sampling methods for graphs

Statistics and networks : motivations and methodsSurvey sampling

Extending the sampling designResults and future work

An overview of sampling methods for graphs -Application to Twitter

Antoine Rebecq

Universite Paris X - INSEE

6/15/16

Antoine Rebecq Sampling the Twitter graph



1 Statistics and networks : motivations and methodsGraphs and statsMethods

2 Survey samplingEstimatesSampling design

3 Extending the sampling designSnowball samplingAdaptive sampling

4 Results and future workResultsSample sizeFuture work




Graphs and statsMethods

Section 1

Statistics and networks : motivations andmethods





Subsection 1

Graphs and stats





Examples of statistics on graphs

Official statistics : measuring “hidden populations”





Examples of statistics on graphs

Rise of “big graphs”





Statistics of interest

Degree

Centrality

Clustering

Communities

. . .





Subsection 2

Methods





Methods for graph statistics

Algorithms (computer science, “big data”)

Model-based estimation

Sampling (“Design-based estimation”)





Methods for graph statistics





Computer science methods

Efficient algorithms (speed / memory).





Big data begets big graph

Twitter in 2013

Image from [1]





Computer science methods

Efficient algorithms (speed / memory).

Sometimes require sampling.





Model-based estimation

Famous graph models :

Erdos-Renyi

Price / Barabasi-Albert (High tailed degree distribution)

Watts-Strogatz / “small-world” (short path lengths)

Stochastic block models (communities)

Images from [9]





Model-based estimation : Erdos-Renyi (“random graphs”)





Model-based estimation : Barabasi-Albert (“preferentialattachment”)





Model-based estimation : Watts-Strogatz (“small world”)





Model-based estimation : Stochastic Block Models





Example : Star Wars : The Force Awakens

Star Wars : The Force Awakens





Example : Star Wars : The Force Awakens

How many (real) users behind these tweets ?





Example : “Star Wars, The Force Awakens”

Let’s write :

yk = Number of tweets @starwars by user k

between 10/29/15, 7 :48 - 10 :48 PM EST

zk = 1{yk ≥ 1}

Goal : estimate NC = T (Z )

Additionally, we write : nC =∑k∈s

zk





The Twitter graph

The Twitter graph ([6]) :

Is directed

Degree distribution is heavy-tailed





The Twitter graph

Has small path lengths





Sampling / Design-based estimation

Sampling : select a few vertices/edges and compute estimatorsusing sample data. Very little exists about design-based statisticalinference on networks (Kolaczyk 2009 , [5])

We try survey sampling methods used in official StatisticsInstitutes to make design-based inference about “big graphs”




EstimatesSampling design

Section 2

Survey sampling





Subsection 1

Estimates





Horvitz-Thompson estimator

Population U : vertices of the Twitter graph.Assign all k ∈ U an inclusion probability P(k ∈ s) = πk






Classic unbiased estimator for totals and means :Horvitz-Thompson

T (Y )HT =∑k∈s

ykπk

ˆy =1

N

∑k∈s

ykπk






Variance of the Horvitz-Thompson estimator depends on the firstand second-order inclusion probabilities :

πk = P(k ∈ s)

πkl = P(k , l ∈ s)

V(T (Y )HT ) =∑k∈U

∑l∈U

(πkl − πkπl)ykπk

ylπl





Calibrated estimator

Deville-Sarndal, 1992 ([2]). Modification of the Horvitz-Thompsonestimator to take auxiliary information into account. For example :

T (Y ) = Number of tweets @StarWars

N = Number of users in scope

Structure of number of followers

Number of verified users

. . .

Very similar to empirical likelihood methods ([8]).





Subsection 2

Sampling design





Sampling design : Bernoulli

Poisson sampling : For each k ∈ U , run a πk -Bernoulli experimentto decide whether to include unit k in the sample.

Bernoulli sampling : ∀k, πk = p

Sampling design of non-fixed sample size. We set the expectedsample size to 20000.





Sampling design : Stratified Bernoulli

More efficient estimators/design : use of external information.






We write : U = U1⊕U2 (h = 1, 2 being called “strata”) and

draw two independant Bernoulli samples in U1 and U2.

Here :

U1 = Followers of official @starwars account

U2 = Rest of Twitter users





Sampling design : Neyman allocation

Optimal variance of the Horvitz-Thompson estimator is obtainedfor (Neyman, [7]) :

nh =NhS2

h∑h

NhS2h

Given the expected values, we set :

n1 = 9700

n2 = 10300






Estimators for the two “simple” designs :

NC1 =nC

p

NC2 =N1

n1nC1 +

N − N1

n2nC2





Variance estimators

V(T (Y ))1 =1

p(

1

p− 1)

∑k∈s

y 2k

V(T (Y ))2 =2∑

h=1

1

ph(

1

ph− 1)

∑k∈sh

y 2k




Snowball samplingAdaptive sampling

Section 3

Extending the sampling design





Snowball sampling

From now on, our sampling designs will include extensions :s = s0 ∪ sext

s0 is still selected using stratified Bernoulli, but with expectedsample size of 1000, so that the expected sample size of s is moreor less 20000.





Subsection 1

Snowball sampling





Snowball sampling

Population U





Snowball sampling

Initial sample s0





Snowball sampling

One stage snowball extension s = A(s0)





Snowball sampling

Formally, we write :

Bi = {i} ∪ {j ∈ V ,Eji 6= ∅}Ai = {i} ∪ {j ∈ V ,Eij 6= ∅}

s = A(s0)





Snowball sampling

NC3 =∑k∈s

zi1− π(Bi )

where :

π(Bi ) = P(Bi ⊂ s)

=∏k∈Bi

(1− P(k ∈ s))

= q#(Bi∩U1)S1 · q#(Bi∩U2)

S2





Snowball sampling

V(NC3) =∑i∈s

∑j∈s

zizjπ(Bi ∪ Bj)

γ′ij

where :

γ′ij =π(Bi ∪ Bj)− π(Bi )π(Bj)

[1− π(Bi )][1− π(Bj)]





Subsection 2

Adaptive sampling





Adaptive sampling

In adaptive sampling, when (Thompson, [10])

Used in official statistics to measure number of drugs users orHIV-positive people

Sampling design often compared to the video game“minesweeper”





Adaptive sampling

Image from [11]





Adaptive sampling

Once a unit bearing the characteristic of interest (i.e. a user whotweeted about the Star Wars trailer) is found, all its network (i.e.its friends and friends of friends, etc. who have tweeted about StarWars) is included in the sample.





Adaptive sampling

Estimator :

NC4 =K∑

k=1

n∗CkJkπgk

where :

K = number of networks

y∗k = total of Y in the network k

n∗Ck= Number of people with yk ≥ 1 in the network k

Jk = 1{k ∈ C}πgk = probability that the initial sample intersects k





Adaptive sampling

When using an adaptive design, it is often better to use theRao-Blackwell of the previous estimate. It has a very simple closedform in the case of the adaptive stratified.

NC5 = n0 +K∑

k=1

nr

1− (1− p)nr

where : n0 = #s0 and s0 = ∪r{k ∈ s, δ(k ,C ) = 1} is the union ofthe sides of C.





Adaptive sampling - Variance

V(NC4) =K∑

k=1

K∑k ′=1

ykyk ′

πgkk ′

(πgkk ′

πgkπgk ′− 1

)where :

πgkk ′ = 1− πgk − πgk ′ + (1− p)ngk+ngk′





Adaptive sampling - Variance

Variance estimation for the Rao-Blackwell can be done by selectingm samples :

V(NC5) = V(NC4)− 1

m − 1

m∑i=1

(NC5i − NC4)2




ResultsSample sizeFuture work

Section 4

Results and future work





Subsection 1

Results





Results

Design n nscope n0 NC CV ˆDeff

Bernoulli 20013 3946 354121 0.231 1.04

Stratified 20094 9832 316889 0.097 0.68

1-snowball 159957 73570 1000 331097 0.031 0.60





Results

Mean number of tweets @StarWars per user : 1.18± 0.07

Suggests that bots are not responsible for this very large number oftweets (see [4], [3]) !





Subsection 2

Sample size





Snowball sampling - sample size

Expected sample size ≈ 20000.

Actual sample size : > 150000 !





Adaptive sampling

With our test subject (tweets @AmericanIdol), average networksize was no greater than a few units (≈ 10000 tweets in the scope)

With Star Wars (≈ 300000 tweets in the scope, with much lesstweets per people), we couldn’t get to the end of every network !





Subsection 3

Future work





Future work

Control sample size

Estimates and calibration for other statistics (centrality,clustering coefficients, path length, etc.)

Take advantage of graph description using models





Auxiliary information for Barabasi-Albert model :

Degree Centrality Local clustering Mean path Max pathDegree ++ - - - -Centrality - - - -Local clustering + +Mean path ++Max path





Conclusion

Thank you !

http://nc233.com/isnps2016

@nc233


http://nc233.com/isnps2016

https://twitter.com/nc233

Statistics and networks: motivations and methodsSurvey sampling



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Statistics and networks: motivations and methodsSurvey sampling



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