Small-world phenomenon: An Algorithmic Perspective Jon Kleinberg

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 such that the probability to choose a long range contact at lattice distance d is proportional to 1/d r r = 2 p = 1, q = 2 n n

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   n 2/3

Observation 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? j + 4. (j-1) + 4. (j-2) + … = j. (j+1)/2 = 1 + 2j(j+1) ≃ 2j 2 +2j + 1

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

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   (log n) 2

Proof of theorem 2 Phase j means t usource target Distance from t is between 2 j and 2 j+1 Phase 0 Phase 1

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 2 j of t is O(log n); at this point, phase j will come to an end. As there are at most log n phases, a bound proportional to log 2 n follows.

Proof Probability (u chooses v as a long-range contact) is But Thus, the probability that v is chosen is at least There are 4j nodes at distance j (Note: 1= ln e)

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

Proof Ball Bj consists of all nodes within lattice distance 2 j from the target u v So each has a probability of of being a long-distance contact of u, So, the search enters the ball Bj with a probability of at least 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