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Analyzing and Characterizing Small-World Graphs

Van Nguyen and Chip MartelComputer Science, UC Davis

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Contents

Small-world phenomenon & Models

The diameter of Kleinberg’s grid

A Framework for Small-world graphs

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Small-world phenomenon

Nebraska

BostonTwo 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)

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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

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Kleinberg’s Model

Based on an n by n, 2-D grid, where each node has 4 local undirected links

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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

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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 whereDefine 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.

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Kleinberg’s SW networkis Greedy Routable iff r=2

Greedy routing algorithmusing 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

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Kleinberg’s SW networkis Greedy Routable iff r=2

A greedy routing algorithmusing local information only,

find a short path from s to tu

t

v

s

This greedy routing achieves expected `delivery time’ of O(log2n),

i.e. the st paths have expected length O(log2n).

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Kleinberg’s SW networkis Greedy Routable iff r=2

A greedy routing algorithmusing local information only,

find a short path from s to tu

t

v

s

This greedy routing achieves expected `delivery time’ of O(log2n), i.e. the st paths have

expected length O(log2n). This does not work unless r=2 : for r2, >0 such that the

expected delivery time of any decentralized algorithm is (n).

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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

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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

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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

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Our ResultsA 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.

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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

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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[ij] ~ |i-j|-r

For 0 r 1, we showed the diameter is (logn) in PODC’04

Now consider r>1.

0 12

n-1.

..

...

i

j

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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.

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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).

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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).

D(n)D(na)

D(na2)

D(x0)

…

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Partitioning:A segment of length x is divided into multiplesub-segments of length

y=x a for a(0,1).

Partitioning Hierarchy

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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

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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

Partitioning HierarchyD(n)D(na)

D(na2)

D(x0)

…

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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.

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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 st can be upper bounded by two shortest paths within a

sub-segment plus 1

length(st) length(sv)+length(wt)+1for (v,w) 2 max D(y) +1

Supporting Facts

A

u u+x-1s tv w* *

B

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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.

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Poly-log diameter for 1r2

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

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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

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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=2k is still open.

Also generalized for Growth Restricted Graphs (mention more later)

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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

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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 uv with probability d-r (u,v) We want to characterize distributions so most shortest paths are (logn)

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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.

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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

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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)

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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

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Expansion families

Expansion familiesA Random Family (H,) is an

Expansion Family if the distribution satisfies the two expansion criteria.

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Out-ExpansionEach node likely to have a random link out of a neighborhood of certain size

In-ExpansionEach 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

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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-ExpansionEach node likely to have a random link out of a neighborhood of certain size

In-ExpansionEach 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

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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

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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

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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

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More refined classes using distance measures

We add a general distance function d:V2R+ 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

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A phase transition on diameter

Class InvDist(,): We also add random links such that

Pr[uv] ~ 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

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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

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ExpansionEach node has q (,)-EXP links, where q>1

InExpansionSimilar 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[uv] ~ 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 (log2n)

small-world with D=(logn)

FRG Hierarchy

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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