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Analyzing and Characterizing Small-World Graphs

Analyzing and Characterizing Small-World Graphs. Van Nguyen and Chip Martel Computer Science, UC Davis. Contents. Small-world phenomenon & Models The diameter of Kleinberg’s grid A Framework for Small-world graphs. Small-world phenomenon.

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Analyzing and Characterizing Small-World Graphs

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  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 • Two strangers meet and discover that they are two ends of a short chain of acquaintances Boston Nebraska • 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. Based on an n by n, 2-D grid, where each node has 4 local undirected links Kleinberg’s Model

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

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

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

  9. t Kleinberg’s SW networkis Greedy Routable iff r=2 v • A greedy routing algorithm using local information only, find a short path from s to t u s • This greedy routing achieves • expected `delivery time’ of O(log2n), i.e. the st paths have expected length O(log2n).

  10. t Kleinberg’s SW networkis Greedy Routable iff r=2 v • A greedy routing algorithm using local information only, find a short path from s to t u s • This greedy routing achieves • expected `delivery time’ of O(log2n), i.e. the st paths have expected length O(log2n). • 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< r <4: diameter < logcn for c>1 • If 4< r: diameter > nc 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< r <4: diameter <logcn for c>1 • If 4< r: diameter> nc for 0<c<1 • Can be generalized for k-D grid, say if k< r <2k: diameter < logcn for c>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 0 n-1 1 • 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 . 2 . . . . . j i • For 0 r 1, we showed the diameter is (logn) in PODC’04 Now consider r>1.

  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=xa for some a(0,1). • Intuitively, w.h.p, D(x)is bounded by a constant multiple of D(y).

  19. D(n) D(na) D(na2) … D(x0) 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(x0) for some x0 small enough (a poly-log function of n).

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

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

  22. D(n) D(na) D(na2) … D(x0) Partitioning Hierarchy • We iterate this partitioning from x=n to some small x0 (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

  23. Supporting Facts • Fact 1: For a fixed a s.t.r/2< a <1and for xlarge 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. * * w t u+x-1 s v u B A Supporting Facts • Fact 2: If a partitionof 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)+1for (v,w)  2 max D(y) +1

  25. Supporting Facts • Fact 2: If a partitionof 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. • Fact 2 still true if we redefine D(x) as the maximum value of the diameters of all segments of length x

  26. Poly-log diameter for 1r2 • Consider the sequence of maximum diameter values in our partitioning hierarchy D(n), D(na), … ,D(x0) Where almost surely, D(x)  2D(x a)+1 • The # of terms is (loglog n) • D(x0) x0, bounded by a poly-log(n) • So, D(n)= O(logcn) 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 (nc), 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< r <2k: diameter < logcn , c>1 • If 2k< r: diameter> nc for 0<c<1 • The case r=2kis 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. C u 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.

  32. C u 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“

  33. C u 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. • We define (in paper) precisely the two parameters: the size of set C and the probability (for escaping from C)

  34. u C 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.

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

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

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

  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:V2R+ and hence,defineour base graphs as growth restricted graphs, wherethe 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 • ClassInvDist(,): 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, but changes to polynomial (n) for >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. FRG Hierarchy From a base graph (of a collection H) generate independent random links, using distribution  FRG (H,) InExpansion Expansion InvDist(,) Similar to Expansionbut for incoming links Each node has q (,)-EXP links, where q>1 Growth restricted graphs degree  +random links: Pr[uv] ~ d-(u,v) 0: small-world with D=(logn) <<2: SW, D=poly-log(n) 2<:`large’ world, D= poly(n) Expansion family -symmetric InvDist with logn-neighbored base graphs Exp-family with logn-neighbored base graph METR() where  = and some easy conditions small-world with D=(logn) Greedy-routable with short paths(log2n)

  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

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