Small World and Decentralized Algorithms

A class presentation for

S. Milgram J. Travers. “An Experimental Study of the SmallWorld Problem”. In: Sociometry 32 (1969), p. 425

keywords: network, social

J. Kleinberg. “The Small-World Phenomenon: An AlgorithmicPerspective”. In: Proceedings of the 32nd ACM Symposium on

Theory of Computing (2000), pp. 163–170keywords: network, decentralized algorithm, lower/upper bounds

Anirban MitraDepartment of Computer Science

Stony Brook University6 September, 2012

Milgram’s Experiment

I letters to be delievered from a source s to a target t

I starting with a few, a sender could have only sent it to onewhom he knew by first name

I each sender had some basic information about t such asaddress, occupation

I each successful chain was found to have 5–6 hops on anaverage

I the birth of six degree of separation

Findings

I short chain of acquaintances connecting random people

I people were able to find short paths using only localinformation

Algorithmic Perspective

I small world graphs have very short diameter and highclustering coefficient

I random graphs have short diameter but very low clusteringcoefficient [5]

I a different model is than random graph required to studythe small world phenomenon

I Watts–Strogatz proposed models with these propertieswhere a uniform graph is superimposed with sparserandom graph [1]

Watts–Strogatz Lattice Ring Model

Source: http://en.wikipedia.org/wiki/Watts_and_Strogatz_model

Figure : Watts – Strogatz Model

A Note About Clustering Coefficient

I clustering coefficient is defined as with any threeimmediately connected nodes called a triplet

C =3× number of triangles

number of connected triples of vertices

=number of closed triplets

total number of triplets

I extended definition in case of a weighted graph whichcaptures the connectedness of graph better

C =sum of weights of closed triplets

total sum of weights of triplets

Generalized Grid Model

Source: http://www.cs.cornell.edu/home/kleinber/swn.d/swn.html

Figure : (A) A two-dimensional grid network with n = 6, p = 1, andq = 0 (B) The contacts of a node u with p = 1 and q = 2

Continued ...

I consider nodes arranged in a nxn grid

I all nodes v within manhattan distance p of u are connectedby short range contacts i.e. d(u, v) ≤ p

I q long range contacts where the probability of a long rangecontacts is directly proportional to d(u, v)−r, rth inversepower

P =d(u, v)−r∑w d(u,w)−r

I a model is completely characterized by n, p, q, r

I very likely that a path of O(logn) between random nodes

Motivation for Kleinberg Results

I in Milgram’s experiment people were able to use localinformation to send the letter to correct destination in fewhops

I so, is there a decentralized algorithm capable of findingshort paths between any source s to target t with nonnegligible probabilities?

I what are the lower and upper bounds of such decentralizedalgorithms for the general model?

I note that it proves the lower/upper bounds, so terms arereplaced by their lower/upper bound functions in proofs

Decentralised Algorithm

I all nodes know about the grid structure i.e. all short rangecontacts

I the position of the t in the grid

I a node in a path know about all the long range contacts ofall the previous nodes

I the last assumption may seem inappropriate, but only usedto prove lower bounds

I otherwise, if one had full knowledge of network, shortestpath can be easily found by breadth first search

Result 1

I the lower bound for the case when r = 0

I uniformly distributed long range contacts

I the expected path length between any pair should bebounded by a polynomial in m = logn

I theorem proves that the lower bound of expected hops forany decentralised algorithm is αn2/3)

I α depends on p, q but not on n

I hence any decentralised algorithm takes time exponential inm

Intuitive Proof

I lattice distance is the number of short range hops betweentwo nodes

I let U be the set of nodes within lattice distance of n2/3

I most likely s will be outside of U

|U | = (p× lattice distance)2

= p2n4/3 (1)

I Probability that any node u has a long range contact in U

=|U |

total nodes

=p2n4/3

n2

= 4p2n−2/3 (2)

Continued ...

I let 1/c = 4p2n−2/3

I the expected number of steps for reaching a node having along range contact in U

= 1× 1/c+ 2× (1− 1/c)1/c+ 3× (1− 1/c)21/c...

= (1/c)(1 + 2x+ 3x2 + 4x3 + ...)x = (1− 1/c)

=1

c

d

dx(1 + x+ x2 + x3 + ...)

=1

c

d

dx

1

(1− x)2= (1/c)

1

(1− x)2

= c =n2/3

4p2(3)

Result 2

I the upper bound for the case when p = q = 1, r = 2

I does not uses the third assumption

I theorem proves that there exists a decentralised algorithmwhich is at most α(logn)2

I hence any decentralised algorithm takes time exponential inp

I for this case the long range contacts are formed in a specificway related to the geometry of grid

Result 3

I the general lower bound result

I lower bound for the case 0 ≤ r < 2

I lower bound runtime of decentralised algorithm is αn(2−r)/3

I for the runtime r > 2

I lower bound runtime of decentralised algorithm isαn(r−2)/(r−1)

I model with r = 2 is the unique one for which effectivedecentralized algorithm exists

The Lower Bound

Source: http://www.cs.cornell.edu/home/kleinber/swn.d/swn.html

Figure : Variation of runtime of decentralized algorithm with r

Summary of Kleinberg’s Result

I for the general class of grid model there is no effectivedecentralized algorithm

I decentralized algorithm takes time exponential in theminimum path length between two nodes

I moreover, it proves a stronger result, there exists a uniquemodel within the family for which a decentralized algorithmis effective

Critique of Milgram’s Experiment [4]

I the chains varied from 2 to 10, with 5 being the median [4]

I since six appeared to be the average length, the use of SixDegree of Separation entered into popular culture

I vast majority of unsuccessful chains or experiments withdifferent results were ignored such as in one case only 3 outof 60 chains were successful and that too, with 8 avragehops

I participants selection biased towards

I no other experimental studies done on a large scale

I an unpublished study suggests that people are dramaticallyseparated by social class

I similarly separation exists by race, culture and geography

Refrences I

S. Strogatz D. Watts. “Collective dynamics of small-worldnetworks”. In: Nature 393 (1998), p. 400.

S. Milgram J. Travers. “An Experimental Study of theSmall World Problem”. In: Sociometry 32 (1969), p. 425.

J. Kleinberg. “The Small-World Phenomenon: AnAlgorithmic Perspective”. In: Proceedings of the 32ndACM Symposium on Theory of Computing (2000),pp. 163–170.

J. Klienfeld. “Could it be a Big World After All? The “SixDegrees of Separation” Myth”. In: Society (2002).

M. Kochen I. de Sola Pool. “Contacts and influence”. In:Social Networks 1 (1978), p. 5.