Proof of Kleinberg’s small-world theorems

Proof of Kleinberg’s small-world theorems

Kleinberg’s Small-World Model Consider an (n x n) grid. Each node has links to every node at lattice distance p (short range neighbors) & q long range links. Choose long-range links s.t. the prob. to have a long range contact at lattice distance d is proportional to 1/dr n p = 1, q = 2 r = 2 n Recall Kleinberg’s results.Is there a justification?

Results Theorem 1There is a constant α0 (depending on p and q but independent of n), so that when r = 0, the expected delivery time of any decentralized algorithm is at least α0.n2/3

Proof of theorem 1 U .. s t Probability that the source s will lie outside U = (n2 - 2. n 4/3) / n2 = 1 - 2/n2/3 (w.h.p) n2/3 Probability that a node outside U has a long distance link inside U = 2. n 4/3 / n2 =2/n -2/3 . So, roughly in O(n2/3) steps, the query will enter U, and thereafter, it can take at most n2/3 steps.

Results Theorem 2. There is a decentralized algorithm A and a constant α2 dependent on p and q but independent of n, so that when r = 2 and p = q = 1, the expected delivery time of A is at most α2.(log n)2

Proof of theorem 2 Phase j means Distance from t is between 2j and 2j+1 21 20 target t 22 source s

Observation How many nodes are at a lattice distance j from a given node? 4.j How many nodes are at a lattice distance jor less from a given node? 1 + 4.j + 4. (j-1) + 4. (j-2) + … = 1 + 4.j.(j+1)/2 = 1 + 2.j.(j+1) = 1 + 2j2 + 2j

Proof Main idea We show that in phase j, the expected no of hops before the current message holder discovers a long-range contact within lattice distance 2j of t is O(log n); After this, phase j comes to an end. As there are at most log n phases, a bound proportional to log2n follows.

Proof Probability (node u chooses node v as its long-range contact) is There are 4j nodes at distance j But So, the probability (u chooses v) is

Proof Zone Bj consists Of all nodes within Lattice distance 2j from the target Phase j2j+1 ≤ (distance to v) < 2j The maximum value of j is log 2n (⋍ log n) When will phase j end? What is the prob that it will end in the next step? v No of nodes in Zone Bj is u each within distance2j+1 +2j <2j+2 from a node like u

Proof Zone Bj consists Of all nodes within Lattice distance 2j from the target So each has a probability of of being a long-distance contact of u, So, the search enters Bj with a probability of at least v u So, the expected number of steps spent in phase j is 128 ln (6n). Since There are at most log n phases, the Expected time to reach v is O(log n)2

Proof of Kleinberg’s small-world theorems