Identity and search in social networks

Presented by Pooja Deodhar

Duncan J. Watts, Peter Sheridan Dodds and M. E. J. Newman

Presentation OutlinePresentation OutlineIntroductionContentions – Social NetworksAlgorithm explanationOur model and Milgram’s findingsFurther ExtensionsApplications

2

IntroductionIntroductionSocial Networks are “Searchable”Our model offers explanation of

searchability in terms of recognizable personal identities

Personal identities - sets of characteristics in different social dimensions

Class of searchable networks and method for searching them applicable to many real world problems

3

IntroductionIntroductionSmall World Network

◦Network in which most nodes are not neighbors of each other but most nodes can be reached from every other node by a number of hops

4

IntroductionIntroduction

Milgram’s Experiment ◦ Short paths exist between individuals in large

social network◦ Ordinary people can find these short paths◦ People rarely have more than local knowledge

about the network

5

Source

IntroductionIntroductionSearchability

◦Property of being able to find a target quickly

Shown to exist in networks◦With certain fraction of hubs (highly

connected nodes which once reached can distribute messages to all parts of the network)

◦Built upon underlying geometric lattice

6

IntroductionIntroductionLimited hubs in social networksSocial Networks are more like a

peer-to-peer networkNeed for a hierarchical modelSome measure of distance

between individualsCan be based on targets identity,

friends identity, friend’s popularity

7

Contentions – Social Contentions – Social NetworksNetworksIndividual identities – sets of

characteristics attributed to them by virtue of association, participation in social groups

Groups – Collection of individuals with well-defined set of social characteristics

8

Contentions – Social Contentions – Social NetworksNetworksBreaking down of world into set

of layersTop layer – whole populationLower layers – specific division

into groups

9

Contentions – Social Contentions – Social NetworksNetworksSimilarity xij – between individuals i, j xij – Height of the lowest common

ancestor level between i and jIndividuals in same group are at

distance of one from each other

10

Contentions – Social Contentions – Social NetworksNetworks

Combined social distance yij = minh xij

In the above figure H = 2In 1st heirarchy, yij = 1 and yjk = 1

in 2nd

But yik = 4 > yij + yjk = 211

Contentions – Social Contentions – Social NetworksNetworksProbability of acquaintance

between i and j decreases with decreasing similarity of groups to which they belong

Link distance x for individual i has probability

p(x) = ce-αx

Measure of homophily – tendency of like to associate with like

12

Contentions – Social Contentions – Social NetworksNetworksIndividuals hierarchically

partition the social world in more than one way.◦h = 1, …, H hierarchies

Node’s identity is the vector ◦ is position of node i in hierarchy

h.Social distance

hiv

hiv

hij

hij x y min

13

Contentions – Social Contentions – Social NetworksNetworksAt each step the holder i of the

message passes it to one of its friends who is closest to the target t in terms of social distance

Individuals know the identity vectors of:◦themselves◦their friends,◦the target

Two kinds of partial information – social distance and network paths

14

Algorithm ExplanationAlgorithm ExplanationPrincipal objective – determine

conditions for average path length L of a message chain is small

Define q as probability of an arbitrary message chain reaching a target.

Searchable network - Any network for which q ≥ rfor a desired r.

15

SearchabilitySearchabilitySearchable networks occupy a

broad region of parameter space <α,H> which are sociologically plausible

Searchability is generic property of social networks

16

Algorithm ExplanationAlgorithm ExplanationIn terms of chain length L,

q = (1 - p)L ≥ rL = length of message chainP = message failure probability

From above, L can be obtained by the approximate inequality,

L <= ln r / ln (1 - p)

17

Our model and Milgram’s Our model and Milgram’s findingsfindingsAll searchable networks have α > 0, H

> 1Individuals are essentially homophilous

but judge similarity along more than one social dimension

Best performance is achieved for H = 2 or 3

Thus, use of 2 or 3 dimensions used by individuals in small world experiments when forwarding a message

18

Searchable NetworksSearchable Networks

19

Solid boundary – N=102,400Dot-dash – N=204800Dash – N=409,600p = 0.25, b = 2, g = 100, r = 0.25

at least

Our model and Milgram’s Our model and Milgram’s findingsfindingsIncreasing number of independent

dimensions from H = 1 yields dramatic reduction in delivery time for α > 0

This improvement lost as H is increased further

Thus, network ties become less correlated as H increases

For large H, network becomes a random graph, search algorithm becomes random walk

20

Searchable NetworksSearchable Networks

21

Probability of message completion when for α = 0 (squares) and for α = 2 (circles) for N = 102,400

Horizontal line – pos of the threshold Open symbols indicate network is

searchable – q <= r

Our model and Milgram’s Our model and Milgram’s datadata

n(L) – no. of completed chains of length L taken from original small world expt. (shown by bar graphs)

Taken for example of our model for N = 10^8 individuals and for 42 completed chains shown by filled circles

22

Our model and Milgram’s Our model and Milgram’s findingsfindingsComparison of distribution of

chain lengths in our model with that of Travers and Milgram

Avg. chain length for Milgrams expt = 6.5

Avg. chain length for our model = 6.7

23

SummarySummarySimple greedy algorithm.Represents properties present

in real social networks:◦Considers local clustering.◦Reflects the notion of locality.

High-level structure + random links.

24

Further ExtensionsFurther ExtensionsShould we consider other parameters such as friend’s popularity information in addition to homophily?◦Allow variation in node degrees?

Assume correlation between hierarchies?

Are all hierarchies equally important?

25

ApplicationsApplicationsBroad class of decentralized

problems◦Peer to peer networking

Any data structure in which data elements can be judged along more than one dimension

Designing of databases◦Eg. Music files – same genre/same

year

26