. 2.With more local knowledge we can improve the path length to O(log 1+1/k n)

21. 21 Prior work on similar (diameter) problems Diameter of a cycle plus a random matching: Bollobas & Chung, 88 Can be seen as a special case of Kleinberg’s grid setting where: 1-D lattice, undirected graph, r=0 (random links are uniform) Diameter of long-range percolation graphs Benjamini & Berger, 2001 Coppersmith et al., 2002 Biskup, 2004: similar to our approach

22. 22 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

23. 23 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

24. 24 T-Tree is the nodes within logn of t in the grid is nodes at distance i (random links) to t

25. 25 Small-worlds: Finding a short path from s to t S2S2 SjSj S1S1 TiTi T2T2 T1T1 t S0S0 T0T0 s t S i = nodes at distance i from S 0, an Initial neighbor set of sufficient size T i is nodes at distance i (random links) to Initial set T 0 We want: the {S i } and {T i } to grow exponentially until big enough, so the two subset chains intersect with high probability.

26. 26 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

27. 27 Proving Exponential Growth Growth rate depends on set size and shape We analyze using an artificial experiment

28. 28 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)

29. 29 Exponential Growth u C Neighbor sets should have exponential growth If a node u is surrounded by a moderate size set of vertices C, a random link from u is likely to “escape” from C.

30. 30 Links into or out of a ball: the facts B L (u) ={nodes within distance L from u } For any 0<  <1, any integer 1  L  n , for n large The Pr[ u->v | v outside of B L ( u ) ] > 1-  -o(1) Similar for a random link to u from outside of B L ( u ) Note that B L ( u ) has about L 2 nodes. For a ball with radius n.51 a random link from the center leaves the ball with probability >.48 With 4 links, expect 4*.48 > 1.9 new nodes.

31. 31 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

32. 32 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)

33. 33 Reducing the Random Links To change from 1/node to 4/node: Collect nodes into super-nodes Each 2x2 square contracted to a super- node New graph has 4 random links /node and diameter differs from old by a constant factor

34. 34 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 Now look at r > k.

35. 35 The diameter of Kleinberg’s SW setting For simplicity, use a 1-D setting Define C(r,n) as an n-node cycle. Each node has 2 local links and One directed random-link: i is connected to j  i with Pr[i  j] ~ |i-j| -r For 0  r  1, we showed the diameter is  (logn) Now consider r>1. 0 1 2 n-1...... i j

36. 36 Upper bound for the diameter of C(r,n) when 1<r<2 We use a probabilistic recurrence approach Our approach is similar to Karp's (STOC’91) We establish a (probabilistic) relation between the diameter of a segment and that of a smaller one.

37. 37 Upper bound for the diameter of C(r,n) when 1<r<2 We use a (probabilistic) relation between the diameter of a segment and a sub-segment.  We relate D(x), the diameter of a segment of length x, to D(y), where y=x a for some a  (0,1).  Intuitively, w.h.p, D(x) is bounded by a constant multiple of D(y).

38. 38 Upper bound for the diameter of C(r,n) when 1<r<2  Iterating the relation, starting with x=n, standard recurrence techniques bound D(n) - the graph's expected diameter - based on D(x 0 ) for some x 0 small enough (a poly-log function of n ). D(n) D( n a ) D(n a 2 ) D(x 0 ) …

39. 39 Partitioning: A segment of length x is divided into multiple sub-segments of length y=x a for a  (0,1). Partitioning Hierarchy

40. 40 A partition is complete when every pair of sub-segments has two random directed edges connecting one to the other. Partitioning A segment of length x is divided into multiple sub-segments of length y=x a for some a  (0,1). Partitioning Hierarchy A B

41. 41 We iterate this partitioning from x=n to some small x 0 ( for fixed a ). We need to specify y ( or a ) s.t. Small enough  # iterations is order loglog (n) Not too small  Almost surely, each phase’s partition is complete Partitioning Hierarchy D(n) D( n a ) D(n a 2 ) D(x 0 ) …

42. 42 Supporting Facts Fact 1: For a fixed a s.t. r/2< a <1 and for x large enough, almost surely, all partitions of length x segments are complete Note: 0<r<1 and y=x a Implies that all sub-segments are large enough so can get to another by one link.

43. 43 Fact 2: If a partition of a segment of length x is complete, then almost surely D(x) is at most twice the maximum diameter of a subsegment, plus 1. Basically, any shortest path s  t can be upper bounded by two shortest paths within a sub-segment plus 1 length(s  t)  length(s  v)+length(w  t)+1 for (v,w)  2 max D(y) +1 Supporting Facts A u u+x-1 s t v w * * B

44. 44 Fact 2 still true if we redefine D(x) as the maximum value of the diameters of all segments of length x Supporting Facts Fact 2: If a partition of a segment of length x is complete, then almost surely D(x) is at most twice the maximum diameter of a sub-segment, plus 1.

45. 45 Poly-log diameter for 1  r  2 Consider the sequence of maximum diameter values in our partitioning hierarchy D( n), D( n a ), …,D( x 0 ) Where almost surely, D( x)  2 D( x a ) +1 The # of terms is  (loglog n) D( x 0 )  x 0, bounded by a poly-log(n) So, D( n) = O(log c n) for c>0 depending on r only

46. 46 The diameter of C(r,n) For r>2, C(r,n) is a ‘large’ world expected diameter  (n c ), c= r-1 / r  Random links tend to go to close nodes  Few long links

47. 47 Higher dimensions We generalize to k-dimensional grids If 0  r  k: diameter=  (logn) If k 1 If 2k n c for 0<c<1 The case r=2k is still open.

48. 48 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 than1/2, 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.

49. 49 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)

50. 50 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.

51. 51 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

52. 52 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