1. 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 Symposium on Algorithms and Computation (ISAAC 2006)

2. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 Outline Small-World Graphs Features Kleinberg’s model Greedy Routing strategies Our proposals Why to reduce randomization? Restricted Small-World Small-World with communities Conclusions

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

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

5. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 Watts and Strogatz model [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 Randomness 0<p<1

6. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 Watts and Strogatz model [WS98] Clustering coefficient - Average Path length log scale in p p Small World

7. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 Kleinberg’s model  ( n,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

8. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 t Greedy Routing : move to the neighbor that minimizes the distance to the target. s Greedy Routing Strategies

9. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 t Greedy Routing Strategies Indirect Greedy Routing (IR): each node is aware of the long- range contacts of its closest neighbors s

10. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 t Greedy Routing Strategies s Neighbor-of-Neighbor (NoN) Greedy Routing: each node is aware of the long-range contacts of its long-range contacts

11. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 Related Work Kleinberg (2000) showed that each  ( n,s,q,p ) network is navigable (i.e. Greedy routing require O((log 2 n )/ q ) steps) Barrière et al. (2001) showed that Kleinberg’s result is indeed optimal (Greedy routing require  ((log 2 n )/ q ) steps) Fraignaud et al. (2004) analyzed IR Routing (it requires  ((log 1+1/s n )/ q 1/s ) steps) Manku et al. (2004) provided an Overlay network which exploits the NoN Greedy Routing (it requires O(log 2 n /( q log q ))) s = O( 1 ) Optimal for q= log n s = 1

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

13. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 Our Proposals:  ( n,s,q ) Restricted Small World : Long-range connections are allowed only with nodes that differ in exactly one coordinate higher probability darker ~

14. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 Our Results:  ( n,s,q ) Theorem The average path length is O((log 2 n )/ q ) for the greedy routing on  ( n,s,q ) when 1  q  log n. Corollary The average path length is O((log 1+1/s n )/ q 1/s ) for the indirect routing on  ( n,s,q ) when each node is aware of the long-range contacts of its ( e s ln n )/ q closest neighbors and 1 < q  log n. Theorem The average path length is O((log 2 n )/( q log q )) for the NoN greedy routing on  ( n,s,q ) when 1 < q  log n. s could be non-constant

15. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 Our Results:  ( n,s,q ) Greedy Routing Theorem The average path length is O((log 2 n )/ q ) for the greedy routing on  ( n,s,q ) when 1  q  log n. Proof (Sketch) For each i = 1,…,s; let d i 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 d i to at most d i /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) ct didi d i /2

16. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 Our Results:  ( n,s,q ) Greedy Routing Theorem The average path length is O((log 2 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 n 1/s ))= O((log 2 n )/ q ) ct didi d i /2

17. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 Our Proposals:  c ( n,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.

18. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 Our Results:  c ( n,s,q ) Theorem The average path length is O((log 2 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((log 1+1/s n )/ q 1/s ) for the indirect routing on  c ( n,s,q ) when each node is aware of the long-range contacts of its ( e s ln n )/ q closest neighbors, 1 < q  log n and c  ( 2e s ln n )/ q. Theorem The average path length is O((log 2 n )/( q log q )) for the NoN greedy routing on  ( n,s,q ) when 1 log n.

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

20. Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006 Thanks for your attention Any questions? Salerno ITALY Gennaro Cordasco - How Much Independent Should Individual Contacts be to Form a Small-World? - 19/12/2006