Sampling graphs efficiently - MAD Stat (TSE)

Statistics and networksSurvey sampling

Extending the sampling designApplication to Twitter data

Sampling graphs efficiently: model assisted designsand application to Twitter data

Antoine Rebecq

Universite Paris X - INSEE

3/23/17

Antoine Rebecq Sampling designs for graphs



1 Statistics and networksGraphs and statsMethods - algorithms - models

2 Survey samplingEstimatesUse of auxiliary information

3 Extending the sampling designSnowball samplingAdaptive sampling

4 Application to Twitter dataThe problemResultsModel-assisted sampling




Graphs and statsMethods - algorithms - models

Section 1

Statistics and networks





Subsection 1

Graphs and stats





Graphs

Graph G, set of vertices and edges : G = (V ,E )





Directed graphs





Statistics of interest - graphs

Size

Degree

Centrality

Clustering

Communities

. . .





Degree

dv = number of edges incident upon vertex v





Degree / scale-free property





Path lengths





Centrality

Measure of “importance” of a node.

Examples : Google Pagerank, betweenness centrality (number oftimes a node acts as a bridge along the shortest path between twoother nodes)





Betweenness centrality





Clustering

Global clustering coefficient =3 · number of triangles

number of connected triplets

Local clustering coefficient of a vertex = how close its neighboursare to being a clique (complete graph).





Local clustering coefficient





The rise of “big graphs”

Rise of “big graphs”





The rise of “big graphs”

Example : The Graph500 benchmark(http://www.graph500.org). Size of data sets up to 1.1 PBadjacency list (human connectome size)


http://www.graph500.org




Subsection 2

Methods - algorithms - models





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).





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 [8]





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





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”




EstimatesUse of auxiliary information

Section 2

Survey sampling





Subsection 1

Estimates





Horvitz-Thompson estimator

Population U (here vertices of the 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





Bernoulli sampling

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





Subsection 2

Use of auxiliary information





Auxiliary information

If πk ∝ yk then V(T (Y )HT ) = 0

In practice, use auxiliary variable : X which is well correlated to Y .





Stratified sampling

We write : U = U1⊕U2⊕. . .⊕UH and draw independant

samples in each Uh.

Strata should be formed so that intra dispersion of yk is the lowestpossible.





Stratified sampling : Neyman allocation

Given a set of strata and a sample size n, optimal variance isobtained for :

nh =NhS2

h∑h

NhS2h





Calibrated estimator

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

Very similar to empirical likelihood methods ([7]).

Computing variances for calibrated estimators is easy.




Snowball samplingAdaptive sampling

Section 3

Extending the sampling design





Official statistics

Measuring “hidden populations”





Community structure

When trying to measure the size of a community (NC ), use ofedges as auxiliary variables.





Snowball sampling

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





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))





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

Adaptive sampling (Thompson, [9])

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 [10]





Adaptive sampling

Once a unit bearing the characteristic of interest is found, all itsnetwork 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




The problemResultsModel-assisted sampling

Section 4

Application to Twitter data





Subsection 1

The problem





The Twitter graph

Twitter in 2013

Image from [1]





The Twitter API

Access to the Twitter data through an API (Applicationprogramming interface), which limits the number of calls per hour.





Example : Star Wars : The Force Awakens

How many (real) users behind tweets talking about the new StarWars movie ?





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 designs

1 Bernoulli sample

2 Stratified Bernoulli

3 Snowball over the stratified Bernoulli

4 Adaptive over the stratified Bernoulli

5 (Rao-blackwell of the adaptive estimator)





Stratification

U1 = Followers of official @starwars account

U2 = Rest of Twitter users





Stratification : Neyman allocation

Given some preliminary exploratory data, we get (for n = 2000) :

n1 = 9700

n2 = 10300





Sample size - extension

Size of s0 : 1000 (so that total sample size, with extensions, wouldbe about n = 20000).





Calibration variables

N = Number of users in scope

Structure of number of followers

Number of verified users

. . .





Estimators

NC1 =nC

p

NC2 =N1

n1nC1 +

N − N1

n2nC2

NC3 =∑k∈s

zi1− π(Bi )

NC4 =K∑

k=1

n∗CkJkπgk

NC5 = n0 +K∑

k=1

nr

1− (1− p)nr





Exclusion probabilities

π(Bi ) = P(Bi ⊂ s)

=∏k∈Bi

(1− P(k ∈ s))

= q#(Bi∩U1)S1 · q#(Bi∩U2)

S2





Subsection 2

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]) !

Adaptive sampling did not converge.





Subsection 3

Model-assisted sampling





Auxiliary information for Barabasi-Albert model :

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





Future work

Combine all these (optimal allocations, etc.)

Asymptotics





Conclusion

Thank you !

http://nc233.com/madstat2017

@nc233


http://nc233.com/madstat2017

https://twitter.com/nc233




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Sampling graphs efficiently - MAD Stat (TSE)