1. Navigation in small worlds Social Networks: Models and Applications Seminar Toronto, Fall 2007 (based on a presentation by Stratis Ioannidis)

2. 2 the small-world phenomenon “most people are linked by short chains of acquaintances”

3. 3 Milgram’s experiment (1960s) ► people in Omaha, Nebraska, were each given a letter addressed to a target person in Boston, Massachusetts, along with demographic information (name, address, profession) on this person. ► they were asked to send the letter to the target person, by forwarding it to other people that they knew on a first-name basis, instructing them to do the same. ► median number of hops to get the letter to the target: 6 -> six degrees of separation

4. 4 significance of small-world phenomenon ► qualitatively similar results by subsequent experiments on e.g. [Dodds et al. ‘03] ► small-world phenomenon also appears in other networks:  powergrid  actor collaboration graph  WWW  neural network of C. elegans  semantic networks of languages  food webs  …

5. 5 modeling the small-world phenomenon ► small-world network model: 1.short paths between almost all pairs of nodes 2.small node degree (on average) 3.locally clustered: a node’s neighbors are likely to be neighbors of each other ► a graph selected at random from all n-node graphs where each node has degree =3, has diameter O(logn), whp ► but, does not satisfy clustering requirement

6. 6 [Watts-Strogats ‘98]’s model ► d-dimensional lattice of n d nodes [here, d=2] ► for each node u:  local edges to nodes v, s.t. dist ρ(u,v) ≤ p[p=2]  long-range directed edges to q random nodes selected independently & uniformly over all nodes [q=3] ► expected diameter O(logn)

7. 7 [Kleinberg’00]: a new perspective on Milgram’s experiment ► “short paths not only exist, but can be found by individuals using only local information !” ► proposed a simple extension to Watts-Strogats’ model ► used that to demonstrated that:  ability to route efficiently with local information ≠ network diameter  this ability is affected by the correlation between local structure and long-range connections; efficient routing is possible only when this correlation is near a critical threshold; as we move away from this threshold routing deteriorates rapidly.

8. 8 Kleinberg’s (grid-based) model extends model of [Watts-Strogats ‘98]: ► d-dimensional lattice of n d nodes ► for each node u:  local edges to nodes v, s.t. dist ρ(u,v) ≤ p  long-range directed edges to q random nodes selected independently & uniformly over all nodes s.t. Pr(u->v) ~ ρ(u,v) -a ► a: concentration of long-range neighbors around u  a: smallconnections close to uniformly random -> a = 0[Watts-Strogats ‘98] model  a: largestrong preference for close connections -> a = ∞ long-range neighbors = local neighbors

9. 9 decentralized algorithms decentralized algorithm for transmitting messages: ► at each step the holder u of the message passes it to one of its neighbors (local or long-range) ► u knows only  the underlying grid structure  the location of the target on the lattice  the location and the long-range neighbors of all nodes that have touched the message so far delivery time T: ► expected number of steps to forward a message from a random source to a random target

10. 10 Kleinberg’s results when d = 2 1. for 0 ≤ a < 2, any decentralized algorithm has T = Ω(n (2-a)/3 ) 2. for a = 2, there is a decentralized algorithm s.t. T = O(log 2 n) 3. for a > 2, any decentralized algorithm has T = Ω(n (a-2)/(a-1) ) can be extended for d ≠ 2, with 0 ≤ a d, respectively the upper bound when a = 2 is achieved by greedy algorithm: ► a node forwards a message for t to its neighbor v such that ρ(v,t) is minimum corresponding diameter results [Martel-Nguyen ’04+’05]: 1. for a ≤ d, Θ(logn) 2. for d < a < 2d, Polylog(n) 3. for a = 2d,?? 4. for a > 2d, Poly(n)

11. 11 outline of proof of the upper bound ► a = 2, p = q = 1 ► in each step, the dist. from current node u to target t is halved with prob. ~1/logn [Ω(1/logn)] ► so, the expected number of steps until from u we reach a node u’ such that ρ(u’,t) ≤ ρ(u,t)/2 is at most ~logn[O(logn)] ► the target is reached after at most logn+1 halvings, so, in ~log 2 n expected steps [O(log 2 n)] ► crucial property of a = 2: it produces long-range neighbors approx. uniformly distributed over “distance scales”: for u’s long-range neighbor v, the probability that 2 j ≤ρ(u,v) ≤2 j+1, is the same for all j

12. 12 proof of the upper bound ► a = 2, p = q = 1 ► Prob(u->v) = ρ(u,v) -2 /Σ w ρ(u,w) -2 Σ w ρ(u,w) -2 ≤ Σ j<2n (4j)j -2 = O(logn) so, Prob(u->v) = Ω( 1/ρ(u,v) 2 logn ) ► choose random s,t; in each step we move to a node closer to t, so, each node is visited exactly once ► phase j: 2 j < ρ(u,t) ≤ 2 j+1 [u: current node]  B j : set of nodes w, s.t. ρ(t,w) ≤ 2 j ► |B j | = Ω(2 2j ) ► for all w in B j, ρ(u,w) < 2 j+2, so, Prob(u->w) = Ω( 1/ 2 2j logn )  prob. of entering phase j+i in next step Prob(u->|B j |) = |B j | · Ω( 1/ 2 2j logn ) = Ω( 1/logn )  Xj: steps in phase j  E[Xj] = 1/ Ω( 1/logn ) = O(logn) ► T = Σ j E[X j ] = O(log 2 n) j

13. 13 outline of proof of lower bounds ► a = 0, p = q = 1 ► Let U: set of nodes w s.t. ρ(w,t) < n 2/3 ► |U|~ n 4/3 ► Prob(s  U) ~ |U|/n 2 ~ 1/n 2/3 -> almost certainly s  U ► if s  U and no node u in the path to t has a long-range neighbor in U, then the number of steps to t are ≥ n 2/3 ► for any u, Prob(u->U) = |U|/n 2 = 1/n 2/3 -> starting from s  U, the expected number of steps to reach a node with a long-range neighbor in U is ~ 1/ Prob(u->U) = n 2/3 ► expected number of steps to t is ≥ n 2/3 ► a = ∞, p = q = 1 ► the random graph is the grid; expected number of steps ~n

14. 14 hierarchical model [Kleinberg 01] ρ(u,v) = 2 vu ► natural model for categorizing occupation, web pages,… ► ρ(u,v): height of lowest common ancestor of u,v ► polylog(n) long-range neighbors from distr ~ b -aρ(u,v) ; efficient routing only for a = 1 ► [Watts et al. 02]: many indep. trees; ρ the min of dist. in any tree b = 3

15. 15 group-based model [Kleinberg 01] ► set of groups {S 1,S 2,…} “bounded growth”: if S j,S k,… have sizes < g and all contain u, their union’s size is O(g) ► ρ(u,v): size of smallest group containing u,v ► polylog(n) long-range neighbors from distr ~ ρ(u,v) -a ; efficient routing only for a = 1 (and a > 1 in some cases) v u ρ(u,v) = 6

16. 16 rank-based model [Liben-Nowel 05] ► based on data from LiveJournal ► variation of grid-based model to handle non-uniformity ► each lattice point has ≥1 people associated with it ► local edges to one of people in each neighboring lattice point ► long-range edges to random nodes selected from distr. ~1/rank u (v) rank u (v): rank of v when nodes sorted in increasing dist. from u ► delivery time (to lattice point) O(log 3 n)

17. 17 related results ► decentralized search with additional information: a node may “consult” a small number of nearby nodes for free  [Lebhar-Schabanel ‘04]: paths of O(logn(loglogn) 2 ) steps with O(log 2 n) nodes consulted  [Fraigniaud et al.’05], [Martel-Nguyen ’04]: paths of O(log 3/2 n) steps by consulting neighborhood of size log(n) of current node  [Manku et al.’04]: neighbor of neighbor approach: optimal for some settings ► alternative distributions for choosing long-range neighbors: can we improve routing by  choosing long-range neighbors from a distribution other than ~1/ρ a  allowing variation in node degrees  allowing dependence between long-range neighbors of same node? - (almost certainly) no: [Aspnes et al. ‘02], [Flamini et al.’05], [Giakkoupis-Hadzilacos ‘07], [Woelfel ’08?] - what if we make edges to long-range neighbors bidirectional?

18. 18 related results ► small-world networks on arbitrary underlying graphs: is it possible to augment any graph such that greedy routing is efficient ? (greedy: wrt initial graph)  [Fraigniaud ‘05]: yes, for graphs of bounded tree-width  [Duchon et al.’06]: yes, for bounded growth rate  [Slivkin ‘05]: yes, for doubling dimension O(loglonn)  [Fraigniaud ‘05]: no, for doubling dimension >> loglonn

19. 19 applications ► peer-to-peer networks  file sharing ► searching the web  focused web crawling ► sensor networks ► on-line communities

20. 20 Thank you!