Navigating Social Networks: Geographic and Hierarchical Searching in CS 249B, Study notes of Computer Science

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Navigating Social NW:Geographic and H-Dim Searching

CS 249B: Science of NetworksWeek 14: Monday, 04/28/08Daniel BilarWellesley CollegeSpring 2008

Goals today
^ How to search Small Worlds networks with onlylocal information
^ Two Models^ 

Kleinberg’s single criterion “geography” model  Watts and Dodds’s multiple criterion “hierarchy”model

^ Serves also as examples of two schools ofscientific inquiry^ 

Abstracted, rationalist, proof-oriented (Kleinberg)  Empirical, experimental, data-oriented (Watts-Dodds)

Some material and slides gratefully acknowledgedfrom Kearns (U Penn)

Local vs Global information
^ Common problem^ 

Need to find path, search for nodes, people  Have only local (partial) view of network, notglobal (total) overview

^ How to we search and find globally with^ only local information

?

^ Two examples^ 

Kleinberg’s geography grid model  Watts and Dodd’s hierarchy model

Milgram’s experiment revisited
^ What did Milgram’s experiment show?^ (a) There are short paths in large networks that

connect individuals(b) People are able to find these short pathsusing a simple, greedy, decentralizedalgorithm

^ Small world models take care of (a)
^ Kleinberg: what about (b)?

Kleinberg’s model
Model parameters

p,^ q

and^

^ Start with an

n^ by

n^ grid

of vertices (so N = n^2)^ ^ add

local connections

: all

vertices within griddistance

p^ (e.g. 2) ^ add

distant connections: q additional connections;probability of connectionat distance

r: ~ 1/r^ α

large α

: heavy bias towards “more local” long-distance

connections small α

: approach uniformlyrandom

Kleinberg’s question
^ Assume parties knowonly:^ ^

grid address of target  addresses of their owndirect links

^ Algorithm: passmessage to neighborclosest to target
^ Fine-tuning

^ what value of

α^ permits effective search?

large α

: heavy bias towards “more local” long-distance

connections small α

: approach uniformlyrandom

Kleinberg’s results
α = 2

is the only value that permits rapidnavigation(~log

2 (N)

steps)
^ Any other value of

α^ will

result in time polynomialin n: n

β ^ Locality ofinformation

crucial to

this argument^ ^ Centralized algorithm maycompute short paths^ ^ Can recognize when“backwards” steps arebeneficial

add

local connections

: all

vertices within grid distance p^ (here <= 2 steps away)
add one

distant connection: q^ ; probability of connectionat distance

r: ~ 1/r^α

Searching in a small world
^ For

α < 2

, the graph has paths of logarithmic length (small world), but a greedy algorithm cannot find them
For^ α > 2

, the graph does not have short paths – no small world exists
For^ α= 2

is the only case where there are short paths, and the greedy algorithm is able to find them

y-axis is exponent^ β^ of the deliverytime T lowerbound cn

β

x-axis is exponentα of long rangelinks

Interpretation of r
^ Given node

u^ if we can partition the remaining node into sets^ A

, A, A 12

, … , A 3

,^ logN where

A,^ consists of all nodesi

whose distance from

u^ is between

i^2 and

i+1,^2 i=0..logN-1.

^ Then given

r = dim

each long range contact of

u^ is nearly equally

likely to belong to any of the sets

Ai

^ Roughly “same number of friends on each scale”

“View of the Worldfrom 9

th^ Ave”

A^ A^4

A^1 A^2

Recap: Searching in a small world
^ Given a source

s^ and a destination

t, define a greedy local search

algorithm that 1 knows the positions of the nodes on the grid 2 knows the neighbors and shortcuts of the current node 3 knows the neighbors and shortcuts of all nodes seen so far 4 operates greedily, each time moving as close to t as possible
Kleinberg proved the following ^ When

α=2, an algorithm that uses only local information at eachnode (not^2 ) can reach the destination in expected time

(^2) O(log n).

^ When

α<2^ a local greedy algorithm (

1-4) needs expected time

(2-α)/3 Ω(n ). ^ When

r>2^ a local greedy algorithm (

1-4) needs expected time

(α-2)/(α-1) Ω(n

). ^ Generalizes for a

d-dimensional lattice, when

α=d^ (query time is

independent of the lattice dimension)
^ d = 1

, the Watts-Strogatz model

H-Dim Searching
^ Follow-up to on Columbia Small Worldinvestigation^ 

http://smallworld.columbia.edu/

^ With respect to how/why social networks aresearchable, W&D say^ 

Kleinberg’s model not a satisfactory model of society^
^ Based exclusively on

geography

^ We don’t navigate social networks by purely“geographic” information (Kleinberg’s distance) ^ We don’t use any

single

criterion

^ Different criteria

used a

different points

in the chain

Six contentions about social NW W&D model start with plausible assumptions/observations about society andindividuals: 1.^ Individual don’t just have ties, but H-dimensional

identities

in^ groups

2.^ Hierarchical tree-like cognitive partition

of humanity into size-

wise^ manageable groups 3. Group

membership

primary

basis for interaction

4.^ Cognitive partition is done along

simultaneous H dimensions via

attributes.

Attribute values have distances between them (tree- structured) 5. Social Distance

between individuals: minimum distance in

any

attribute 6. Individuals use

social distance and network ties

to direct messages

efficientlyAlgorithm: given attribute vector of target, forward message to neighborclosest to target

Navigation via social distance
^ distance h(i,j) = height of “leastcommon ancestor”
^ Individuals in the same group aredistance x = 1 apart, and themaximum separation of twoindividuals is x = l
^ Individuals i and j belong to acategory two levels above that oftheir respective groups, dis-tancebetween them is x

= 3.ij

^ Individuals each have z friends ;are more likely to be connectedwith each other the closer theirgroups are (see contention (3) inpaper) ^ Permits fast, decentralizednavigation under

broader conditions^ ^ Not as sensitive as Kleinberg’smodel

multiple independenthierarchies coexist Hierarchical organization of groups

Main Results (Fig. 2.A is a good “bad example” –why?)
Simulation model of H-dimensional hierarchicaldecentralized search and empiricalsmall world experimental resultsconverge
Best performance for fast,decentralized search algorithmwhen 2 or 3 dimensions are use tonavigate
W &D say model is applicable todecentralized search on

any

network that has elementswith quantifiablecharacteristics akin toidentities^ ^ People, music files, webpages,news, research reports can bejudges along more than onedimension - can you name some?

H = dimensionsα = measure of homophily (tendencyto group with like)