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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

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

- 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)

- 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

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

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

q=2

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

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.

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

v

- Greedy routing algorithm
using local information only,

find a short path from s to t

u

s

t

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 st paths have expected length O(log2n).

t

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 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).

- 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

- 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

- 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

- 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.

- 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

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[ij] ~ |i-j|-r

.

2

.

.

.

.

.

j

i

- For 0 r 1, we showed the diameter is (logn) in PODC’04
Now consider r>1.

- 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.

- 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).

D(n)

D(na)

D(na2)

…

D(x0)

- 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).

Partitioning Hierarchy

Partitioning:

A segment of length x

is divided into multiple

sub-segments of length

y=x afor a(0,1).

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.

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

- 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.

*

*

w

t

u+x-1

s

v

u

B

A

- 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 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

- 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

- 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

- 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

- 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)

- 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

- 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)

C

u

- 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.

C

u

- 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“

- 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.

C

u

- 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)

u

C

- 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
A Random Family (H,)is an

Expansion Family

if the distribution satisfies

the two expansion criteria.

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

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

- 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

- 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

- 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

- If we add in local links to make an appropriate base graph, then the graph becomes a small-world:

- We add a general distance function d:V2R+ 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

- ClassInvDist(,): 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

- 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

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[uv] ~ 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)

- 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