Social Search in Small-

World ExperimentsSharad Goel, Roby Muhamad and Duncan Watts

2009

Akhil Jain

2011CS50274

Outline

Algorithmic Small World Hypothesis

Work done until now

A new model of attrition

Heterogeneity

Unbiasedness

Results

Algorithmic Small World Hypothesis

Topological For a randomly chosen pair of individuals,

there exists with high probability a short chain of

intermediaries that connects them.

Algorithmic Ordinary individuals can effectively

navigate these short chains themselves with every

individual having only local knowledge of the social

network in question.

Topological Studies

A majority of individuals in a given population are

connected in a single giant component.

The typical shortest path length connecting pairs of

nodes within the giant component is of the order of the

logarithm of the system size.

By Daniel' (User:Dannie-walker) (Own work) [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0-

2.5-2.0-1.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

The problems with algorithmic studies

Reliance on simulations hence not generalizable.

Homogeneity assumption!

Attrition Low rates of chain completion with length

Lack of motivation

Failure to receive the message

Stochastic modelling of attrition the

story so far

Parameter is the probability of termination at each step.

Two estimators for true length in absence of attrition:

Problems with such attrition modelling

Existence of multiple small worlds bowl of lumpy

oatmeal!

Search ability is not uniform but a function of social

capital Homogeneity!

Most source-target pairs live in separate populations that

can only be reached via long or undiscoverable paths

Unbiasedness.

A new model of attritionUnbiased and Heterogeneous estimators

Experiments

The use of email to allow larger access at reduced cost

1. Dec 2001 Aug 2003; 98865 people from 168 countries

initiated 106295 chains directed at 18 targets in 13

countries 491 (0.5%) reached the target.

2. Aug 2003 Dec 2007; 85621 people from 163 countries

initiated 56033 chains directed at 21 targets in 13

countries 61 (0.1%) reached the target.

Dealing with heterogeneity

Next step continuance probability

Logistic multi-level regression

Dealing with heterogeneity

9 categories 66 parameters

Each category is modelled as a normal distribution with 0 mean

Obtain regression coefficients for each attribute value in each category

A few interesting results

Dealing with Biasedness

Space of paths that can be observed

P() probability of choosing a path in

Q() probability that path is observed

A missing value is observed with probability ()

Let denote the ith chain observed in the experiment

Without attrition, expected true length is given by = () where f() is the length of path

Dealing with Biasedness

Let m is the number of paths observed with non missing

values, then an unbiased estimator for the man path

length is given by

This can also be used to find an unbiased estimate for pi the probability that path length is i

Estimating Chain Lengths

Homogenous Attrition

Uniform for all

Different for the first link in the chain

Estimating Chain Lengths

Heterogeneous attrition

Estimating Chain Lengths

Randomised attrition

Results

Thank youQuestions?