Small world
-
Upload
anirban-mitra -
Category
Documents
-
view
1.203 -
download
0
description
Transcript of Small world
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.