1 Analyzing and Characterizing Small-World Graphs Van Nguyen and Chip Martel Computer Science, UC Davis

2 Contents Small-world phenomenon & Models The diameter of Kleinberg’s grid A Framework for Small-world graphs

3 Small-world phenomenon Nebraska Boston Two strangers meet and discover that they are two ends of a short chain of acquaintances Milgram’s pioneering work (1967): “six degrees of separation between any two Americans” Source person in Nebraska, target at person in Boston Chained people supposed to forward to someone they knew based on a first-name basis Here, we often use `small-world graphs’ for graphs with small diameter (poly-log functions of size)

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

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

6 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 each node

7 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.

8 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

9 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).

10 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  ).

11 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

12 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

13 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

14 Our Results A framework to construct classes of random graphs with  (logn) expected diameter We start with a general framework where random arcs are added to a fixed base graph. Then we refine this setting adding additional properties. A more refined class of random graphs where with local information only we find paths of expected poly-log length.

15 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

16 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) in PODC’04 Now consider r> n i j

17 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.

18 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).

19 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 ) …

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

21 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

22 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 ) …

23 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.

24 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

25 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.

26 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

27 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

28 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. Also generalized for Growth Restricted Graphs ( mention more later )

29 Building Small-World Graphs We start with a general framework where random arcs are added to a fixed base graph. Then we refine this setting adding additional properties. Create Families of Random Graphs - FRG (H,  ): H: set of base graphs (e.g. grids)  : a distribution for adding random links

30 Building Small World Graphs Based on a random assignment operation: For a given node u, make a random trial under distribution  to find another node v Each assignment performs an independent trial E.g. in Kleinberg’s grid setting,  Base graphs are grids   is defined as having u  v with probability  d -r (u,v) We want to characterize distributions  so most shortest paths are  (logn)

31 Our small-world graphs: the distribution of random links 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.

32 Our small-world graphs: the distribution of random links 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.  diversity and fairness: no small set takes most of chance to be hit  “don't give too much to a small group“ u C

33 Our small-world graphs: the distribution of random links 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. u C  We define (in paper) precisely the two parameters: the size of set C and the probability (for escaping from C)

34 Our small-world graphs: the distribution of random links Similar criterion for the `inverse direction’ If a node u is surrounded by a moderate size set of vertices C, there likely exists a random link coming to u from outside of C. u C

35 Expansion families A Random Family (H,  ) is an Expansion Family if the distribution  satisfies the two expansion criteria.

36 Out-Expansion Each node likely to have a random link out of a neighborhood of certain size In-Expansion Each node is likely to be reached by a random link from outside of a neighborhood of certain size FRG (H,  ) From a base graph (of a collection H ) generate independent random links, using distribution  Expansion family Refining for small-worlds

37 Expansion family log n-neighbored base graphs small-world with expected diameter =  (logn) Refining for small-worlds Includes many well-known SW settings, such as Kleinberg’s grid and hierarchy model Out-Expansion Each node likely to have a random link out of a neighborhood of certain size In-Expansion Each node is likely to be reached by a random link from outside of a neighborhood of certain size FRG (H,  ) From a base graph (of a collection H ) generate independent random links, using distribution  Expansion family

38 Applications of the framework To obtain diameter bounds for some small-world models, E.g. Kleinberg’s k-dimension grid model for any k  1 ( as in our earlier PODC’04 paper ) To augment certain settings to become graphs with small diameters Example is next on Kleinberg’s Tree-based setting Also more: show later

39 Kleinberg’s Tree-based setting Quite different to grid setting Nodes are leaves of a full b-ary tree T A distance measure: h(u,v) – the height of the least common ancestor of u and v  That tree T is only used for defining this distance A random link from a node u can go to v with probability  b -h(u,v). No local links  possibly unconnected

40 Kleinberg’s Tree-based setting Quite different to grid setting Nodes are leaves of a full b-ary tree T A distance measure: h(u,v) – the height of the least common ancestor of u and v A random link from a node u can go to v with probability  b -h(u,v). No local links  possibly unconnected If there is at least 3 random links going out from each node, this setting is an Expansion Family If we add in local links to make an appropriate base graph, then the graph becomes a small-world:  A way to do so, say, ring the nodes within a sub-tree of size logn

41 More refined classes using distance measures We add a general distance function d:V 2  R + and hence, define our base graphs as growth restricted graphs, where the growth of a neighborhood ( nodes within distance r from u ) is  (r  ). E.g. think of a  -D grid but  can be any positive real value

42 A phase transition on diameter Class InvDist( ,  ): We also add random links such that Pr[u  v] ~ d -  (u,v) E.g. Kleinberg’s 2-D setting for greedy- routing is InvDist(2,2) The diameter is poly-log(n) if  2 

43 A Design for Greedy-like Routing We further refining, adding  =  and some condition on the connectivity of small neighborhoods to gain a class of random graphs where Greedy-like Routing is possible: Each node doesn’t have the global topology, but `knows’ a small neighborhood (i.e. knows the random links coming from there)  Choose the random link which goes closest to the destination All Kleinberg’s settings (grid, tree, group-induced) are (or after some easy augment) of this class

44 Expansion Each node has q ( ,  )-EXP links, where q  >1 InExpansion Similar to Expansion but for incoming links FRG (H,  ) From a base graph (of a collection H ) generate independent random links, using distribution  Expansion family Exp-family with logn- neighbored base graph InvDist( ,  ) Growth restricted graphs degree  +random links: Pr[u  v] ~ d -  (u,v) METR(  ) where  =  and some easy conditions  -symmetric InvDist with logn-neighbored base graphs 0  : small-world with D=  (logn)  <  <2  : SW, D=poly-log(n) 2  <  :`large’ world, D= poly(n) Greedy-routable with short paths  (log 2 n) small-world with D=  (logn) FRG Hierarchy

45 Future Work Many known Network graphs follow some `growth restricted’ rules. E.g. wireless networks can be modeled using the unit disk graph (  =2) The Internet network distance defined by round-trip propagation and transmission delay forms growth restricted metrics (Ng&Zhang, SPAA’02) Idea: Using our framework, consider adding long links to certain Network graphs to shrink these graphs (into small-worlds, ideally) E.g., how to add in long links (fixed long wire) to a wireless network (unit disk) to best shrink the graph diameter