How Much Independent Should Individual Contacts be to Form a Small-World?

Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006

How Much Independent Should Individual

Contacts be to Form a

Small-World?

Gennaro Cordasco and Luisa Gargano

University of Salerno

The 17th International The 17th International Symposium on Algorithms Symposium on Algorithms and Computation (ISAAC and Computation (ISAAC

2006)2006)


OutlineOutline

• Small-World Graphs• Features• Kleinberg’s model• Greedy Routing strategies

• Our proposals• Why to reduce randomization?• Restricted Small-World• Small-World with communities

• Conclusions


Traditional modelsTraditional models

• Random Network (Erdös and Rényi)• edges are generated completely at

random• low avg. path length L ≤ log n/log • small clustering coefficient C ~ /n

• Regular Network• edges follow a structure• high avg. path length • high clustering coefficient

where C(v) = clustering coefficient of node v (number of real links between neighbours of v divided by number of possible links)

V

vCC Vv

)(


Small-World NetworksSmall-World Networks

Unfortunately, neither Random nor Regular Networks capture reality…

• According to Watts and Strogatz [WS98]:• Large networks (n >> 1)• Sparse connectivity (avg degree << n)• Large clustering coefficient (larger than in an

equivalent random graph)• Short average paths length (~log n, close to

those of an equivalent random graph)


Watts and Strogatz model Watts and Strogatz model [WS98][WS98]

• Start with a ring, where every node is connected to the next nodes

• With probability p, rewire every edge to a uniformly chosen destination.

Order Randomness0<p<1


Watts and Strogatz model Watts and Strogatz model [WS98][WS98]

• Clustering coefficient - Average Path length

log scale in p

p

Small World


Kleinberg’s model Kleinberg’s model ((n,s,q,pn,s,q,p))• Consider n nodes lying on a toroidal s-

dimensional grid, for each node• (2s) short-range contacts• q long range contacts (Each node v establishes q

directed links independently according to the probability distribution p(d(u,v)))

Usually p(d) is proportional to d-s with normalization factor =v d(u,v)-s


t

Greedy Routing: move to the neighbor that minimizes the distance to the target.

s

Greedy Routing StrategiesGreedy Routing Strategies


t

Greedy Routing StrategiesGreedy Routing StrategiesIndirect Greedy Routing (IR): each node is aware of the long-range contacts of its closest neighbors

s


t

Greedy Routing StrategiesGreedy Routing Strategies

s

Neighbor-of-Neighbor (NoN) Greedy Routing: each node is aware of the long-range contacts of its long-range contacts


Related WorkRelated Work

• Kleinberg (2000) showed that each (n,s,q,p) network is navigable (i.e. Greedy routing require O((log2 n)/q) steps)

• Barrière et al. (2001) showed that Kleinberg’s result is indeed optimal (Greedy routing require ((log2 n)/q) steps)

• Fraignaud et al. (2004) analyzed IR Routing (it requires ((log1+1/s n)/q1/s) steps)

• Manku et al. (2004) provided an Overlay network which exploits the NoN Greedy Routing (it requires O(log2n /(q log q)))

s = O(1)

Optimal for q=log n s = 1


Why to reduce randomization?Why to reduce randomization?

• The use of randomization increases the difficulties in the implementation and testing of applications.

• The smaller is the randomization the higher is the clustering coefficient of the considered network• Clustering represents a fundamental feature that a

network model, designed to describe complex network, must hold

• The resilience of a network grows with the clustering coefficient

• An high clustering implies an improved ability to handle heavy traffic workload


Our Proposals: Our Proposals: ((n,s,qn,s,q))

Restricted Small World:

Long-range connections are allowed only with nodes that differ in exactly one coordinate

higher probabilitydarker ~


Our Results: Our Results: ((n,s,qn,s,q))Theorem

The average path length is O((log2 n)/q) for the greedy routing on (n,s,q) when 1 q log n.

Corollary The average path length is O((log1+1/s n)/q1/s) for

the indirect routing on (n,s,q) when each node is aware of the long-range contacts of its (es ln n)/q closest neighbors and 1<q log n.

Theorem The average path length is O((log2 n)/(q log q)) for the NoN greedy routing on (n,s,q) when 1<q log n.

s could be non-constants could be non-constant


Our Results: Our Results: ((n,s,qn,s,q) Greedy ) Greedy RoutingRouting

Theorem The average path length is O((log2 n)/q) for the greedy routing on (n,s,q) when 1 q log n.

Proof (Sketch)For each i = 1,…,s; let di the distance between the current (c)

and target node (t) on dimension i.

Let denote the event that the current node is able to diminish the remaining distance, from di to at most di/2 in one hop

Considering that each node has q long range, it is easy to show that that the probability that the event occurs is (q/log n)

c t

di

di/2


Our Results: Our Results: ((n,s,qn,s,q) Greedy ) Greedy RoutingRouting

Theorem The average path length is O((log2 n)/q) for the greedy routing on (n,s,q) when 1 q log n.

Proof (Sketch)

The expected number of nodes encountered before a successful event occurs is O((log n) / q)

By repeating for each dimension we have that the expected number of hops is O(s ((log n)/ q) (log n1/s))= O((log2 n)/q)

c t

di

di/2


Our Proposals: Our Proposals: cc((n,s,qn,s,q))Small World with

communities:

same community means same long-range distances

Same community means same long-range distances

Each node randomly chooses one of the communities to belong to and selects its long-range contacts only among a subset of nodes depending on the chosen community.


Our Results: Our Results: cc((n,s,qn,s,q))Theorem

The average path length is O((log2 n)/q) for the greedy routing on c(n,s,q) when 1 q log n and c (4 ln n)/q.

Corollary The average path length is O((log1+1/s n)/q1/s) for the

indirect routing on c(n,s,q) when each node is aware of the long-range contacts of its (es ln n)/q closest neighbors, 1<q log n and c (2es ln n)/q.

Theorem The average path length is O((log2 n)/(q log q)) for the NoN greedy routing on (n,s,q) when 1<q log n and c > log n.


ConclusionsConclusions

• We showed that it is not necessary to use a completely eclectic network in order to obtain a Small World environment.

• Our networks presents a higher clustering coefficient, hence they can be used to model “real” complex networks.

• Moreover, our networks can be used toward the design of efficient as well as easy to implement overlay network infrastructures based on the SW approach.


Thanks for your attention

Any questions?

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How Much Independent Should Individual Contacts be to Form a Small-World?