Analyzing and characterizing small world graphs
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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

Analyzing and Characterizing Small-World Graphs

Van Nguyen and Chip Martel

Computer Science, UC Davis


Contents

Contents

  • Small-world phenomenon & Models

  • The diameter of Kleinberg’s grid

  • A Framework for Small-world graphs


Small world phenomenon

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)


Modeling small worlds

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


Kleinberg s model

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

Kleinberg’s Model


Kleinberg s model1

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


Kleinberg s model2

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.


Kleinberg s sw network is greedy routable iff r 2

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


Kleinberg s sw network is greedy routable iff r 21

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).


Kleinberg s sw network is greedy routable iff r 22

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).


Our results

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


Our results1

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


Our results2

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


Our results3

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.


Prior work on similar diameter problems

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


The diameter of kleinberg s sw setting

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.


Upper bound for the diameter of c r n when 1 r 2

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.


Upper bound for the diameter of c r n when 1 r 21

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).


Upper bound for the diameter of c r n when 1 r 22

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).


Analyzing and characterizing small world graphs

Partitioning Hierarchy

Partitioning:

A segment of length x

is divided into multiple

sub-segments of length

y=x afor a(0,1).


Analyzing and characterizing small world graphs

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.


Analyzing and characterizing small world graphs

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


Supporting facts

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.


Supporting facts1

*

*

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


Supporting facts2

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


Poly log diameter for 1 r 2

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


The diameter of c r n

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


Higher dimensions

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)


Building small world graphs

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


Building small world graphs1

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)


Our small world graphs the distribution of random links

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.


Our small world graphs the distribution of random links1

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“


Our small world graphs the distribution of random links2

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)


Our small world graphs the distribution of random links3

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.


Expansion families

Expansion families

  • Expansion families

    A Random Family (H,)is an

    Expansion Family

    if the distribution  satisfies

    the two expansion criteria.


Refining for small worlds

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


Refining for small worlds1

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


Applications of the framework

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


Kleinberg s tree based setting

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


Kleinberg s tree based setting1

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


More refined classes using distance measures

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


A phase transition on diameter

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


A design for greedy like routing

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


Frg hierarchy

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)


Future work

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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