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 1 There is a constant (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

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

Proof of theorem 1 U .. u v Probability that the source u 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 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

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

Proof Main idea We show that in phase j, the expected time before the current message holder has a long-range contact within lattice distance 2j of t is bounded proportionally to log n; at this point, phase j will come to an end. As there are at most log n phases, a bound proportional to log2n follows.

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

Proof Phase j  2j+1 ≤ (distance to v) < 2j Ball 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 n. When will phase j end? What is the prob that it will end in the next step? v No of nodes in Ball Bj is u each within distance 2j+1 + 2j < 2j+1 from a node like u

Proof So each has a probability of Ball Bj consists Of all nodes within Lattice distance 2j from the target So each has a probability 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